3.12.7 \(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^3} \, dx\)
Optimal. Leaf size=394 \[ -\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2} \]
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Rubi [A] time = 0.90, antiderivative size = 394, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{822, 826, 1166, 208} \begin {gather*} \frac {\sqrt {d+e x} \left (c x \left (b^2 c d e (6 A e+19 B d)+b^3 \left (-e^2\right ) (4 B d-3 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}+\frac {c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]
[Out]
-(Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(2*b^2*d*(c*d - b*e)*(b*x + c*x^2)^2) + (Sq
rt[d + e*x]*(b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d - 3*A*e) - b*c*d*(6*B*d + 7*A*e)) + c*(24*A*c^3*d^3 -
b^3*e^2*(4*B*d - 3*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(19*B*d + 6*A*e))*x))/(4*b^4*d^2*(c*d - b*e)^
2*(b*x + c*x^2)) - ((48*A*c^2*d^2 - b^2*e*(4*B*d - 3*A*e) - 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt
[d]])/(4*b^5*d^(5/2)) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d
+ 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(5/2))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 826
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 1166
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )-\frac {5}{2} c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^2 \left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )+\frac {1}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^2 \left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )-\frac {1}{4} c d e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right )+\frac {1}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^2 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\left (c \left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^2}-\frac {\left (c^2 \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 1.48, size = 408, normalized size = 1.04 \begin {gather*} \frac {\frac {c \sqrt {d+e x} \left (b^2 e (3 A e-4 B d)+b c d (7 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b+c x)^2 (b e-c d)}+\frac {-\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 e (3 A e-4 B d)+12 b c d (A e-2 B d)+48 A c^2 d^2\right )-\frac {b c \sqrt {d} \sqrt {d+e x} \left (b^3 e^2 (4 B d-3 A e)-b^2 c d e (6 A e+19 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )}{(b+c x) (c d-b e)^2}+\frac {c^{3/2} d^{5/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{(c d-b e)^{5/2}}}{b^4 d^{3/2}}+\frac {\sqrt {d+e x} (3 A b e+8 A c d-4 b B d)}{b d x (b+c x)^2}-\frac {2 A \sqrt {d+e x}}{x^2 (b+c x)^2}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]
[Out]
((c*(-12*A*c^2*d^2 + b^2*e*(-4*B*d + 3*A*e) + b*c*d*(6*B*d + 7*A*e))*Sqrt[d + e*x])/(b^2*d*(-(c*d) + b*e)*(b +
c*x)^2) - (2*A*Sqrt[d + e*x])/(x^2*(b + c*x)^2) + ((-4*b*B*d + 8*A*c*d + 3*A*b*e)*Sqrt[d + e*x])/(b*d*x*(b +
c*x)^2) + (-((b*c*Sqrt[d]*(-24*A*c^3*d^3 + b^3*e^2*(4*B*d - 3*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e) - b^2*c*d*e*(1
9*B*d + 6*A*e))*Sqrt[d + e*x])/((c*d - b*e)^2*(b + c*x))) - (48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + A*e) + b^2*e*(-
4*B*d + 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (c^(3/2)*d^(5/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2
*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(c*d - b*e)^(5/2)
)/(b^4*d^(3/2)))/(4*b*d)
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IntegrateAlgebraic [B] time = 2.47, size = 888, normalized size = 2.25 \begin {gather*} \frac {\sqrt {d+e x} \left (-24 A c^5 d^6+12 b B c^4 d^6+72 A b c^4 e d^5-37 b^2 B c^3 e d^5+72 A c^5 (d+e x) d^5-36 b B c^4 (d+e x) d^5-69 A b^2 c^3 e^2 d^4+37 b^3 B c^2 e^2 d^4-72 A c^5 (d+e x)^2 d^4+36 b B c^4 (d+e x)^2 d^4-180 A b c^4 e (d+e x) d^4+93 b^2 B c^3 e (d+e x) d^4+18 A b^3 c^2 e^3 d^3-16 b^4 B c e^3 d^3+24 A c^5 (d+e x)^3 d^3-12 b B c^4 (d+e x)^3 d^3+144 A b c^4 e (d+e x)^2 d^3-75 b^2 B c^3 e (d+e x)^2 d^3+136 A b^2 c^3 e^2 (d+e x) d^3-74 b^3 B c^2 e^2 (d+e x) d^3+4 b^5 B e^4 d^2+8 A b^4 c e^4 d^2-36 A b c^4 e (d+e x)^3 d^2+19 b^2 B c^3 e (d+e x)^3 d^2-73 A b^2 c^3 e^2 (d+e x)^2 d^2+41 b^3 B c^2 e^2 (d+e x)^2 d^2-24 A b^3 c^2 e^3 (d+e x) d^2+24 b^4 B c e^3 (d+e x) d^2-5 A b^5 e^5 d+6 A b^2 c^3 e^2 (d+e x)^3 d-4 b^3 B c^2 e^2 (d+e x)^3 d+A b^3 c^2 e^3 (d+e x)^2 d-8 b^4 B c e^3 (d+e x)^2 d-4 b^5 B e^4 (d+e x) d-10 A b^4 c e^4 (d+e x) d+3 A b^3 c^2 e^3 (d+e x)^3+6 A b^4 c e^4 (d+e x)^2+3 A b^5 e^5 (d+e x)\right )}{4 b^4 d^2 e (b e-c d)^2 x^2 (-c d+b e+c (d+e x))^2}+\frac {\left (48 A d^2 c^{9/2}-24 b B d^2 c^{7/2}-108 A b d e c^{7/2}+63 A b^2 e^2 c^{5/2}+56 b^2 B d e c^{5/2}-35 b^3 B e^2 c^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 (c d-b e)^2 \sqrt {b e-c d}}+\frac {\left (-3 A e^2 b^2+4 B d e b^2+24 B c d^2 b-12 A c d e b-48 A c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]
