3.16.28 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=425 \[ -\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^3}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7 (a+b x)}+\frac {5 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)} \]

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Rubi [A]  time = 0.43, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} -\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7 (a+b x)}+\frac {5 b^3 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6 (a+b x)}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (a+b x) (d+e x)^2}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^3}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d - a*e)
^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^3) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*
e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) - (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*
e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*
x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) + (b^5*B*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e
^7*(a + b*x)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e
^7*(a + b*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^4} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^4}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^3}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^2}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)}{e^6}+\frac {b^{10} B (d+e x)^2}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac {b^5 B (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 504, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (2 A e+B (d+3 e x))+5 a^4 b e^4 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+3 d e x+3 e^2 x^2\right )-B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-10 a^2 b^3 e^2 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )-5 a b^4 e \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )+60 b^2 (d+e x)^3 (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)+b^5 \left (A e \left (-47 d^5-81 d^4 e x+9 d^3 e^2 x^2+63 d^2 e^3 x^3+15 d e^4 x^4-3 e^5 x^5\right )+2 B \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )\right )}{6 e^7 (a+b x) (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

-1/6*(Sqrt[(a + b*x)^2]*(a^5*e^5*(2*A*e + B*(d + 3*e*x)) + 5*a^4*b*e^4*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x +
3*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*(d^2 + 3*d*e*x + 3*e^2*x^2) - B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) - 10*
a^2*b^3*e^2*(A*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 -
3*e^4*x^4)) - 5*a*b^4*e*(2*A*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + B*(47*d^5 +
81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)) + b^5*(A*e*(-47*d^5 - 81*d^4*e*x + 9
*d^3*e^2*x^2 + 63*d^2*e^3*x^3 + 15*d*e^4*x^4 - 3*e^5*x^5) + 2*B*(37*d^6 + 51*d^5*e*x - 39*d^4*e^2*x^2 - 73*d^3
*e^3*x^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6)) + 60*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)
^3*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^3)

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IntegrateAlgebraic [F]  time = 13.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

Defer[IntegrateAlgebraic][((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4, x]

