3.3.97 \(\int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=213 \[ \frac {a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}-\frac {a^4 (5 b c-2 a d)}{b^3 (a+b x) (b c-a d)^4}+\frac {c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}-\frac {a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}-\frac {c^5}{2 d^3 (c+d x)^2 (b c-a d)^3}+\frac {c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4} \]

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Rubi [A]  time = 0.27, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \begin {gather*} -\frac {a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac {c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}+\frac {a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}-\frac {a^4 (5 b c-2 a d)}{b^3 (a+b x) (b c-a d)^4}+\frac {c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4}-\frac {c^5}{2 d^3 (c+d x)^2 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)^3*(c + d*x)^3),x]

[Out]

a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) - (a^4*(5*b*c - 2*a*d))/(b^3*(b*c - a*d)^4*(a + b*x)) - c^5/(2*d^3*(b*c
- a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/(d^3*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d + a
^2*d^2)*Log[a + b*x])/(b^3*(b*c - a*d)^5) + (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^3*(b*c -
a*d)^5)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx &=\int \left (-\frac {a^5}{b^2 (b c-a d)^3 (a+b x)^3}-\frac {a^4 (-5 b c+2 a d)}{b^2 (b c-a d)^4 (a+b x)^2}-\frac {a^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right )}{b^2 (b c-a d)^5 (a+b x)}-\frac {c^5}{d^2 (-b c+a d)^3 (c+d x)^3}-\frac {c^4 (2 b c-5 a d)}{d^2 (-b c+a d)^4 (c+d x)^2}-\frac {c^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right )}{d^2 (-b c+a d)^5 (c+d x)}\right ) \, dx\\ &=\frac {a^5}{2 b^3 (b c-a d)^3 (a+b x)^2}-\frac {a^4 (5 b c-2 a d)}{b^3 (b c-a d)^4 (a+b x)}-\frac {c^5}{2 d^3 (b c-a d)^3 (c+d x)^2}+\frac {c^4 (2 b c-5 a d)}{d^3 (b c-a d)^4 (c+d x)}-\frac {a^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac {c^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^5}\\ \end {align*}

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Mathematica [A]  time = 0.44, size = 213, normalized size = 1.00 \begin {gather*} \frac {a^5}{2 b^3 (a+b x)^2 (b c-a d)^3}+\frac {a^4 (2 a d-5 b c)}{b^3 (a+b x) (b c-a d)^4}-\frac {c^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (a d-b c)^5}-\frac {a^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (a+b x)}{b^3 (b c-a d)^5}+\frac {c^5}{2 d^3 (c+d x)^2 (a d-b c)^3}+\frac {c^4 (2 b c-5 a d)}{d^3 (c+d x) (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x)^3*(c + d*x)^3),x]

[Out]

a^5/(2*b^3*(b*c - a*d)^3*(a + b*x)^2) + (a^4*(-5*b*c + 2*a*d))/(b^3*(b*c - a*d)^4*(a + b*x)) + c^5/(2*d^3*(-(b
*c) + a*d)^3*(c + d*x)^2) + (c^4*(2*b*c - 5*a*d))/(d^3*(b*c - a*d)^4*(c + d*x)) - (a^3*(10*b^2*c^2 - 5*a*b*c*d
+ a^2*d^2)*Log[a + b*x])/(b^3*(b*c - a*d)^5) - (c^3*(b^2*c^2 - 5*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^3*(-(
b*c) + a*d)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{(a+b x)^3 (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5/((a + b*x)^3*(c + d*x)^3),x]

[Out]

IntegrateAlgebraic[x^5/((a + b*x)^3*(c + d*x)^3), x]

