3.12.6 \(\int \frac {(A+B x) \sqrt {d+e x}}{(b x+c x^2)^3} \, dx\)
Optimal. Leaf size=317 \[ -\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )}{4 b^5 d^{3/2}}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}+\frac {\sqrt {c} \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}} \]
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Rubi [A] time = 0.70, antiderivative size = 317, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{820, 822, 826, 1166, 208} \begin {gather*} \frac {\sqrt {c} \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 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)^{3/2}}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )}{4 b^5 d^{3/2}}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
[Out]
-((A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(2*b^2*(b*x + c*x^2)^2) - (Sqrt[d + e*x]*(b*(c*d - b*e)*(6*b*B*d - 12
*A*c*d + A*b*e) - c*(24*A*c^2*d^2 + b^2*e*(11*B*d + A*e) - 12*b*c*d*(B*d + 2*A*e))*x))/(4*b^4*d*(c*d - b*e)*(b
*x + c*x^2)) - ((48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 12*b*c*d*(2*B*d + A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
(4*b^5*d^(3/2)) + (Sqrt[c]*(48*A*c^3*d^2 - 15*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 7*A*e) + 5*b^2*c*e*(8*B*d + 7*A*
e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))
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 820
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])
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 {(A b-(b B-2 A c) x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} (12 A c d-b (6 B d+A e))-\frac {5}{2} (b B-2 A c) e x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(A b-(b B-2 A c) x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e) \left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right )+\frac {1}{4} c e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac {(A b-(b B-2 A c) x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e) \left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right )-\frac {1}{4} c d e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right )+\frac {1}{4} c e \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 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 (c d-b e)}\\ &=-\frac {(A b-(b B-2 A c) x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac {\left (c \left (48 A c^2 d^2+b^2 e (4 B d-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}-\frac {\left (c \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 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)}\\ &=-\frac {(A b-(b B-2 A c) x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) (6 b B d-12 A c d+A b e)-c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2+b^2 e (4 B d-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^{3/2}}+\frac {\sqrt {c} \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 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)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 2.22, size = 449, normalized size = 1.42 \begin {gather*} \frac {\frac {(b+c x) \left ((b+c x) \left (2 c^{3/2} (c d-b e)^2 \left (\sqrt {d+e x}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )+2 c^2 d^2 \left (-5 b^2 c e (7 A e+8 B d)+12 b c^2 d (7 A e+2 B d)-48 A c^3 d^2+15 b^3 B e^2\right ) \left (\sqrt {c} \sqrt {d+e x}-\sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )\right )+2 b c^{5/2} (d+e x)^{3/2} \left (b^3 e^2 (A e-4 B d)+b^2 c d e (10 A e+17 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )-2 b^2 c^{5/2} (d+e x)^{3/2} (c d-b e) \left (b^2 e (A e-4 B d)+3 b c d (3 A e+2 B d)-12 A c^2 d^2\right )}{b^4 c^{3/2} d (c d-b e)^2}-\frac {2 (d+e x)^{3/2} (-A b e-8 A c d+4 b B d)}{b d x}-\frac {4 A (d+e x)^{3/2}}{x^2}}{8 b d (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
