3.12.4 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^3} \, dx\)
Optimal. Leaf size=344 \[ -\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}+\frac {\sqrt {c d-b e} \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}-\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{4 b^4 c \left (b x+c x^2\right )} \]
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Rubi [A] time = 0.70, antiderivative size = 344, 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 =
{818, 820, 826, 1166, 208} \begin {gather*} -\frac {\sqrt {d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{4 b^4 c \left (b x+c x^2\right )}+\frac {\sqrt {c d-b e} \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+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^{3/2}}-\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^3,x]
[Out]
-((d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(2*b^2*c*(b*x + c*x^2)^2) - (Sqrt[d +
e*x]*(b*d*(6*b*B*c*d - 12*A*c^2*d - 2*b^2*B*e + 9*A*b*c*e) - (24*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(B*d + 2*
A*e) + b^2*c*e*(7*B*d + 3*A*e))*x))/(4*b^4*c*(b*x + c*x^2)) - (Sqrt[d]*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e)
- 12*b*c*d*(2*B*d + 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (Sqrt[c*d - b*e]*(48*A*c^3*d^2 + b^3*B*
e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/
(4*b^5*c^(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 818
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
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 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) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\frac {1}{2} e \left (6 A c^2 d-b^2 B e-3 b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac {(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{4} c d \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right )-\frac {1}{4} e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c}\\ &=-\frac {(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} d e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right )-\frac {1}{4} c d e \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right )-\frac {1}{4} e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 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 c}\\ &=-\frac {(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\left ((c d-b e) \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 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}+\frac {\left (c d \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 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}\\ &=-\frac {(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac {\sqrt {d} \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {\sqrt {c d-b e} \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 3.04, size = 516, normalized size = 1.50 \begin {gather*} \frac {\frac {30 c (d+e x)^{7/2} \left (b^2 (-e) (3 A e+4 B d)+b c d (13 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {(b+c x) \left ((b+c x) \left (15 c^{7/2} (c d-b e)^2 \left (-2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+\frac {2}{3} d \sqrt {d+e x} (4 d+e x)+\frac {2}{5} (d+e x)^{5/2}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )-2 c^2 d^2 \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )\right )-30 b c^{9/2} (d+e x)^{7/2} \left (b^3 e^2 (3 A e+4 B d)-b^2 c d e (18 A e+13 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{7/2} d (c d-b e)^2}-\frac {30 (d+e x)^{7/2} (3 A b e-8 A c d+4 b B d)}{b d x}-\frac {60 A (d+e x)^{7/2}}{x^2}}{120 b d (b+c x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^3,x]
[Out]
((30*c*(-12*A*c^2*d^2 - b^2*e*(4*B*d + 3*A*e) + b*c*d*(6*B*d + 13*A*e))*(d + e*x)^(7/2))/(b^2*d*(-(c*d) + b*e)
) - (60*A*(d + e*x)^(7/2))/x^2 - (30*(4*b*B*d - 8*A*c*d + 3*A*b*e)*(d + e*x)^(7/2))/(b*d*x) + ((b + c*x)*(-30*
b*c^(9/2)*(-24*A*c^3*d^3 + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 3*A*e) - b^2*c*d*e*(13*B*d + 18*A*e))
*(d + e*x)^(7/2) + (b + c*x)*(15*c^(7/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e) - 12*b*c*d*(2*B