[Out]
(Sqrt[d + e*x]*(12*b*B*c^4*d^6 - 24*A*c^5*d^6 - 37*b^2*B*c^3*d^5*e + 72*A*b*c^4*d^5*e + 37*b^3*B*c^2*d^4*e^2 -
69*A*b^2*c^3*d^4*e^2 - 16*b^4*B*c*d^3*e^3 + 18*A*b^3*c^2*d^3*e^3 + 4*b^5*B*d^2*e^4 + 8*A*b^4*c*d^2*e^4 - 5*A*
b^5*d*e^5 - 36*b*B*c^4*d^5*(d + e*x) + 72*A*c^5*d^5*(d + e*x) + 93*b^2*B*c^3*d^4*e*(d + e*x) - 180*A*b*c^4*d^4
*e*(d + e*x) - 74*b^3*B*c^2*d^3*e^2*(d + e*x) + 136*A*b^2*c^3*d^3*e^2*(d + e*x) + 24*b^4*B*c*d^2*e^3*(d + e*x)
- 24*A*b^3*c^2*d^2*e^3*(d + e*x) - 4*b^5*B*d*e^4*(d + e*x) - 10*A*b^4*c*d*e^4*(d + e*x) + 3*A*b^5*e^5*(d + e*
x) + 36*b*B*c^4*d^4*(d + e*x)^2 - 72*A*c^5*d^4*(d + e*x)^2 - 75*b^2*B*c^3*d^3*e*(d + e*x)^2 + 144*A*b*c^4*d^3*
e*(d + e*x)^2 + 41*b^3*B*c^2*d^2*e^2*(d + e*x)^2 - 73*A*b^2*c^3*d^2*e^2*(d + e*x)^2 - 8*b^4*B*c*d*e^3*(d + e*x
)^2 + A*b^3*c^2*d*e^3*(d + e*x)^2 + 6*A*b^4*c*e^4*(d + e*x)^2 - 12*b*B*c^4*d^3*(d + e*x)^3 + 24*A*c^5*d^3*(d +
e*x)^3 + 19*b^2*B*c^3*d^2*e*(d + e*x)^3 - 36*A*b*c^4*d^2*e*(d + e*x)^3 - 4*b^3*B*c^2*d*e^2*(d + e*x)^3 + 6*A*
b^2*c^3*d*e^2*(d + e*x)^3 + 3*A*b^3*c^2*e^3*(d + e*x)^3))/(4*b^4*d^2*e*(-(c*d) + b*e)^2*x^2*(-(c*d) + b*e + c*
(d + e*x))^2) + ((-24*b*B*c^(7/2)*d^2 + 48*A*c^(9/2)*d^2 + 56*b^2*B*c^(5/2)*d*e - 108*A*b*c^(7/2)*d*e - 35*b^3
*B*c^(3/2)*e^2 + 63*A*b^2*c^(5/2)*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*
(c*d - b*e)^2*Sqrt[-(c*d) + b*e]) + ((24*b*B*c*d^2 - 48*A*c^2*d^2 + 4*b^2*B*d*e - 12*A*b*c*d*e - 3*A*b^2*e^2)*
ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(5/2))
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fricas [B] time = 34.57, size = 4310, normalized size = 10.94
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^3*
e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*b^
3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5*c
- 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(
c/(c*d - b*e)))/(c*x + b)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^
3*e - (16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24
*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 -
2*(2*B*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 -
21*A*b^3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(d)*log((e
*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2
*d*e^3 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*
d^2*e^2)*x^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(
4*B*b^5*c - 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b
^4*c^2)*d^3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3
*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2
)*x^2), -1/8*(2*((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*
c^4)*d^3*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2
- 9*A*b^3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(
5*B*b^5*c - 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b
*e))/(c*e*x + c*d)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (
16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2
*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B
*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^
3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(d)*log((e*x - 2*
sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3
- 12*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2
)*x^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5
*c - 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)
*d^3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x
^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2),
1/8*(2*((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3 -
27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b*c
^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A*b
^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e -
(16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(
-d)/d) - ((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^
3*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*
b^3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5
*c - 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqr
t(c/(c*d - b*e)))/(c*x + b)) - 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3 - 1
2*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2)*x
^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5*c
- 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)*d^
3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4
+ 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2), -1
/4*(((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^3*e^2
)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*b^3*c
^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5*c -
9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x
+ c*d)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3