[Out]
((-4*A*(d + e*x)^(3/2))/x^2 - (2*(4*b*B*d - 8*A*c*d - A*b*e)*(d + e*x)^(3/2))/(b*d*x) + (-2*b^2*c^(5/2)*(c*d -
b*e)*(-12*A*c^2*d^2 + b^2*e*(-4*B*d + A*e) + 3*b*c*d*(2*B*d + 3*A*e))*(d + e*x)^(3/2) + (b + c*x)*(2*b*c^(5/2
)*(24*A*c^3*d^3 + b^3*e^2*(-4*B*d + A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(17*B*d + 10*A*e))*(d + e*x)
^(3/2) + (b + c*x)*(2*c^(3/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 12*b*c*d*(2*B*d + A*e))*(Sqr
t[d + e*x] - Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) + 2*c^2*d^2*(-48*A*c^3*d^2 + 15*b^3*B*e^2 + 12*b*c^2*d*(2
*B*d + 7*A*e) - 5*b^2*c*e*(8*B*d + 7*A*e))*(Sqrt[c]*Sqrt[d + e*x] - Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d - b*e]]))))/(b^4*c^(3/2)*d*(c*d - b*e)^2))/(8*b*d*(b + c*x)^2)
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IntegrateAlgebraic [B] time = 3.10, size = 730, normalized size = 2.30 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (A b^2 e^2+12 A b c d e-48 A c^2 d^2-4 b^2 B d e+24 b B c d^2\right )}{4 b^5 d^{3/2}}+\frac {\left (35 A b^2 c^{3/2} e^2-84 A b c^{5/2} d e+48 A c^{7/2} d^2-15 b^3 B \sqrt {c} e^2+40 b^2 B c^{3/2} d e-24 b B c^{5/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 (c d-b e) \sqrt {b e-c d}}+\frac {\sqrt {d+e x} \left (-A b^4 e^4 (d+e x)-A b^4 d e^4-9 A b^3 c d^2 e^3+13 A b^3 c d e^3 (d+e x)-2 A b^3 c e^3 (d+e x)^2+46 A b^2 c^2 d^3 e^2-85 A b^2 c^2 d^2 e^2 (d+e x)+40 A b^2 c^2 d e^2 (d+e x)^2-A b^2 c^2 e^2 (d+e x)^3-60 A b c^3 d^4 e+144 A b c^3 d^3 e (d+e x)-108 A b c^3 d^2 e (d+e x)^2+24 A b c^3 d e (d+e x)^3+24 A c^4 d^5-72 A c^4 d^4 (d+e x)+72 A c^4 d^3 (d+e x)^2-24 A c^4 d^2 (d+e x)^3+4 b^4 B d^2 e^3-4 b^4 B d e^3 (d+e x)-21 b^3 B c d^3 e^2+38 b^3 B c d^2 e^2 (d+e x)-17 b^3 B c d e^2 (d+e x)^2+29 b^2 B c^2 d^4 e-69 b^2 B c^2 d^3 e (d+e x)+51 b^2 B c^2 d^2 e (d+e x)^2-11 b^2 B c^2 d e (d+e x)^3-12 b B c^3 d^5+36 b B c^3 d^4 (d+e x)-36 b B c^3 d^3 (d+e x)^2+12 b B c^3 d^2 (d+e x)^3\right )}{4 b^4 d e x^2 (b e-c d) (b e+c (d+e x)-c d)^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^3*d^5 + 24*A*c^4*d^5 + 29*b^2*B*c^2*d^4*e - 60*A*b*c^3*d^4*e - 21*b^3*B*c*d^3*e^2 +
46*A*b^2*c^2*d^3*e^2 + 4*b^4*B*d^2*e^3 - 9*A*b^3*c*d^2*e^3 - A*b^4*d*e^4 + 36*b*B*c^3*d^4*(d + e*x) - 72*A*c^4
*d^4*(d + e*x) - 69*b^2*B*c^2*d^3*e*(d + e*x) + 144*A*b*c^3*d^3*e*(d + e*x) + 38*b^3*B*c*d^2*e^2*(d + e*x) - 8
5*A*b^2*c^2*d^2*e^2*(d + e*x) - 4*b^4*B*d*e^3*(d + e*x) + 13*A*b^3*c*d*e^3*(d + e*x) - A*b^4*e^4*(d + e*x) - 3
6*b*B*c^3*d^3*(d + e*x)^2 + 72*A*c^4*d^3*(d + e*x)^2 + 51*b^2*B*c^2*d^2*e*(d + e*x)^2 - 108*A*b*c^3*d^2*e*(d +
e*x)^2 - 17*b^3*B*c*d*e^2*(d + e*x)^2 + 40*A*b^2*c^2*d*e^2*(d + e*x)^2 - 2*A*b^3*c*e^3*(d + e*x)^2 + 12*b*B*c
^3*d^2*(d + e*x)^3 - 24*A*c^4*d^2*(d + e*x)^3 - 11*b^2*B*c^2*d*e*(d + e*x)^3 + 24*A*b*c^3*d*e*(d + e*x)^3 - A*
b^2*c^2*e^2*(d + e*x)^3))/(4*b^4*d*e*(-(c*d) + b*e)*x^2*(-(c*d) + b*e + c*(d + e*x))^2) + ((-24*b*B*c^(5/2)*d^
2 + 48*A*c^(7/2)*d^2 + 40*b^2*B*c^(3/2)*d*e - 84*A*b*c^(5/2)*d*e - 15*b^3*B*Sqrt[c]*e^2 + 35*A*b^2*c^(3/2)*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)*Sqrt[-(c*d) + b*e]) + ((2
4*b*B*c*d^2 - 48*A*c^2*d^2 - 4*b^2*B*d*e + 12*A*b*c*d*e + A*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^
(3/2))
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fricas [B] time = 7.84, size = 3396, normalized size = 10.71
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^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e
