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

Optimal. Leaf size=421 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}-\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]

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Rubi [A]  time = 0.39, antiderivative size = 421, 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 x \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e 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)}{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)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (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)^5,x]

[Out]

-((b^4*(5*b*B*d - A*b*e - 5*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^5*B*x^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b
*x)*(d + e*x)^4) + ((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])/(3*e^7*(a + b*x)*
(d + e*x)^3) - (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])/(2*e^7*(a + b*x)*
(d + e*x)^2) + (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])/(e^7*(a + b*x)*(
d + e*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*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)^5} \, 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)^5} \, 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 {b^9 (-5 b B d+A b e+5 a B e)}{e^6}+\frac {b^{10} B x}{e^5}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^5}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^4}-\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)^3}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^2}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {b^4 (5 b B d-A b e-5 a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\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}}{3 e^7 (a+b x) (d+e x)^3}-\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}}{2 e^7 (a+b x) (d+e x)^2}+\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}}{e^7 (a+b x) (d+e x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 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 = 0.30, size = 497, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 e^5 (3 A e+B (d+4 e x))+5 a^4 b e^4 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+10 a^2 b^3 e^2 \left (3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-5 a b^4 e \left (A d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-B \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )-60 b^3 (d+e x)^4 (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)+b^5 \left (A e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-3 B \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^4} \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)^5,x]

[Out]

-1/12*(Sqrt[(a + b*x)^2]*(a^5*e^5*(3*A*e + B*(d + 4*e*x)) + 5*a^4*b*e^4*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x +
6*e^2*x^2)) + 10*a^3*b^2*e^3*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3
)) + 10*a^2*b^3*e^2*(3*A*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - B*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*
x^2 + 48*e^3*x^3)) - 5*a*b^4*e*(A*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4
*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + b^5*(A*e*(77*d^5 + 248*d^4*e*x + 252*d
^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - 3*B*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^
3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6)) - 60*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d +
e*x)^4*Log[d + e*x]))/(e^7*(a + b*x)*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 180.42, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \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)^5,x]

[Out]

$Aborted

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fricas [B]  time = 0.46, size = 871, normalized size = 2.07 \begin {gather*} \frac {6 \, B b^{5} e^{6} x^{6} + 171 \, B b^{5} d^{6} - 3 \, A a^{5} e^{6} - 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 125 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} - 5 \, {\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} - 12 \, {\left (3 \, B b^{5} d e^{5} - {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} - 12 \, {\left (17 \, B b^{5} d^{2} e^{4} - 4 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5}\right )} x^{4} - 24 \, {\left (4 \, B b^{5} d^{3} e^{3} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} - 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 6 \, {\left (66 \, B b^{5} d^{4} e^{2} - 42 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 90 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} - 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 4 \, {\left (126 \, B b^{5} d^{5} e - 62 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 110 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 30 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} - 5 \, {\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 (3 \, B b^{5} d^{6} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + {\left (3 \, B b^{5} d^{2} e^{4} - {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 4 \, {\left (3 \, B b^{5} d^{3} e^{3} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5}\right )} x^{3} + 6 \, {\left (3 \, B b^{5} d^{4} e^{2} - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (3 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(6*B*b^5*e^6*x^6 + 171*B*b^5*d^6 - 3*A*a^5*e^6 - 77*(5*B*a*b^4 + A*b^5)*d^5*e + 125*(2*B*a^2*b^3 + A*a*b^
4)*d^4*e^2 - 30*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 5*(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 - 12*(3*B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 - 12*(17*B*b^5*d^2*e^4 - 4*(5*B*a*b^4 + A*b^5)*d*e^5)*x^4
- 24*(4*B*b^5*d^3*e^3 + 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 - 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 5*(B*a^3*b^2 + A*a
^2*b^3)*e^6)*x^3 + 6*(66*B*b^5*d^4*e^2 - 42*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 90*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 -
30*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 - 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 4*(126*B*b^5*d^5*e - 62*(5*B*a*b^4 +
A*b^5)*d^4*e^2 + 110*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 - 5*(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*(3*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + (2*B*a^2*b^3 + A*a*
b^4)*d^4*e^2 + (3*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + (2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 4*(3*B*b^5*d^
3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + (2*B*a^2*b^3 + A*a*b^4)*d*e^5)*x^3 + 6*(3*B*b^5*d^4*e^2 - (5*B*a*b^4 + A
*b^5)*d^3*e^3 + (2*B*a^2*b^3 + A*a*b^4)*d^2*e^4)*x^2 + 4*(3*B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4*e^2 + (2*B*a
^2*b^3 + A*a*b^4)*d^3*e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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giac [B]  time = 0.23, size = 870, normalized size = 2.07