- 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b
*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A
*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e
- (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqr
t(-d)/d) + (2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3 - 12*(B*b^2*c^4 - 2*A*b*c
^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2)*x^3 - (6*A*b^5*c*d*e^3
- 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5*c - 5*A*b^4*c^2)*d^2*e^2
)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)*d^3*e - 2*(2*B*b^6 - A*b
^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b
^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2)]
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giac [B] time = 0.31, size = 1046, normalized size = 2.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
1/4*(24*B*b*c^4*d^2 - 48*A*c^5*d^2 - 56*B*b^2*c^3*d*e + 108*A*b*c^4*d*e + 35*B*b^3*c^2*e^2 - 63*A*b^2*c^3*e^2)
*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^2*d^2 - 2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) - 1
/4*(12*(x*e + d)^(7/2)*B*b*c^4*d^3*e - 24*(x*e + d)^(7/2)*A*c^5*d^3*e - 36*(x*e + d)^(5/2)*B*b*c^4*d^4*e + 72*
(x*e + d)^(5/2)*A*c^5*d^4*e + 36*(x*e + d)^(3/2)*B*b*c^4*d^5*e - 72*(x*e + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(x*e
+ d)*B*b*c^4*d^6*e + 24*sqrt(x*e + d)*A*c^5*d^6*e - 19*(x*e + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(x*e + d)^(7/2)*
A*b*c^4*d^2*e^2 + 75*(x*e + d)^(5/2)*B*b^2*c^3*d^3*e^2 - 144*(x*e + d)^(5/2)*A*b*c^4*d^3*e^2 - 93*(x*e + d)^(3
/2)*B*b^2*c^3*d^4*e^2 + 180*(x*e + d)^(3/2)*A*b*c^4*d^4*e^2 + 37*sqrt(x*e + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(x*e
+ d)*A*b*c^4*d^5*e^2 + 4*(x*e + d)^(7/2)*B*b^3*c^2*d*e^3 - 6*(x*e + d)^(7/2)*A*b^2*c^3*d*e^3 - 41*(x*e + d)^(
5/2)*B*b^3*c^2*d^2*e^3 + 73*(x*e + d)^(5/2)*A*b^2*c^3*d^2*e^3 + 74*(x*e + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 136*(x*
e + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 37*sqrt(x*e + d)*B*b^3*c^2*d^4*e^3 + 69*sqrt(x*e + d)*A*b^2*c^3*d^4*e^3 - 3*(
x*e + d)^(7/2)*A*b^3*c^2*e^4 + 8*(x*e + d)^(5/2)*B*b^4*c*d*e^4 - (x*e + d)^(5/2)*A*b^3*c^2*d*e^4 - 24*(x*e + d
)^(3/2)*B*b^4*c*d^2*e^4 + 24*(x*e + d)^(3/2)*A*b^3*c^2*d^2*e^4 + 16*sqrt(x*e + d)*B*b^4*c*d^3*e^4 - 18*sqrt(x*
e + d)*A*b^3*c^2*d^3*e^4 - 6*(x*e + d)^(5/2)*A*b^4*c*e^5 + 4*(x*e + d)^(3/2)*B*b^5*d*e^5 + 10*(x*e + d)^(3/2)*
A*b^4*c*d*e^5 - 4*sqrt(x*e + d)*B*b^5*d^2*e^5 - 8*sqrt(x*e + d)*A*b^4*c*d^2*e^5 - 3*(x*e + d)^(3/2)*A*b^5*e^6
+ 5*sqrt(x*e + d)*A*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d +
c*d^2 + (x*e + d)*b*e - b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e - 3*A*b^2*e
^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^2)
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maple [B] time = 0.08, size = 1009, normalized size = 2.56 \begin {gather*} -\frac {15 \left (e x +d \right )^{\frac {3}{2}} A \,c^{4} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}-\frac {63 A \,c^{3} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A \,c^{5} d e}{\left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{4}}+\frac {27 A \,c^{4} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 A \,c^{5} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{5}}+\frac {11 \left (e x +d \right )^{\frac {3}{2}} B \,c^{3} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2}}+\frac {35 B \,c^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,c^{4} d e}{\left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}-\frac {14 B \,c^{3} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {6 B \,c^{4} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {17 \sqrt {e x +d}\, A \,c^{3} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{3}}+\frac {3 \sqrt {e x +d}\, A \,c^{4} d e}{\left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{4}}+\frac {13 \sqrt {e x +d}\, B \,c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{2}}-\frac {2 \sqrt {e x +d}\, B \,c^{3} d e}{\left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{3}}-\frac {3 A \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {5}{2}}}-\frac {3 A c e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} d^{\frac {3}{2}}}-\frac {12 A \,c^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5} \sqrt {d}}+\frac {B e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} d^{\frac {3}{2}}}+\frac {6 B c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} \sqrt {d}}-\frac {5 \sqrt {e x +d}\, A}{4 b^{3} d \,x^{2}}-\frac {3 \sqrt {e x +d}\, A c}{b^{4} e \,x^{2}}+\frac {\sqrt {e x +d}\, B}{b^{3} e \,x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A}{4 b^{3} d^{2} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A c}{b^{4} d e \,x^{2}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B}{b^{3} d e \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x)