^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c
^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*
b^4*c)*d^2*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)) - ((A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b
^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^
2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*
b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) +
2*d)/x) - 2*(2*A*b^4*c*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2
- 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2
)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/(
(b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), -1/8*(2*
((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^
4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^
2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)
*d^2*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)) + ((
A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)
*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^
4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*
d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c
*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e
)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*
d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2
*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/8*(2*((A*b^3*c^2*e^3 - 24*(B
*b*c^4 - 2*A*c^5)*d^3 + 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^
4*c*e^3 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d
*e^2)*x^3 + (A*b^5*e^3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11
*A*b^4*c)*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2*c
^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*(1
0*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d
^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*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)) - 2*(2*A*b^4*c*d^3 - 2*A*b^5*d^2
*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c
*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c
- 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b
^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), -1/4*(((24*(B*b*c^4 - 2*A*c^5)*d^4 - 4*(10*B*b^2
*c^3 - 21*A*b*c^4)*d^3*e + 5*(3*B*b^3*c^2 - 7*A*b^2*c^3)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^4 - 4*
(10*B*b^3*c^2 - 21*A*b^2*c^3)*d^3*e + 5*(3*B*b^4*c - 7*A*b^3*c^2)*d^2*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)
*d^4 - 4*(10*B*b^4*c - 21*A*b^3*c^2)*d^3*e + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*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)) - ((A*b^3*c^2*e^3 - 24*(B*b*c^4 - 2*A*c^5)*d^3
+ 4*(7*B*b^2*c^3 - 15*A*b*c^4)*d^2*e - (4*B*b^3*c^2 - 11*A*b^2*c^3)*d*e^2)*x^4 + 2*(A*b^4*c*e^3 - 24*(B*b^2*c
^3 - 2*A*b*c^4)*d^3 + 4*(7*B*b^3*c^2 - 15*A*b^2*c^3)*d^2*e - (4*B*b^4*c - 11*A*b^3*c^2)*d*e^2)*x^3 + (A*b^5*e^
3 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 + 4*(7*B*b^4*c - 15*A*b^3*c^2)*d^2*e - (4*B*b^5 - 11*A*b^4*c)*d*e^2)*x^2)
*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*c*d^3 - 2*A*b^5*d^2*e - (A*b^3*c^2*d*e^2 - 12*(B*b^2*c^3
- 2*A*b*c^4)*d^3 + (11*B*b^3*c^2 - 24*A*b^2*c^3)*d^2*e)*x^3 - (2*A*b^4*c*d*e^2 - 18*(B*b^3*c^2 - 2*A*b^2*c^3)
*d^3 + (17*B*b^4*c - 37*A*b^3*c^2)*d^2*e)*x^2 - (A*b^5*d*e^2 - 4*(B*b^4*c - 2*A*b^3*c^2)*d^3 + (4*B*b^5 - 9*A*
b^4*c)*d^2*e)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*
c*d^3 - b^8*d^2*e)*x^2)]