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)^5,x, algorithm="giac")

[Out]

5*(3*B*b^5*d^2*sgn(b*x + a) - 5*B*a*b^4*d*e*sgn(b*x + a) - A*b^5*d*e*sgn(b*x + a) + 2*B*a^2*b^3*e^2*sgn(b*x +
a) + A*a*b^4*e^2*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/2*(B*b^5*x^2*e^5*sgn(b*x + a) - 10*B*b^5*d*x*e^4*s
gn(b*x + a) + 10*B*a*b^4*x*e^5*sgn(b*x + a) + 2*A*b^5*x*e^5*sgn(b*x + a))*e^(-10) + 1/12*(171*B*b^5*d^6*sgn(b*
x + a) - 385*B*a*b^4*d^5*e*sgn(b*x + a) - 77*A*b^5*d^5*e*sgn(b*x + a) + 250*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 1
25*A*a*b^4*d^4*e^2*sgn(b*x + a) - 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) - 5*B*
a^4*b*d^2*e^4*sgn(b*x + a) - 10*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*sg
n(b*x + a) - 3*A*a^5*e^6*sgn(b*x + a) + 120*(2*B*b^5*d^3*e^3*sgn(b*x + a) - 5*B*a*b^4*d^2*e^4*sgn(b*x + a) - A
*b^5*d^2*e^4*sgn(b*x + a) + 4*B*a^2*b^3*d*e^5*sgn(b*x + a) + 2*A*a*b^4*d*e^5*sgn(b*x + a) - B*a^3*b^2*e^6*sgn(
b*x + a) - A*a^2*b^3*e^6*sgn(b*x + a))*x^3 + 30*(21*B*b^5*d^4*e^2*sgn(b*x + a) - 50*B*a*b^4*d^3*e^3*sgn(b*x +
a) - 10*A*b^5*d^3*e^3*sgn(b*x + a) + 36*B*a^2*b^3*d^2*e^4*sgn(b*x + a) + 18*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 + 4*(141*B*b^5*d^5*e*sgn(b*x + a) - 325*B*a*b^4*d^4*e^2*sgn(b*x + a) - 65*A*b^5*d^4*e^2*sgn(b*x
+ a) + 220*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 110*A*a*b^4*d^3*e^3*sgn(b*x + a) - 30*B*a^3*b^2*d^2*e^4*sgn(b*x +
a) - 30*A*a^2*b^3*d^2*e^4*sgn(b*x + a) - 5*B*a^4*b*d*e^5*sgn(b*x + a) - 10*A*a^3*b^2*d*e^5*sgn(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)^4