*d + 5*A*e))*((2*(d + e*x)^(5/2))/5 + (2*d*Sqrt[d + e*x]*(4*d + e*x))/3 - 2*d^(5/2)*ArcTanh[Sqrt[d + e*x]/Sqrt
[d]]) - 2*c^2*d^2*(48*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*(3*c^(5/2)
*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d - b*e)^(3/2)*ArcTanh[
(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))/(b^4*c^(7/2)*d*(c*d - b*e)^2))/(120*b*d*(b + c*x)^2)
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IntegrateAlgebraic [B] time = 2.21, size = 746, normalized size = 2.17 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-15 A b^2 \sqrt {d} e^2+60 A b c d^{3/2} e-48 A c^2 d^{5/2}-20 b^2 B d^{3/2} e+24 b B c d^{5/2}\right )}{4 b^5}-\frac {\sqrt {d+e x} \left (-12 A b^3 c d^2 e^3+19 A b^3 c d e^3 (d+e x)-5 A b^3 c e^3 (d+e x)^2+48 A b^2 c^2 d^3 e^2-91 A b^2 c^2 d^2 e^2 (d+e x)+46 A b^2 c^2 d e^2 (d+e x)^2-3 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+b^4 B d^2 e^3-2 b^4 B d e^3 (d+e x)+b^4 B e^3 (d+e x)^2-14 b^3 B c d^3 e^2+23 b^3 B c d^2 e^2 (d+e x)-8 b^3 B c d e^2 (d+e x)^2-b^3 B c e^2 (d+e x)^3+25 b^2 B c^2 d^4 e-57 b^2 B c^2 d^3 e (d+e x)+39 b^2 B c^2 d^2 e (d+e x)^2-7 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 c e x^2 (b e+c (d+e x)-c d)^2}+\frac {\left (-3 A b^3 c e^3+39 A b^2 c^2 d e^2-84 A b c^3 d^2 e+48 A c^4 d^3+b^4 (-B) e^3-7 b^3 B c d e^2+32 b^2 B c^2 d^2 e-24 b B c^3 d^3\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^{3/2} \sqrt {b e-c d}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^3,x]
[Out]
-1/4*(Sqrt[d + e*x]*(-12*b*B*c^3*d^5 + 24*A*c^4*d^5 + 25*b^2*B*c^2*d^4*e - 60*A*b*c^3*d^4*e - 14*b^3*B*c*d^3*e
^2 + 48*A*b^2*c^2*d^3*e^2 + b^4*B*d^2*e^3 - 12*A*b^3*c*d^2*e^3 + 36*b*B*c^3*d^4*(d + e*x) - 72*A*c^4*d^4*(d +
e*x) - 57*b^2*B*c^2*d^3*e*(d + e*x) + 144*A*b*c^3*d^3*e*(d + e*x) + 23*b^3*B*c*d^2*e^2*(d + e*x) - 91*A*b^2*c^
2*d^2*e^2*(d + e*x) - 2*b^4*B*d*e^3*(d + e*x) + 19*A*b^3*c*d*e^3*(d + e*x) - 36*b*B*c^3*d^3*(d + e*x)^2 + 72*A
*c^4*d^3*(d + e*x)^2 + 39*b^2*B*c^2*d^2*e*(d + e*x)^2 - 108*A*b*c^3*d^2*e*(d + e*x)^2 - 8*b^3*B*c*d*e^2*(d + e
*x)^2 + 46*A*b^2*c^2*d*e^2*(d + e*x)^2 + b^4*B*e^3*(d + e*x)^2 - 5*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 - 7*b^2*B*c^2*d*e*(d + e*x)^3 + 24*A*b*c^3*d*e*(d + e*x)^3 - b^3*B*c*e^2*
(d + e*x)^3 - 3*A*b^2*c^2*e^2*(d + e*x)^3))/(b^4*c*e*x^2*(-(c*d) + b*e + c*(d + e*x))^2) + ((-24*b*B*c^3*d^3 +
48*A*c^4*d^3 + 32*b^2*B*c^2*d^2*e - 84*A*b*c^3*d^2*e - 7*b^3*B*c*d*e^2 + 39*A*b^2*c^2*d*e^2 - b^4*B*e^3 - 3*A
*b^3*c*e^3)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*c^(3/2)*Sqrt[-(c*d) + b*e])
+ ((24*b*B*c*d^(5/2) - 48*A*c^2*d^(5/2) - 20*b^2*B*d^(3/2)*e + 60*A*b*c*d^(3/2)*e - 15*A*b^2*Sqrt[d]*e^2)*Arc
Tanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5)
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fricas [B] time = 4.03, size = 2754, normalized size = 8.01
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[-1/8*(((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2
*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*
(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt((c*d - b*
e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - ((15*A*b^2*c^3*e^2 - 24*(
B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4
)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4
*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c*d^2 + (12*(B*b
^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2
- 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b
^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), -1/8*(2*((24*(B*b*c^4 -
2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b
*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^
3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x
+ d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^2*c^3 -
3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)
*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(d)*log((
e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 -
24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*
b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d)
)/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^