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giac [B] time = 0.29, size = 841, normalized size = 2.65 \begin {gather*} \frac {{\left (24 \, B b c^{3} d^{2} - 48 \, A c^{4} d^{2} - 40 \, B b^{2} c^{2} d e + 84 \, A b c^{3} d e + 15 \, B b^{3} c e^{2} - 35 \, A b^{2} c^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c d - b^{6} e\right )} \sqrt {-c^{2} d + b c e}} - \frac {12 \, {\left (x e + d\right )}^{\frac {7}{2}} B b c^{3} d^{2} e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{4} d^{2} e - 36 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c^{3} d^{3} e + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{4} d^{3} e + 36 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{3} d^{4} e - 72 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{4} d^{4} e - 12 \, \sqrt {x e + d} B b c^{3} d^{5} e + 24 \, \sqrt {x e + d} A c^{4} d^{5} e - 11 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{2} c^{2} d e^{2} + 24 \, {\left (x e + d\right )}^{\frac {7}{2}} A b c^{3} d e^{2} + 51 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} c^{2} d^{2} e^{2} - 108 \, {\left (x e + d\right )}^{\frac {5}{2}} A b c^{3} d^{2} e^{2} - 69 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{2} d^{3} e^{2} + 144 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{3} d^{3} e^{2} + 29 \, \sqrt {x e + d} B b^{2} c^{2} d^{4} e^{2} - 60 \, \sqrt {x e + d} A b c^{3} d^{4} e^{2} - {\left (x e + d\right )}^{\frac {7}{2}} A b^{2} c^{2} e^{3} - 17 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} c d e^{3} + 40 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{2} c^{2} d e^{3} + 38 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c d^{2} e^{3} - 85 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d^{2} e^{3} - 21 \, \sqrt {x e + d} B b^{3} c d^{3} e^{3} + 46 \, \sqrt {x e + d} A b^{2} c^{2} d^{3} e^{3} - 2 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} c e^{4} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e^{4} + 13 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c d e^{4} + 4 \, \sqrt {x e + d} B b^{4} d^{2} e^{4} - 9 \, \sqrt {x e + d} A b^{3} c d^{2} e^{4} - {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{5} - \sqrt {x e + d} A b^{4} d e^{5}}{4 \, {\left (b^{4} c d^{2} - b^{5} d e\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} - \frac {{\left (24 \, B b c d^{2} - 48 \, A c^{2} d^{2} - 4 \, B b^{2} d e + 12 \, A b c d e + A b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d} \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="giac")
[Out]
1/4*(24*B*b*c^3*d^2 - 48*A*c^4*d^2 - 40*B*b^2*c^2*d*e + 84*A*b*c^3*d*e + 15*B*b^3*c*e^2 - 35*A*b^2*c^2*e^2)*ar
ctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c*d - b^6*e)*sqrt(-c^2*d + b*c*e)) - 1/4*(12*(x*e + d)^(7/2)*
B*b*c^3*d^2*e - 24*(x*e + d)^(7/2)*A*c^4*d^2*e - 36*(x*e + d)^(5/2)*B*b*c^3*d^3*e + 72*(x*e + d)^(5/2)*A*c^4*d
^3*e + 36*(x*e + d)^(3/2)*B*b*c^3*d^4*e - 72*(x*e + d)^(3/2)*A*c^4*d^4*e - 12*sqrt(x*e + d)*B*b*c^3*d^5*e + 24
*sqrt(x*e + d)*A*c^4*d^5*e - 11*(x*e + d)^(7/2)*B*b^2*c^2*d*e^2 + 24*(x*e + d)^(7/2)*A*b*c^3*d*e^2 + 51*(x*e +
d)^(5/2)*B*b^2*c^2*d^2*e^2 - 108*(x*e + d)^(5/2)*A*b*c^3*d^2*e^2 - 69*(x*e + d)^(3/2)*B*b^2*c^2*d^3*e^2 + 144
*(x*e + d)^(3/2)*A*b*c^3*d^3*e^2 + 29*sqrt(x*e + d)*B*b^2*c^2*d^4*e^2 - 60*sqrt(x*e + d)*A*b*c^3*d^4*e^2 - (x*
e + d)^(7/2)*A*b^2*c^2*e^3 - 17*(x*e + d)^(5/2)*B*b^3*c*d*e^3 + 40*(x*e + d)^(5/2)*A*b^2*c^2*d*e^3 + 38*(x*e +
d)^(3/2)*B*b^3*c*d^2*e^3 - 85*(x*e + d)^(3/2)*A*b^2*c^2*d^2*e^3 - 21*sqrt(x*e + d)*B*b^3*c*d^3*e^3 + 46*sqrt(
x*e + d)*A*b^2*c^2*d^3*e^3 - 2*(x*e + d)^(5/2)*A*b^3*c*e^4 - 4*(x*e + d)^(3/2)*B*b^4*d*e^4 + 13*(x*e + d)^(3/2
)*A*b^3*c*d*e^4 + 4*sqrt(x*e + d)*B*b^4*d^2*e^4 - 9*sqrt(x*e + d)*A*b^3*c*d^2*e^4 - (x*e + d)^(3/2)*A*b^4*e^5
- sqrt(x*e + d)*A*b^4*d*e^5)/((b^4*c*d^2 - b^5*d*e)*((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 + A*b^2*e^2)*arctan(sqrt(x*e + d)/s