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fricas [B]  time = 1.06, size = 1267, normalized size = 5.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(3*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 9*a^4*b^3*c^5*d^2 - 9*a^5*b^2*c^4*d^3 + 12*a^6*b*c^3*d^4 - 3*a^7*c^2*d
^5 + 2*(2*b^7*c^6*d - 7*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 7*a^5*b^2*c*d^6 - 2*a^6*b*d^7)
*x^3 + (3*b^7*c^7 - 4*a*b^6*c^6*d - 19*a^2*b^5*c^5*d^2 + 20*a^3*b^4*c^4*d^3 - 20*a^4*b^3*c^3*d^4 + 19*a^5*b^2*
c^2*d^5 + 4*a^6*b*c*d^6 - 3*a^7*d^7)*x^2 + 2*(3*a*b^6*c^7 - 10*a^2*b^5*c^6*d + 2*a^3*b^4*c^5*d^2 - 2*a^5*b^2*c
^3*d^4 + 10*a^6*b*c^2*d^5 - 3*a^7*c*d^6)*x - 2*(10*a^5*b^2*c^4*d^3 - 5*a^6*b*c^3*d^4 + a^7*c^2*d^5 + (10*a^3*b
^4*c^2*d^5 - 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^4 + 2*(10*a^3*b^4*c^3*d^4 + 5*a^4*b^3*c^2*d^5 - 4*a^5*b^2*c*d^6
+ a^6*b*d^7)*x^3 + (10*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 - a^6*b*c*d^6 + a^7*d^7)*x^2 +
2*(10*a^4*b^3*c^4*d^3 + 5*a^5*b^2*c^3*d^4 - 4*a^6*b*c^2*d^5 + a^7*c*d^6)*x)*log(b*x + a) + 2*(a^2*b^5*c^7 - 5
*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4)*x^4 + 2*(b^7*c^6*d
- 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 +
35*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 + 10*a^4*b^
3*c^4*d^3)*x)*log(d*x + c))/(a^2*b^8*c^7*d^3 - 5*a^3*b^7*c^6*d^4 + 10*a^4*b^6*c^5*d^5 - 10*a^5*b^5*c^4*d^6 + 5
*a^6*b^4*c^3*d^7 - a^7*b^3*c^2*d^8 + (b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8
+ 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^4 + 2*(b^10*c^6*d^4 - 4*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 - 5*a^4*b^6*c^2
*d^8 + 4*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^3 + (b^10*c^7*d^3 - a*b^9*c^6*d^4 - 9*a^2*b^8*c^5*d^5 + 25*a^3*b^7*c^
4*d^6 - 25*a^4*b^6*c^3*d^7 + 9*a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^2 + 2*(a*b^9*c^7*d^3 - 4*a^2*
b^8*c^6*d^4 + 5*a^3*b^7*c^5*d^5 - 5*a^5*b^5*c^3*d^7 + 4*a^6*b^4*c^2*d^8 - a^7*b^3*c*d^9)*x)

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giac [B]  time = 0.97, size = 484, normalized size = 2.27 \begin {gather*} -\frac {{\left (10 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8} c^{5} - 5 \, a b^{7} c^{4} d + 10 \, a^{2} b^{6} c^{3} d^{2} - 10 \, a^{3} b^{5} c^{2} d^{3} + 5 \, a^{4} b^{4} c d^{4} - a^{5} b^{3} d^{5}} + \frac {{\left (b^{2} c^{5} - 5 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{3} - 5 \, a b^{4} c^{4} d^{4} + 10 \, a^{2} b^{3} c^{3} d^{5} - 10 \, a^{3} b^{2} c^{2} d^{6} + 5 \, a^{4} b c d^{7} - a^{5} d^{8}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (2 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{4} b^{2} c d^{5} + 2 \, a^{5} b d^{6}\right )} x^{3} + {\left (3 \, b^{6} c^{6} - a b^{5} c^{5} d - 20 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b c d^{5} + 3 \, a^{6} d^{6}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} - 5 \, a^{4} b^{2} c^{3} d^{3} - 7 \, a^{5} b c^{2} d^{4} + 3 \, a^{6} c d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

-(10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*log(abs(b*x + a))/(b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10
*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5) + (b^2*c^5 - 5*a*b*c^4*d + 10*a^2*c^3*d^2)*log(abs(d*x + c))
/(b^5*c^5*d^3 - 5*a*b^4*c^4*d^4 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) + 1/2*(3*
a^2*b^4*c^6 - 9*a^3*b^3*c^5*d - 9*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + 2*(2*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^4*b^2
*c*d^5 + 2*a^5*b*d^6)*x^3 + (3*b^6*c^6 - a*b^5*c^5*d - 20*a^2*b^4*c^4*d^2 - 20*a^4*b^2*c^2*d^4 - a^5*b*c*d^5 +
3*a^6*d^6)*x^2 + 2*(3*a*b^5*c^6 - 7*a^2*b^4*c^5*d - 5*a^3*b^3*c^4*d^2 - 5*a^4*b^2*c^3*d^3 - 7*a^5*b*c^2*d^4 +
3*a^6*c*d^5)*x)/((b*c - a*d)^4*(b*x + a)^2*(d*x + c)^2*b^3*d^3)

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maple [A]  time = 0.01, size = 315, normalized size = 1.48 \begin {gather*} \frac {a^{5} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{3}}-\frac {5 a^{4} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{2}}+\frac {10 a^{3} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b}-\frac {10 a^{2} c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d}+\frac {5 a b \,c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{2}}-\frac {b^{2} c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{3}}+\frac {2 a^{5} d}{\left (a d -b c \right )^{4} \left (b x +a \right ) b^{3}}-\frac {5 a^{4} c}{\left (a d -b c \right )^{4} \left (b x +a \right ) b^{2}}-\frac {5 a \,c^{4}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d^{2}}+\frac {2 b \,c^{5}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d^{3}}-\frac {a^{5}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} b^{3}}+\frac {c^{5}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^3/(d*x+c)^3,x)