2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c
^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(
-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3
*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*
c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A
*b^4*c)*e^2)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x +
b)) - 2*(2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*
b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*
e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^
7*c*x^2), -1/4*(((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2
)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x
^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt(
-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 -
2*A*c^5)*d^2 + 20*(B*b^2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 2
0*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*
b^3*c^2)*d*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^
2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 -
(11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^
2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2)]
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giac [B] time = 0.34, size = 841, normalized size = 2.44 \begin {gather*} -\frac {{\left (24 \, B b c d^{3} - 48 \, A c^{2} d^{3} - 20 \, B b^{2} d^{2} e + 60 \, A b c d^{2} e - 15 \, A b^{2} d e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} + \frac {{\left (24 \, B b c^{3} d^{3} - 48 \, A c^{4} d^{3} - 32 \, B b^{2} c^{2} d^{2} e + 84 \, A b c^{3} d^{2} e + 7 \, B b^{3} c d e^{2} - 39 \, A b^{2} c^{2} d e^{2} + B b^{4} e^{3} + 3 \, A b^{3} c e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5} c} - \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 - 7 \, {\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} + 39 \, {\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} - 57 \, {\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} + 25 \, \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}} B b^{3} c e^{3} - 3 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{2} c^{2} e^{3} - 8 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} c d e^{3} + 46 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{2} c^{2} d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c d^{2} e^{3} - 91 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{2} d^{2} e^{3} - 14 \, \sqrt {x e + d} B b^{3} c d^{3} e^{3} + 48 \, \sqrt {x e + d} A b^{2} c^{2} d^{3} e^{3} + {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} e^{4} - 5 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} c e^{4} - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e^{4} + 19 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c d e^{4} + \sqrt {x e + d} B b^{4} d^{2} e^{4} - 12 \, \sqrt {x e + d} A b^{3} c d^{2} e^{4}}{4 \, {\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} b^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
-1/4*(24*B*b*c*d^3 - 48*A*c^2*d^3 - 20*B*b^2*d^2*e + 60*A*b*c*d^2*e - 15*A*b^2*d*e^2)*arctan(sqrt(x*e + d)/sqr
t(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^3*d^3 - 48*A*c^4*d^3 - 32*B*b^2*c^2*d^2*e + 84*A*b*c^3*d^2*e + 7*B*b^3*c
*d*e^2 - 39*A*b^2*c^2*d*e^2 + B*b^4*e^3 + 3*A*b^3*c*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c
^2*d + b*c*e)*b^5*c) - 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 - 7*(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 + 39*(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 - 57*(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 + 25*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)*B*b^3*c*e^3 - 3*(x*e + d)^(7/2)*A*b^2*c^2*e^3 - 8*(
x*e + d)^(5/2)*B*b^3*c*d*e^3 + 46*(x*e + d)^(5/2)*A*b^2*c^2*d*e^3 + 23*(x*e + d)^(3/2)*B*b^3*c*d^2*e^3 - 91*(x
*e + d)^(3/2)*A*b^2*c^2*d^2*e^3 - 14*sqrt(x*e + d)*B*b^3*c*d^3*e^3 + 48*sqrt(x*e + d)*A*b^2*c^2*d^3*e^3 + (x*e
+ d)^(5/2)*B*b^4*e^4 - 5*(x*e + d)^(5/2)*A*b^3*c*e^4 - 2*(x*e + d)^(3/2)*B*b^4*d*e^4 + 19*(x*e + d)^(3/2)*A*b
^3*c*d*e^4 + sqrt(x*e + d)*B*b^4*d^2*e^4 - 12*sqrt(x*e + d)*A*b^3*c*d^2*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c)