[Out]
-3/e/b^4/x^2*(e*x+d)^(1/2)*A*c-3*e/b^4/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-1/e/b^3/x^2/d*(e*x+d)^(3/2)*
B-2*e*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B*d+3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*A*d+
3*e*c^5/b^4/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*A*d-2*e*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b
*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*B*d-14*e*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^
(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d+27*e*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(
1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-5/4/b^3/x^2/d*(e*x+d)^(1/2)*A+3/4/b^3/x^2/d^2*(e*x+d)^(3/2)*A+6/b^4/d^(1/2)*ar
ctanh((e*x+d)^(1/2)/d^(1/2))*B*c+e/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B-12/b^5/d^(1/2)*arctanh((e*x+d)
^(1/2)/d^(1/2))*A*c^2+1/e/b^3/x^2*(e*x+d)^(1/2)*B-3/4*e^2/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A-63/4*e^
2*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+35/4*e
^2*c^2/b^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+6*c^4
/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2-12*c^5/
b^5/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+3/e/b^4/
x^2/d*(e*x+d)^(3/2)*A*c-15/4*e^2*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*A+11/4*e^2*c^
3/b^2/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*B-17/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+
d)^(1/2)*A+13/4*e^2*c^2/b^2/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
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mupad [B] time = 7.11, size = 11338, normalized size = 28.78
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((b*x + c*x^2)^3*(d + e*x)^(1/2)),x)
[Out]
log((((((c^2*e^3*(3*A*b^4*e^4 + 24*A*c^4*d^4 - 12*B*b*c^3*d^4 - 4*B*b^4*d*e^3 + 25*B*b^2*c^2*d^3*e - 12*B*b^3*
c*d^2*e^2 + 21*A*b^2*c^2*d^2*e^2 - 48*A*b*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^2*d^2*(b*e - c*d)^2) - b^2*c^2*e^2*
(b*e - 2*c*d)*(d + e*x)^(1/2)*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^1
0*d^5))^(1/2))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/
8 - ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c
^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d
^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4
*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 46
08*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A
*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*b^8*d^4
*(b*e - c*d)^4))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2)
)/8 - (567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*d^5*e^5 - 77760*A^3*b^3*c^9*d^4*e^6
- 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*d^7*e^3 + 25920*B^3*b^4*c^8*d^6*e^4
- 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*d^3*e^7 - 560*B^3*b^8*c^4*d^2*e^8 -
315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^9 + 41472*A*B^2*b^2*c^10*d^7*e^3 -
152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b^5*c^7*d^4*e^6 - 4764*A*B^2*b^6*c^6
*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357696*A^2*B*b^3*c^9*d^5*e^5 + 136368*
A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^2*e^8 + 840*A*B^2*b^8*c^4*d*e^9 - 8
2944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(64*b^12*d^4*(b*e - c*d)^4))*((9*A^2*b^4*e^4 + 2304*A^2*
c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 432*A^2*b^2*c^2*d^2*e^2 + 1152*A^2*b*c^3*d^3*e + 72*A^2*b
^3*c*d*e^3 + 192*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 24*A*B*b^4*d*e^3 - 960*A*B*b^2*c^2*d^3*e - 240*A*B*b^3
*c*d^2*e^2)/(64*b^10*d^5))^(1/2) - log((((((c^2*e^3*(3*A*b^4*e^4 + 24*A*c^4*d^4 - 12*B*b*c^3*d^4 - 4*B*b^4*d*e
^3 + 25*B*b^2*c^2*d^3*e - 12*B*b^3*c*d^2*e^2 + 21*A*b^2*c^2*d^2*e^2 - 48*A*b*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^
2*d^2*(b*e - c*d)^2) + b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 -
4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e +
12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/8 + ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^
2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b
^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5
*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*
d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 -
28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 1
44*A*B*b^7*c^4*d^2*e^8))/(8*b^8*d^4*(b*e - c*d)^4))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e
+ 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/8 - (567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*
d^5*e^5 - 77760*A^3*b^3*c^9*d^4*e^6 - 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*