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fricas [B]  time = 0.43, size = 899, normalized size = 2.12 \begin {gather*} \frac {2 \, B b^{5} e^{6} x^{6} - 74 \, B b^{5} d^{6} - 2 \, A a^{5} e^{6} + 47 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e - 130 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 110 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 3 \, {\left (2 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 15 \, {\left (2 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + {\left (146 \, B b^{5} d^{3} e^{3} - 63 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5}\right )} x^{3} + 3 \, {\left (26 \, B b^{5} d^{4} e^{2} - 3 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} - 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 60 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - 3 \, {\left (34 \, B b^{5} d^{5} e - 27 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 90 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x - 60 \, {\left (2 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + {\left (2 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (2 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5}\right )} x^{2} + 3 \, {\left (2 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 2 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*e^6*x^6 - 74*B*b^5*d^6 - 2*A*a^5*e^6 + 47*(5*B*a*b^4 + A*b^5)*d^5*e - 130*(2*B*a^2*b^3 + A*a*b^4)
*d^4*e^2 + 110*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 10*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - (B*a^5 + 5*A*a^4*b)*d*e^
5 - 3*(2*B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(2*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + 2*(2*B
*a^2*b^3 + A*a*b^4)*e^6)*x^4 + (146*B*b^5*d^3*e^3 - 63*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 90*(2*B*a^2*b^3 + A*a*b^4
)*d*e^5)*x^3 + 3*(26*B*b^5*d^4*e^2 - 3*(5*B*a*b^4 + A*b^5)*d^3*e^3 - 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 60*(
B*a^3*b^2 + A*a^2*b^3)*d*e^5 - 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 3*(34*B*b^5*d^5*e - 27*(5*B*a*b^4 + A*b^5
)*d^4*e^2 + 90*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 90*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 10*(B*a^4*b + 2*A*a^3*b^
2)*d*e^5 + (B*a^5 + 5*A*a^4*b)*e^6)*x - 60*(2*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + 2*(2*B*a^2*b^3 + A*a*b^4
)*d^4*e^2 - (B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + (2*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + 2*(2*B*a^2*b^3
+ A*a*b^4)*d*e^5 - (B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 3*(2*B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3 + 2*(2*
B*a^2*b^3 + A*a*b^4)*d^2*e^4 - (B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 + 3*(2*B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4
*e^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - (B*a^3*b^2 + A*a^2*b^3)*d^2*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e
^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [B]  time = 0.22, size = 874, normalized size = 2.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-10*(2*B*b^5*d^3*sgn(b*x + a) - 5*B*a*b^4*d^2*e*sgn(b*x + a) - A*b^5*d^2*e*sgn(b*x + a) + 4*B*a^2*b^3*d*e^2*sg
n(b*x + a) + 2*A*a*b^4*d*e^2*sgn(b*x + a) - B*a^3*b^2*e^3*sgn(b*x + a) - A*a^2*b^3*e^3*sgn(b*x + a))*e^(-7)*lo
g(abs(x*e + d)) + 1/6*(2*B*b^5*x^3*e^8*sgn(b*x + a) - 12*B*b^5*d*x^2*e^7*sgn(b*x + a) + 60*B*b^5*d^2*x*e^6*sgn
(b*x + a) + 15*B*a*b^4*x^2*e^8*sgn(b*x + a) + 3*A*b^5*x^2*e^8*sgn(b*x + a) - 120*B*a*b^4*d*x*e^7*sgn(b*x + a)
- 24*A*b^5*d*x*e^7*sgn(b*x + a) + 60*B*a^2*b^3*x*e^8*sgn(b*x + a) + 30*A*a*b^4*x*e^8*sgn(b*x + a))*e^(-12) - 1
/6*(74*B*b^5*d^6*sgn(b*x + a) - 235*B*a*b^4*d^5*e*sgn(b*x + a) - 47*A*b^5*d^5*e*sgn(b*x + a) + 260*B*a^2*b^3*d
^4*e^2*sgn(b*x + a) + 130*A*a*b^4*d^4*e^2*sgn(b*x + a) - 110*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 110*A*a^2*b^3*d^
3*e^3*sgn(b*x + a) + 10*B*a^4*b*d^2*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + B*a^5*d*e^5*sgn(b*x
+ a) + 5*A*a^4*b*d*e^5*sgn(b*x + a) + 2*A*a^5*e^6*sgn(b*x + a) + 30*(3*B*b^5*d^4*e^2*sgn(b*x + a) - 10*B*a*b^
4*d^3*e^3*sgn(b*x + a) - 2*A*b^5*d^3*e^3*sgn(b*x + a) + 12*B*a^2*b^3*d^2*e^4*sgn(b*x + a) + 6*A*a*b^4*d^2*e^4*
sgn(b*x + a) - 6*B*a^3*b^2*d*e^5*sgn(b*x + a) - 6*A*a^2*b^3*d*e^5*sgn(b*x + a) + B*a^4*b*e^6*sgn(b*x + a) + 2*
A*a^3*b^2*e^6*sgn(b*x + a))*x^2 + 3*(54*B*b^5*d^5*e*sgn(b*x + a) - 175*B*a*b^4*d^4*e^2*sgn(b*x + a) - 35*A*b^5
*d^4*e^2*sgn(b*x + a) + 200*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 100*A*a*b^4*d^3*e^3*sgn(b*x + a) - 90*B*a^3*b^2*d
^2*e^4*sgn(b*x + a) - 90*A*a^2*b^3*d^2*e^4*sgn(b*x + a) + 10*B*a^4*b*d*e^5*sgn(b*x + a) + 20*A*a^3*b^2*d*e^5*s
gn(b*x + a) + B*a^5*e^6*sgn(b*x + a) + 5*A*a^4*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^3

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maple [B]  time = 0.07, size = 1233, normalized size = 2.90

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x)

[Out]