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maple [B] time = 0.08, size = 1004, normalized size = 2.92
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x)
[Out]
15*e*d^(3/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+6/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c
*d)*c)^(1/2)*c)*B*d^3+7/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d-12/b^5*c
^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^3+3/4*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/
2)*A*c+1/4*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B-5*e*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B+3/4*e^3/b^2/((
b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+5/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A+7/4*d^
2/b^3/x^2*(e*x+d)^(1/2)*A-12*d^(5/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6*d^(5/2)/b^4*arctanh((e*x+d)^(1
/2)/d^(1/2))*B*c-9/4*d/b^3/x^2*(e*x+d)^(3/2)*A-1/4*e^4/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)*B-15/4*e^2*d^(1/2)/b^3*ar
ctanh((e*x+d)^(1/2)/d^(1/2))*A-1/e*d^2/b^3/x^2*(e*x+d)^(3/2)*B+1/e*d^3/b^3/x^2*(e*x+d)^(1/2)*B+5/2*e^3/b/(c*e*
x+b*e)^2*(e*x+d)^(1/2)*B*d+1/4*e^3/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B-39/4*
e^2/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*c*d+21*e/b^4*c^2/((b*e-c*d)*c)^(1/2)
*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2-8*e/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*B*d^2*c-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*c^2*d^2+2*e/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(1/2)*B*d
^3+3/e*d^2/b^4/x^2*(e*x+d)^(3/2)*A*c-3/e*d^3/b^4/x^2*(e*x+d)^(1/2)*A*c-17/4*e^2/b^2/(c*e*x+b*e)^2*(e*x+d)^(1/2
)*B*d^2*c+7/4*e^2/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*c*d-11/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*c*d+3*e/b^4
/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^3*d^2+29/4*e^2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(1/2)*A*d^2-3*e/b^4/(c*e*x+b*e)^
2*c^3*(e*x+d)^(1/2)*A*d^3-15/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^2*d
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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)^(5/2)/(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 = 3.85, size = 7001, normalized size = 20.35
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^3,x)
[Out]
(((d + e*x)^(7/2)*(B*b^3*e^3 + 3*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 + 7*B*b^2*
c*d*e^2))/(4*b^4) - ((d + e*x)^(5/2)*(B*b^4*e^4 - 5*A*b^3*c*e^4 + 72*A*c^4*d^3*e - 108*A*b*c^3*d^2*e^2 + 46*A*
b^2*c^2*d*e^3 + 39*B*b^2*c^2*d^2*e^2 - 36*B*b*c^3*d^3*e - 8*B*b^3*c*d*e^3))/(4*b^4*c) - ((d + e*x)^(1/2)*(24*A
*c^4*d^5*e + B*b^4*d^2*e^4 - 60*A*b*c^3*d^4*e^2 - 12*A*b^3*c*d^2*e^4 - 14*B*b^3*c*d^3*e^3 + 48*A*b^2*c^2*d^3*e
^3 + 25*B*b^2*c^2*d^4*e^2 - 12*B*b*c^3*d^5*e))/(4*b^4*c) + ((d + e*x)^(3/2)*(72*A*c^4*d^4*e + 2*B*b^4*d*e^4 -
144*A*b*c^3*d^3*e^2 - 23*B*b^3*c*d^2*e^3 + 91*A*b^2*c^2*d^2*e^3 + 57*B*b^2*c^2*d^3*e^2 - 19*A*b^3*c*d*e^4 - 36
*B*b*c^3*d^4*e))/(4*b^4*c))/(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) -
(d^(1/2)*atan(((d^(1/2)*((d^(1/2)*((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11
*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) - (d^(1/2)*(64*b^11*c^3*e^3 - 128*b^10*
c^4*d*e^2)*(d + e*x)^(1/2)*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(64*b^1
3*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(8*b^5) - ((d + e*x)^(1/2)*(
B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5
+ 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 -
400*B^2*b^5*c^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*
c^3*d*e^7 + 6*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 1104
0*A*B*b^3*c^5*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8*b^8*c))*(15*A*b^2*e^2 + 48*A*c
^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e)*1i)/(8*b^5) - (d^(1/2)*((d^(1/2)*((12*A*b^12*c^3*d*e^5 -
B*b^13*c^2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e
^4)/(b^12*c) + (d^(1/2)*(64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(d + e*x)^(1/2)*(15*A*b^2*e^2 + 48*A*c^2*d^2 -