d^7*e^3 + 25920*B^3*b^4*c^8*d^6*e^4 - 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*
d^3*e^7 - 560*B^3*b^8*c^4*d^2*e^8 - 315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^
9 + 41472*A*B^2*b^2*c^10*d^7*e^3 - 152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b
^5*c^7*d^4*e^6 - 4764*A*B^2*b^6*c^6*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357
696*A^2*B*b^3*c^9*d^5*e^5 + 136368*A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^
2*e^8 + 840*A*B^2*b^8*c^4*d*e^9 - 82944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(64*b^12*d^4*(b*e - c
*d)^4))*(((9*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 + (B^2*b^4*d^2*e^2)/4 + (27*A^2*b^2*c^2*d^2*
e^2)/4 + 18*A^2*b*c^3*d^3*e + (9*A^2*b^3*c*d*e^3)/8 + 3*B^2*b^3*c*d^3*e - 36*A*B*b*c^3*d^4 - (3*A*B*b^4*d*e^3)
/8 - 15*A*B*b^2*c^2*d^3*e - (15*A*B*b^3*c*d^2*e^2)/4)/(b^10*d^5))^(1/2) - atan(((((6144*A*b^11*c^7*d^7*e^4 - 1
536*A*b^10*c^8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b
^15*c^3*d^3*e^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*
d^6*e^5 - 2880*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*
d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) - ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 +
3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c
^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e -
3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b
^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 -
5*b^14*c*d*e^4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^
6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11
*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2
*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d
^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*
A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11
*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*b^8
*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d
^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e
^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7
+ 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*
b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b
^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^
3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^
7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 1
0368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*
c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*
c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*1i - (((6144*
A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14
*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*
e^4 + 4736*B*b^13*c^5*d^6*e^5 - 2880*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64
*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + ((d + e*x)^(1/2)*
(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^
2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 26
88*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c
^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 +
10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7
*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*
b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*
b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e
^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*
A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 -
b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) - ((d +
e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^
5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 -
4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 12
8*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*
c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^
7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b
^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^
5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 44
10*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c
^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)
/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)
))^(1/2)*1i)/((567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*d^5*e^5 - 77760*A^3*b^3*c^9
*d^4*e^6 - 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*d^7*e^3 + 25920*B^3*b^4*c^8
*d^6*e^4 - 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*d^3*e^7 - 560*B^3*b^8*c^4*d
^2*e^8 - 315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^9 + 41472*A*B^2*b^2*c^10*d^
7*e^3 - 152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b^5*c^7*d^4*e^6 - 4764*A*B^2
*b^6*c^6*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357696*A^2*B*b^3*c^9*d^5*e^5 +
136368*A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^2*e^8 + 840*A*B^2*b^8*c^4*d
*e^9 - 82944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(32*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^