qrt(-d))/(b^5*sqrt(-d)*d)
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maple [B] time = 0.08, size = 829, normalized size = 2.62 \begin {gather*} \frac {11 \left (e x +d \right )^{\frac {3}{2}} A \,c^{3} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{3}}+\frac {35 A \,c^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} A \,c^{4} d e}{\left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{4}}-\frac {21 A \,c^{3} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{4}}+\frac {12 A \,c^{4} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{5}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} B \,c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{2}}-\frac {15 B c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,c^{3} d e}{\left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{3}}+\frac {10 B \,c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {6 B \,c^{3} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{4}}+\frac {13 \sqrt {e x +d}\, A \,c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, A \,c^{3} d e}{\left (c e x +b e \right )^{2} b^{4}}-\frac {9 \sqrt {e x +d}\, B c \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {2 \sqrt {e x +d}\, B \,c^{2} d e}{\left (c e x +b e \right )^{2} b^{3}}+\frac {A \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {3}{2}}}+\frac {3 A c e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} \sqrt {d}}-\frac {12 A \,c^{2} \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}-\frac {B e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} \sqrt {d}}+\frac {6 B c \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {\sqrt {e x +d}\, A}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, A c d}{b^{4} e \,x^{2}}+\frac {\sqrt {e x +d}\, B d}{b^{3} e \,x^{2}}-\frac {\left (e x +d \right )^{\frac {3}{2}} A}{4 b^{3} d \,x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A c}{b^{4} e \,x^{2}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B}{b^{3} 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]
-1/4/b^3/x^2*(e*x+d)^(1/2)*A-1/4/b^3/x^2/d*(e*x+d)^(3/2)*A-7/4*e^2*c^2/b^2/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/
2)*B-3*e*c^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d+2*e*c^2/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d+35/4*e^2*c^2/b^3/
(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A-15/4*e^2*c/b^2/(b*e-c*d)/((b*e-c*d
)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+11/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2
)*A-3/e/b^4/x^2*(e*x+d)^(1/2)*A*c*d+12*c^4/b^5/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c
)^(1/2)*c)*A*d^2-6*c^3/b^4/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2+2*e
*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*B*d-21*e*c^3/b^4/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(
1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*A*d+10*e*c^2/b^3/(b*e-c*d)/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d+3/e/b^4/x^2*(e*x+d)^(3/2)*A*c+1/e/b^3/x^2*(
e*x+d)^(1/2)*B*d+3*e/b^4/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+13/4*e^2*c^2/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(
1/2)-9/4*e^2*c/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-1/e/b^3/x^2*(e*x+d)^(3/2)*B+6/b^4*d^(1/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))*B*c-e/b^3/d^(1/2)*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+1/4*e^2/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A
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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?