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maple [B]  time = 0.07, size = 1163, normalized size = 2.76 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (6 B \,b^{5} e^{6} x^{6}+60 A a \,b^{4} e^{6} x^{4} \ln \left (e x +d \right )-60 A \,b^{5} d \,e^{5} x^{4} \ln \left (e x +d \right )+12 A \,b^{5} e^{6} x^{5}+120 B \,a^{2} b^{3} e^{6} x^{4} \ln \left (e x +d \right )-300 B a \,b^{4} d \,e^{5} x^{4} \ln \left (e x +d \right )+60 B a \,b^{4} e^{6} x^{5}+180 B \,b^{5} d^{2} e^{4} x^{4} \ln \left (e x +d \right )-36 B \,b^{5} d \,e^{5} x^{5}+240 A a \,b^{4} d \,e^{5} x^{3} \ln \left (e x +d \right )-240 A \,b^{5} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+48 A \,b^{5} d \,e^{5} x^{4}+480 B \,a^{2} b^{3} d \,e^{5} x^{3} \ln \left (e x +d \right )-1200 B a \,b^{4} d^{2} e^{4} x^{3} \ln \left (e x +d \right )+240 B a \,b^{4} d \,e^{5} x^{4}+720 B \,b^{5} d^{3} e^{3} x^{3} \ln \left (e x +d \right )-204 B \,b^{5} d^{2} e^{4} x^{4}-120 A \,a^{2} b^{3} e^{6} x^{3}+360 A a \,b^{4} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+240 A a \,b^{4} d \,e^{5} x^{3}-360 A \,b^{5} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-48 A \,b^{5} d^{2} e^{4} x^{3}-120 B \,a^{3} b^{2} e^{6} x^{3}+720 B \,a^{2} b^{3} d^{2} e^{4} x^{2} \ln \left (e x +d \right )+480 B \,a^{2} b^{3} d \,e^{5} x^{3}-1800 B a \,b^{4} d^{3} e^{3} x^{2} \ln \left (e x +d \right )-240 B a \,b^{4} d^{2} e^{4} x^{3}+1080 B \,b^{5} d^{4} e^{2} x^{2} \ln \left (e x +d \right )-96 B \,b^{5} d^{3} e^{3} x^{3}-60 A \,a^{3} b^{2} e^{6} x^{2}-180 A \,a^{2} b^{3} d \,e^{5} x^{2}+240 A a \,b^{4} d^{3} e^{3} x \ln \left (e x +d \right )+540 A a \,b^{4} d^{2} e^{4} x^{2}-240 A \,b^{5} d^{4} e^{2} x \ln \left (e x +d \right )-252 A \,b^{5} d^{3} e^{3} x^{2}-30 B \,a^{4} b \,e^{6} x^{2}-180 B \,a^{3} b^{2} d \,e^{5} x^{2}+480 B \,a^{2} b^{3} d^{3} e^{3} x \ln \left (e x +d \right )+1080 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-1200 B a \,b^{4} d^{4} e^{2} x \ln \left (e x +d \right )-1260 B a \,b^{4} d^{3} e^{3} x^{2}+720 B \,b^{5} d^{5} e x \ln \left (e x +d \right )+396 B \,b^{5} d^{4} e^{2} x^{2}-20 A \,a^{4} b \,e^{6} x -40 A \,a^{3} b^{2} d \,e^{5} x -120 A \,a^{2} b^{3} d^{2} e^{4} x +60 A a \,b^{4} d^{4} e^{2} \ln \left (e x +d \right )+440 A a \,b^{4} d^{3} e^{3} x -60 A \,b^{5} d^{5} e \ln \left (e x +d \right )-248 A \,b^{5} d^{4} e^{2} x -4 B \,a^{5} e^{6} x -20 B \,a^{4} b d \,e^{5} x -120 B \,a^{3} b^{2} d^{2} e^{4} x +120 B \,a^{2} b^{3} d^{4} e^{2} \ln \left (e x +d \right )+880 B \,a^{2} b^{3} d^{3} e^{3} x -300 B a \,b^{4} d^{5} e \ln \left (e x +d \right )-1240 B a \,b^{4} d^{4} e^{2} x +180 B \,b^{5} d^{6} \ln \left (e x +d \right )+504 B \,b^{5} d^{5} e x -3 A \,a^{5} e^{6}-5 A \,a^{4} b d \,e^{5}-10 A \,a^{3} b^{2} d^{2} e^{4}-30 A \,a^{2} b^{3} d^{3} e^{3}+125 A a \,b^{4} d^{4} e^{2}-77 A \,b^{5} d^{5} e -B \,a^{5} d \,e^{5}-5 B \,a^{4} b \,d^{2} e^{4}-30 B \,a^{3} b^{2} d^{3} e^{3}+250 B \,a^{2} b^{3} d^{4} e^{2}-385 B a \,b^{4} d^{5} e +171 B \,b^{5} d^{6}\right )}{12 \left (b x +a \right )^{5} \left (e x +d \right )^{4} e^{7}} \end {gather*}