7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + (((6144*A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^8*d^8*e^3 - 9024*A*
b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e^8 - 192*A*b^16*c^2
*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*d^6*e^5 - 2880*B*b^14*c^4*d^5*e^
6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b
^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) - ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b
^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^
4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A
*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 -
b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*(128*b^1
0*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 6
4*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)
))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7
*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 -
2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^
4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2
+ 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^
2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7
+ 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 -
5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432
*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^
9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5
*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b
^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 1
7712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2
*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*
e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*
b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + (((6144*A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^
8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e
^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*d^6*e^5 - 288
0*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b
^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*
c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 -
4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5
*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^
2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^
4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b
^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 +
6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4
+ 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*
A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d
^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e -
10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4
608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*
A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2
*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*
c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9
+ 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^
6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b
^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*
B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^
8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 105
60*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b
^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)))*(-(2304*A^2*c^9*d^4 + 39
69*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5
*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3
920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3
*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*
b^14*c*d*e^4)))^(1/2)*2i - (((d + e*x)^(3/2)*(4*B*b^5*d*e^5 - 72*A*c^5*d^5*e - 3*A*b^5*e^6 + 180*A*b*c^4*d^4*e
^2 - 24*B*b^4*c*d^2*e^4 - 136*A*b^2*c^3*d^3*e^3 + 24*A*b^3*c^2*d^2*e^4 - 93*B*b^2*c^3*d^4*e^2 + 74*B*b^3*c^2*d
^3*e^3 + 10*A*b^4*c*d*e^5 + 36*B*b*c^4*d^5*e))/(4*b^4*(c*d^2 - b*d*e)^2) - ((d + e*x)^(5/2)*(6*A*b^4*c*e^5 - 7
2*A*c^5*d^4*e + 144*A*b*c^4*d^3*e^2 + A*b^3*c^2*d*e^4 - 73*A*b^2*c^3*d^2*e^3 - 75*B*b^2*c^3*d^3*e^2 + 41*B*b^3
*c^2*d^2*e^3 + 36*B*b*c^4*d^4*e - 8*B*b^4*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)^2) + ((d + e*x)^(1/2)*(24*A*c^4*d^4
*e - 5*A*b^4*e^5 + 4*B*b^4*d*e^4 - 48*A*b*c^3*d^3*e^2 - 12*B*b^3*c*d^2*e^3 + 21*A*b^2*c^2*d^2*e^3 + 25*B*b^2*c
^2*d^3*e^2 + 3*A*b^3*c*d*e^4 - 12*B*b*c^3*d^4*e))/(4*b^4*(c*d^2 - b*d*e)) - (c*(d + e*x)^(7/2)*(3*A*b^3*c*e^4
+ 24*A*c^4*d^3*e - 36*A*b*c^3*d^2*e^2 + 6*A*b^2*c^2*d*e^3 + 19*B*b^2*c^2*d^2*e^2 - 12*B*b*c^3*d^3*e - 4*B*b^3*
c*d*e^3))/(4*b^4*(c*d^2 - b*d*e)^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4
*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*
d^3*e)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(1/2)/(c*x**2+b*x)**3,x)
[Out]
Timed out
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