1/6*((b*x+a)^2)^(5/2)*(-120*A*a*b^4*d^4*e^2*ln(e*x+d)-30*B*a^4*b*d*e^5*x+270*B*a^3*b^2*d^2*e^4*x-540*B*a^2*b^3
*d^3*e^3*x+405*B*a*b^4*d^4*e^2*x+47*A*b^5*d^5*e-240*B*a^2*b^3*d^4*e^2*ln(e*x+d)+300*B*a*b^4*d^5*e*ln(e*x+d)-13
0*A*a*b^4*d^4*e^2-10*B*a^4*b*d^2*e^4-20*A*a^3*b^2*d^2*e^4+110*A*a^2*b^3*d^3*e^3-B*a^5*d*e^5+110*B*a^3*b^2*d^3*
e^3-260*B*a^2*b^3*d^4*e^2+235*B*a*b^4*d^5*e+180*B*a^3*b^2*d^2*e^4*x*ln(e*x+d)-720*B*a^2*b^3*d^3*e^3*x*ln(e*x+d
)+900*B*a*b^4*d^4*e^2*x*ln(e*x+d)+180*A*a^2*b^3*d*e^5*x^2-90*A*a*b^4*d^2*e^4*x^2+60*A*b^5*d^5*e*ln(e*x+d)-15*A
*a^4*b*e^6*x+81*A*b^5*d^4*e^2*x+180*A*a^2*b^3*d^2*e^4*x*ln(e*x+d)-360*A*a*b^4*d^3*e^3*x*ln(e*x+d)+60*B*a^3*b^2
*d^3*e^3*ln(e*x+d)+60*B*ln(e*x+d)*x^3*a^3*b^2*e^6-120*B*ln(e*x+d)*x^3*b^5*d^3*e^3+180*A*b^5*d^4*e^2*x*ln(e*x+d
)-360*B*b^5*d^5*e*x*ln(e*x+d)-2*A*a^5*e^6-74*B*b^5*d^6+60*A*a^2*b^3*d^3*e^3*ln(e*x+d)-75*B*a*b^4*d*e^5*x^4+180
*A*a^2*b^3*d*e^5*x^2*ln(e*x+d)-360*A*a*b^4*d^2*e^4*x^2*ln(e*x+d)+180*B*a^3*b^2*d*e^5*x^2*ln(e*x+d)-5*A*a^4*b*d
*e^5+180*B*a^2*b^3*d*e^5*x^3-315*B*a*b^4*d^2*e^4*x^3-60*A*a^3*b^2*d*e^5*x+270*A*a^2*b^3*d^2*e^4*x-270*A*a*b^4*
d^3*e^3*x+180*B*a^3*b^2*d*e^5*x^2-63*A*b^5*d^2*e^4*x^3+146*B*b^5*d^3*e^3*x^3-60*A*a^3*b^2*e^6*x^2-9*A*b^5*d^3*
e^3*x^2-30*B*a^4*b*e^6*x^2+78*B*b^5*d^4*e^2*x^2+15*B*a*b^4*e^6*x^5+2*B*b^5*e^6*x^6+3*A*b^5*e^6*x^5-3*B*a^5*e^6
*x-120*B*b^5*d^6*ln(e*x+d)+90*A*a*b^4*d*e^5*x^3-120*A*ln(e*x+d)*x^3*a*b^4*d*e^5-240*B*ln(e*x+d)*x^3*a^2*b^3*d*
e^5+300*B*ln(e*x+d)*x^3*a*b^4*d^2*e^4+180*A*b^5*d^3*e^3*x^2*ln(e*x+d)-360*B*b^5*d^4*e^2*x^2*ln(e*x+d)+60*A*ln(
e*x+d)*x^3*a^2*b^3*e^6+60*A*ln(e*x+d)*x^3*b^5*d^2*e^4-45*B*a*b^4*d^3*e^3*x^2-180*B*a^2*b^3*d^2*e^4*x^2-720*B*a
^2*b^3*d^2*e^4*x^2*ln(e*x+d)+900*B*a*b^4*d^3*e^3*x^2*ln(e*x+d)-102*B*b^5*d^5*e*x-6*B*b^5*d*e^5*x^5+30*A*a*b^4*
e^6*x^4-15*A*b^5*d*e^5*x^4+60*B*a^2*b^3*e^6*x^4+30*B*b^5*d^2*e^4*x^4)/(b*x+a)^5/e^7/(e*x+d)^3

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^4,x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**4,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/(d + e*x)**4, x)

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