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)^5,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(60*A*a*b^4*d^4*e^2*ln(e*x+d)-300*B*ln(e*x+d)*x^4*a*b^4*d*e^5-20*B*a^4*b*d*e^5*x-120*B*
a^3*b^2*d^2*e^4*x+880*B*a^2*b^3*d^3*e^3*x-1240*B*a*b^4*d^4*e^2*x-77*A*b^5*d^5*e+120*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)+125*A*a*b^4*d^4*e^2-5*B*a^4*b*d^2*e^4-10*A*a^3*b^2*d^2*e^4-30*A*a^2*b^3*d^3*e^3-
B*a^5*d*e^5-30*B*a^3*b^2*d^3*e^3+250*B*a^2*b^3*d^4*e^2-385*B*a*b^4*d^5*e+480*B*a^2*b^3*d^3*e^3*x*ln(e*x+d)-120
0*B*a*b^4*d^4*e^2*x*ln(e*x+d)-180*A*a^2*b^3*d*e^5*x^2+540*A*a*b^4*d^2*e^4*x^2-60*A*b^5*d^5*e*ln(e*x+d)-20*A*a^
4*b*e^6*x-248*A*b^5*d^4*e^2*x-120*A*a^2*b^3*e^6*x^3+240*A*a*b^4*d^3*e^3*x*ln(e*x+d)+720*B*b^5*d^3*e^3*x^3*ln(e
*x+d)-240*A*b^5*d^4*e^2*x*ln(e*x+d)+720*B*b^5*d^5*e*x*ln(e*x+d)+60*A*ln(e*x+d)*x^4*a*b^4*e^6-60*A*ln(e*x+d)*x^
4*b^5*d*e^5+120*B*ln(e*x+d)*x^4*a^2*b^3*e^6+180*B*ln(e*x+d)*x^4*b^5*d^2*e^4-3*A*a^5*e^6+171*B*b^5*d^6+240*B*a*
b^4*d*e^5*x^4+360*A*a*b^4*d^2*e^4*x^2*ln(e*x+d)-5*A*a^4*b*d*e^5+480*B*a^2*b^3*d*e^5*x^3-240*B*a*b^4*d^2*e^4*x^
3-40*A*a^3*b^2*d*e^5*x-120*A*a^2*b^3*d^2*e^4*x+440*A*a*b^4*d^3*e^3*x-180*B*a^3*b^2*d*e^5*x^2-48*A*b^5*d^2*e^4*
x^3-120*B*a^3*b^2*e^6*x^3-96*B*b^5*d^3*e^3*x^3-60*A*a^3*b^2*e^6*x^2-252*A*b^5*d^3*e^3*x^2-30*B*a^4*b*e^6*x^2+3
96*B*b^5*d^4*e^2*x^2+60*B*a*b^4*e^6*x^5+6*B*b^5*e^6*x^6+12*A*b^5*e^6*x^5-4*B*a^5*e^6*x+180*B*b^5*d^6*ln(e*x+d)
+240*A*a*b^4*d*e^5*x^3+240*A*a*b^4*d*e^5*x^3*ln(e*x+d)+480*B*a^2*b^3*d*e^5*x^3*ln(e*x+d)-1200*B*a*b^4*d^2*e^4*
x^3*ln(e*x+d)-360*A*b^5*d^3*e^3*x^2*ln(e*x+d)+1080*B*b^5*d^4*e^2*x^2*ln(e*x+d)-240*A*b^5*d^2*e^4*x^3*ln(e*x+d)
-1260*B*a*b^4*d^3*e^3*x^2+1080*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)-1800*B*a*b^4*d^3*e^3*
x^2*ln(e*x+d)+504*B*b^5*d^5*e*x-36*B*b^5*d*e^5*x^5+48*A*b^5*d*e^5*x^4-204*B*b^5*d^2*e^4*x^4)/(b*x+a)^5/e^7/(e*
x+d)^4

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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)^5,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 )}^5} \,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)^5,x)

[Out]

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

[Out]

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

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