________________________________________________________________________________________
mupad [B] time = 5.72, size = 8411, normalized size = 26.53
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(1/2))/(b*x + c*x^2)^3,x)
[Out]
(((d + e*x)^(3/2)*(A*b^4*e^5 + 72*A*c^4*d^4*e + 4*B*b^4*d*e^4 - 144*A*b*c^3*d^3*e^2 - 38*B*b^3*c*d^2*e^3 + 85*
A*b^2*c^2*d^2*e^3 + 69*B*b^2*c^2*d^3*e^2 - 13*A*b^3*c*d*e^4 - 36*B*b*c^3*d^4*e))/(4*b^4*(c*d^2 - b*d*e)) - ((d
+ e*x)^(1/2)*(A*b^3*e^4 + 24*A*c^3*d^3*e - 4*B*b^3*d*e^3 - 36*A*b*c^2*d^2*e^2 + 17*B*b^2*c*d^2*e^2 + 10*A*b^2
*c*d*e^3 - 12*B*b*c^2*d^3*e))/(4*b^4) + ((d + e*x)^(5/2)*(2*A*b^3*c*e^4 - 72*A*c^4*d^3*e + 108*A*b*c^3*d^2*e^2
- 40*A*b^2*c^2*d*e^3 - 51*B*b^2*c^2*d^2*e^2 + 36*B*b*c^3*d^3*e + 17*B*b^3*c*d*e^3))/(4*b^4*(c*d^2 - b*d*e)) +
(c*(d + e*x)^(7/2)*(A*b^2*c*e^3 + 24*A*c^3*d^2*e - 24*A*b*c^2*d*e^2 - 12*B*b*c^2*d^2*e + 11*B*b^2*c*d*e^2))/(
4*b^4*(c*d^2 - b*d*e)))/(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) - (ata
n((((-c*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4
*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e
^3 + 3296*B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 +
22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b
^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*
c*d^3*e)) + (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*
b^13*c^3*d^2*e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^
6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(128*b^10*c^5*d^
5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A
*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(64*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*
e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(48*A*c^3*d^2 - 15*B*b
^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6
*c^2*d^2*e - 3*b^7*c*d*e^2)))*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e
+ 40*B*b^2*c*d*e)*1i)/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)) + ((-c*(b*e - c*d)^3)^(1/2
)*(((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3
*e^5 + 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e
^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 460
8*A*B*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b
^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - (((A*b^14*c^2*d
*e^7 - 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*e^6 + 12*B*b^1
1*c^5*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2
*e^2 - 2*b^13*c*d^3*e) + ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^
3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2
- 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(64*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 +
3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 2
4*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)
))*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e)*1i)/(8*(b
^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))/((1728*A^3*c^10*d^6*e^3 - (35*A^3*b^6*c^4*e^9)/32 +
5508*A^3*b^2*c^8*d^4*e^5 - 2376*A^3*b^3*c^7*d^3*e^6 + (1233*A^3*b^4*c^6*d^2*e^7)/4 - 216*B^3*b^3*c^7*d^6*e^3 +
594*B^3*b^4*c^6*d^5*e^4 - 558*B^3*b^5*c^5*d^4*e^5 + (805*B^3*b^6*c^4*d^3*e^6)/4 - (165*B^3*b^7*c^3*d^2*e^7)/8
+ (15*A^2*B*b^7*c^3*e^9)/32 - 5184*A^3*b*c^9*d^5*e^4 + (63*A^3*b^5*c^5*d*e^8)/4 + 1296*A*B^2*b^2*c^8*d^6*e^3
- 3672*A*B^2*b^3*c^7*d^5*e^4 + 3600*A*B^2*b^4*c^6*d^4*e^5 - (2763*A*B^2*b^5*c^5*d^3*e^6)/2 + (1215*A*B^2*b^6*c
^4*d^2*e^7)/8 + 7560*A^2*B*b^2*c^8*d^5*e^4 - 7722*A^2*B*b^3*c^7*d^4*e^5 + (6291*A^2*B*b^4*c^6*d^3*e^6)/2 - (29
97*A^2*B*b^5*c^5*d^2*e^7)/8 + (105*A*B^2*b^7*c^3*d*e^8)/32 - 2592*A^2*B*b*c^9*d^6*e^3 - (465*A^2*B*b^6*c^4*d*e
^8)/32)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) + ((-c*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(A^2*b^6*
c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^2*e
^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5
+ 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*B*b
^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5
*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e^3
+ 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*c^4
*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((-c
*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64
*b^13*c^2*d^2*e^5)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*
c*d*e))/(64*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*
e^2)))*(-c*(b*e - c*d)^3)^(1/2)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*
e + 40*B*b^2*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(48*A*c^3*d^2 - 15*B*b^3*e
^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2
*d^2*e - 3*b^7*c*d*e^2)) - ((-c*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2
+ 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2
- 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 1
3824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c
^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4
+ b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A
*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e
^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) + ((-c*(b*e - c*d)^3)^(1/2)*(d + e*x