24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(64*b^13*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b
^2*d*e - 60*A*b*c*d*e))/(8*b^5) + ((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 1
5840*A^2*b^2*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 24
96*B^2*b^3*c^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*
b^7*c*d*e^7 - 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*
A*B*b^6*c^2*d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B
*b^5*c^3*d^2*e^6))/(8*b^8*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e)*1i)/(
8*b^5))/(((5*B^3*b^9*d^2*e^9)/8 - 1728*A^3*c^9*d^8*e^3 - 11124*A^3*b^2*c^7*d^6*e^5 + 9180*A^3*b^3*c^6*d^5*e^6
- (16389*A^3*b^4*c^5*d^4*e^7)/4 + (1917*A^3*b^5*c^4*d^3*e^8)/2 - (3375*A^3*b^6*c^3*d^2*e^9)/32 + 216*B^3*b^3*c
^6*d^8*e^3 - 594*B^3*b^4*c^5*d^7*e^4 + 558*B^3*b^5*c^4*d^6*e^5 - (725*B^3*b^6*c^3*d^5*e^6)/4 - (59*B^3*b^7*c^2
*d^4*e^7)/8 + (15*A*B^2*b^9*d*e^10)/32 + 6912*A^3*b*c^8*d^7*e^4 + (135*A^3*b^7*c^2*d*e^10)/32 + 8*B^3*b^8*c*d^
3*e^8 - 1296*A*B^2*b^2*c^7*d^8*e^3 + 4104*A*B^2*b^3*c^6*d^7*e^4 - 4788*A*B^2*b^4*c^5*d^6*e^5 + (4815*A*B^2*b^5
*c^4*d^5*e^6)/2 - (3171*A*B^2*b^6*c^3*d^4*e^7)/8 - (1281*A*B^2*b^7*c^2*d^3*e^8)/32 - 9288*A^2*B*b^2*c^7*d^7*e^
4 + 12906*A^2*B*b^3*c^6*d^6*e^5 - (17235*A^2*B*b^4*c^5*d^5*e^6)/2 + (21717*A^2*B*b^5*c^4*d^4*e^7)/8 - (9423*A^
2*B*b^6*c^3*d^3*e^8)/32 - (495*A^2*B*b^7*c^2*d^2*e^9)/32 + (45*A^2*B*b^8*c*d*e^10)/16 + (135*A*B^2*b^8*c*d^2*e
^9)/16 + 2592*A^2*B*b*c^8*d^8*e^3)/(b^12*c) + (d^(1/2)*((d^(1/2)*((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24
*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) - (d^(1/
2)*(64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(d + e*x)^(1/2)*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b
^2*d*e - 60*A*b*c*d*e))/(64*b^13*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e
))/(8*b^5) - ((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*
e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^
3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A
^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 1
1904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8
*b^8*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(8*b^5) + (d^(1/2)*((d^(1
/2)*((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d
^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) + (d^(1/2)*(64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(d + e*x)^(1/2)*(15
*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(64*b^13*c))*(15*A*b^2*e^2 + 48*A*c^2
*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e))/(8*b^5) + ((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^
8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1
152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*B
^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 -
4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*A
*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8*b^8*c))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b
^2*d*e - 60*A*b*c*d*e))/(8*b^5)))*(15*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e)*1
i)/(4*b^5) + (atan((((((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2
*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c
^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7
- 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*
d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2
*e^6))/(8*b^8*c) - (((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 -
12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) - ((64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(-c^3*(b*e
- c*d))^(1/2)*(d + e*x)^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*
B*b^2*c*d*e))/(64*b^13*c^4))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d
^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^3))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^
2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e)*1i)/(8*b^5*c^3) + ((((d + e*x)^(1/2)*(B^2*b^8*e^8 +
9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^