)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(48*A*c^3*d
^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(64*(b^8*c^2*d^4 + b^1
0*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(-c*(b*e - c*d)^3)^(1/2
)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(8*(b^8*e
^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c
^2*d^2 - 84*A*b*c^2*d*e + 40*B*b^2*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2))))*(-c
*(b*e - c*d)^3)^(1/2)*(48*A*c^3*d^2 - 15*B*b^3*e^2 + 35*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 84*A*b*c^2*d*e + 40*B*b
^2*c*d*e)*1i)/(4*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)) - (atan((((((d + e*x)^(1/2)*(A^2*b
^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^
2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*
e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*
B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*
b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e
^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*
c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - (
(d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(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))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d
^2*e^2 - 2*b^9*c*d^3*e)))*(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))/(8*b^5*(d^3)
^(1/2)))*(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)*1i)/(8*b^5*(d^3)^(1/2)) + ((((
d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5
+ 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e^4 -
1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B
*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^
5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - (((A*b^14*c^2*d*e^7
- 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*e^6 + 12*B*b^11*c^5
*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2
- 2*b^13*c*d^3*e) + ((d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*
b^13*c^2*d^2*e^5)*(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))/(64*b^5*(d^3)^(1/2)*
(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(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))/(8*b^5*(d^3)^(1/2)))*(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)*1i)/(8*b^
5*(d^3)^(1/2)))/((1728*A^3*c^10*d^6*e^3 - (35*A^3*b^6*c^4*e^9)/32 + 5508*A^3*b^2*c^8*d^4*e^5 - 2376*A^3*b^3*c^
7*d^3*e^6 + (1233*A^3*b^4*c^6*d^2*e^7)/4 - 216*B^3*b^3*c^7*d^6*e^3 + 594*B^3*b^4*c^6*d^5*e^4 - 558*B^3*b^5*c^5
*d^4*e^5 + (805*B^3*b^6*c^4*d^3*e^6)/4 - (165*B^3*b^7*c^3*d^2*e^7)/8 + (15*A^2*B*b^7*c^3*e^9)/32 - 5184*A^3*b*
c^9*d^5*e^4 + (63*A^3*b^5*c^5*d*e^8)/4 + 1296*A*B^2*b^2*c^8*d^6*e^3 - 3672*A*B^2*b^3*c^7*d^5*e^4 + 3600*A*B^2*
b^4*c^6*d^4*e^5 - (2763*A*B^2*b^5*c^5*d^3*e^6)/2 + (1215*A*B^2*b^6*c^4*d^2*e^7)/8 + 7560*A^2*B*b^2*c^8*d^5*e^4
- 7722*A^2*B*b^3*c^7*d^4*e^5 + (6291*A^2*B*b^4*c^6*d^3*e^6)/2 - (2997*A^2*B*b^5*c^5*d^2*e^7)/8 + (105*A*B^2*b
^7*c^3*d*e^8)/32 - 2592*A^2*B*b*c^9*d^6*e^3 - (465*A^2*B*b^6*c^4*d*e^8)/32)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b
^13*c*d^3*e) + ((((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A
^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296*B^2
*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c
^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e^
4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + (
((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 + 9*A*b^13*c^3*d^2*
e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d^2*e^6)/(b^12*c^2*
d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^1
2*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(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))/(
64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(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))/(8*b^5*(d^3)^(1/2)))*(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))/(8*b^5*(d^3)^(1/2)) - ((((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6*e^2 + 15072*A^2*b^2*c^
7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*
d^5*e^3 + 3296*B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 13824*A^2*b*c^8*d^5*
e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*
A*B*b^3*c^6*d^4*e^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2
*b^9*c*d^3*e)) - (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6*d^5*e^3 + 60*A*b^11*c^5*d^4*e^4 - 46*A*b^12*c^4*d^3*e^5 +
9*A*b^13*c^3*d^2*e^6 + 12*B*b^11*c^5*d^5*e^3 - 29*B*b^12*c^4*d^4*e^4 + 21*B*b^13*c^3*d^3*e^5 - 4*B*b^14*c^2*d
^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) + ((d + e*x)^(1/2)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4
*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5)*(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))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(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))/(8*b^5*(d^3)^(1/2)))*(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))/(8*b^5*(d^3)^(1/2))))*(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)*1i)/(4*b^5*(d^3)^(1/2))
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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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