4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c
^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6
*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5
*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8*b^8*c) + (((12*A*b^12*c^3*d*e^5 - B*b^13*c^
2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12
*c) + ((64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(48*A*c^3*d^2 + B*b^3*e
^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(64*b^13*c^4))*(-c^3*(b*e - c*d))^(1/2)
*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^3))*(-
c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c
*d*e)*1i)/(8*b^5*c^3))/(((5*B^3*b^9*d^2*e^9)/8 - 1728*A^3*c^9*d^8*e^3 - 11124*A^3*b^2*c^7*d^6*e^5 + 9180*A^3*b
^3*c^6*d^5*e^6 - (16389*A^3*b^4*c^5*d^4*e^7)/4 + (1917*A^3*b^5*c^4*d^3*e^8)/2 - (3375*A^3*b^6*c^3*d^2*e^9)/32
+ 216*B^3*b^3*c^6*d^8*e^3 - 594*B^3*b^4*c^5*d^7*e^4 + 558*B^3*b^5*c^4*d^6*e^5 - (725*B^3*b^6*c^3*d^5*e^6)/4 -
(59*B^3*b^7*c^2*d^4*e^7)/8 + (15*A*B^2*b^9*d*e^10)/32 + 6912*A^3*b*c^8*d^7*e^4 + (135*A^3*b^7*c^2*d*e^10)/32 +
8*B^3*b^8*c*d^3*e^8 - 1296*A*B^2*b^2*c^7*d^8*e^3 + 4104*A*B^2*b^3*c^6*d^7*e^4 - 4788*A*B^2*b^4*c^5*d^6*e^5 +
(4815*A*B^2*b^5*c^4*d^5*e^6)/2 - (3171*A*B^2*b^6*c^3*d^4*e^7)/8 - (1281*A*B^2*b^7*c^2*d^3*e^8)/32 - 9288*A^2*B
*b^2*c^7*d^7*e^4 + 12906*A^2*B*b^3*c^6*d^6*e^5 - (17235*A^2*B*b^4*c^5*d^5*e^6)/2 + (21717*A^2*B*b^5*c^4*d^4*e^
7)/8 - (9423*A^2*B*b^6*c^3*d^3*e^8)/32 - (495*A^2*B*b^7*c^2*d^2*e^9)/32 + (45*A^2*B*b^8*c*d*e^10)/16 + (135*A*
B^2*b^8*c*d^2*e^9)/16 + 2592*A^2*B*b*c^8*d^8*e^3)/(b^12*c) - ((((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e
^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 - 8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 +
1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^3 + 1760*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*
B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A^2*b*c^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 -
4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 11904*A*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*
A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8*b^8*c) - (((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A*b
^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) - ((64*b^11*
c^3*e^3 - 128*b^10*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e
^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(64*b^13*c^4))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 +
B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^3))*(-c^3*(b*e - c*d))
^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^
3) + ((((d + e*x)^(1/2)*(B^2*b^8*e^8 + 9*A^2*b^6*c^2*e^8 + 4608*A^2*c^8*d^6*e^2 + 15840*A^2*b^2*c^6*d^4*e^4 -
8640*A^2*b^3*c^5*d^3*e^5 + 2250*A^2*b^4*c^4*d^2*e^6 + 1152*B^2*b^2*c^6*d^6*e^2 - 2496*B^2*b^3*c^5*d^5*e^3 + 17
60*B^2*b^4*c^4*d^4*e^4 - 400*B^2*b^5*c^3*d^3*e^5 - 15*B^2*b^6*c^2*d^2*e^6 + 14*B^2*b^7*c*d*e^7 - 13824*A^2*b*c
^7*d^5*e^3 - 234*A^2*b^5*c^3*d*e^7 + 6*A*B*b^7*c*e^8 - 4608*A*B*b*c^7*d^6*e^2 - 36*A*B*b^6*c^2*d*e^7 + 11904*A
*B*b^2*c^6*d^5*e^3 - 11040*A*B*b^3*c^5*d^4*e^4 + 4320*A*B*b^4*c^4*d^3*e^5 - 570*A*B*b^5*c^3*d^2*e^6))/(8*b^8*c
) + (((12*A*b^12*c^3*d*e^5 - B*b^13*c^2*d*e^5 + 24*A*b^10*c^5*d^3*e^3 - 36*A*b^11*c^4*d^2*e^4 - 12*B*b^11*c^4*
d^3*e^3 + 13*B*b^12*c^3*d^2*e^4)/(b^12*c) + ((64*b^11*c^3*e^3 - 128*b^10*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(
d + e*x)^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(
64*b^13*c^4))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2
*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^3))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2 + 3*A*b^2*c*e^2 - 24*B*
b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*b^5*c^3)))*(-c^3*(b*e - c*d))^(1/2)*(48*A*c^3*d^2 + B*b^3*e^2
+ 3*A*b^2*c*e^2 - 24*B*b*c^2*d^2 - 36*A*b*c^2*d*e + 8*B*b^2*c*d*e)*1i)/(4*b^5*c^3)
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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)**(5/2)/(c*x**2+b*x)**3,x)
[Out]
Timed out
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