[Out]

-5*c^4/d^2/(a*d-b*c)^4/(d*x+c)*a+2*c^5/d^3/(a*d-b*c)^4/(d*x+c)*b+1/2*c^5/d^3/(a*d-b*c)^3/(d*x+c)^2-10*c^3/(a*d
-b*c)^5/d*ln(d*x+c)*a^2+5*c^4/(a*d-b*c)^5/d^2*ln(d*x+c)*a*b-c^5/(a*d-b*c)^5/d^3*ln(d*x+c)*b^2-1/2*a^5/b^3/(a*d
-b*c)^3/(b*x+a)^2+a^5/(a*d-b*c)^5/b^3*ln(b*x+a)*d^2-5*a^4/(a*d-b*c)^5/b^2*ln(b*x+a)*c*d+10*a^3/(a*d-b*c)^5/b*l
n(b*x+a)*c^2+2*a^5/b^3/(a*d-b*c)^4/(b*x+a)*d-5*a^4/b^2/(a*d-b*c)^4/(b*x+a)*c

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maxima [B]  time = 1.21, size = 813, normalized size = 3.82 \begin {gather*} -\frac {{\left (10 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{8} c^{5} - 5 \, a b^{7} c^{4} d + 10 \, a^{2} b^{6} c^{3} d^{2} - 10 \, a^{3} b^{5} c^{2} d^{3} + 5 \, a^{4} b^{4} c d^{4} - a^{5} b^{3} d^{5}} + \frac {{\left (b^{2} c^{5} - 5 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{3} - 5 \, a b^{4} c^{4} d^{4} + 10 \, a^{2} b^{3} c^{3} d^{5} - 10 \, a^{3} b^{2} c^{2} d^{6} + 5 \, a^{4} b c d^{7} - a^{5} d^{8}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (2 \, b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} - 5 \, a^{4} b^{2} c d^{5} + 2 \, a^{5} b d^{6}\right )} x^{3} + {\left (3 \, b^{6} c^{6} - a b^{5} c^{5} d - 20 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{4} b^{2} c^{2} d^{4} - a^{5} b c d^{5} + 3 \, a^{6} d^{6}\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c^{6} - 7 \, a^{2} b^{4} c^{5} d - 5 \, a^{3} b^{3} c^{4} d^{2} - 5 \, a^{4} b^{2} c^{3} d^{3} - 7 \, a^{5} b c^{2} d^{4} + 3 \, a^{6} c d^{5}\right )} x}{2 \, {\left (a^{2} b^{7} c^{6} d^{3} - 4 \, a^{3} b^{6} c^{5} d^{4} + 6 \, a^{4} b^{5} c^{4} d^{5} - 4 \, a^{5} b^{4} c^{3} d^{6} + a^{6} b^{3} c^{2} d^{7} + {\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )} x^{4} + 2 \, {\left (b^{9} c^{5} d^{4} - 3 \, a b^{8} c^{4} d^{5} + 2 \, a^{2} b^{7} c^{3} d^{6} + 2 \, a^{3} b^{6} c^{2} d^{7} - 3 \, a^{4} b^{5} c d^{8} + a^{5} b^{4} d^{9}\right )} x^{3} + {\left (b^{9} c^{6} d^{3} - 9 \, a^{2} b^{7} c^{4} d^{5} + 16 \, a^{3} b^{6} c^{3} d^{6} - 9 \, a^{4} b^{5} c^{2} d^{7} + a^{6} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (a b^{8} c^{6} d^{3} - 3 \, a^{2} b^{7} c^{5} d^{4} + 2 \, a^{3} b^{6} c^{4} d^{5} + 2 \, a^{4} b^{5} c^{3} d^{6} - 3 \, a^{5} b^{4} c^{2} d^{7} + a^{6} b^{3} c d^{8}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

-(10*a^3*b^2*c^2 - 5*a^4*b*c*d + a^5*d^2)*log(b*x + a)/(b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*
b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5) + (b^2*c^5 - 5*a*b*c^4*d + 10*a^2*c^3*d^2)*log(d*x + c)/(b^5*c^5*
d^3 - 5*a*b^4*c^4*d^4 + 10*a^2*b^3*c^3*d^5 - 10*a^3*b^2*c^2*d^6 + 5*a^4*b*c*d^7 - a^5*d^8) + 1/2*(3*a^2*b^4*c^
6 - 9*a^3*b^3*c^5*d - 9*a^5*b*c^3*d^3 + 3*a^6*c^2*d^4 + 2*(2*b^6*c^5*d - 5*a*b^5*c^4*d^2 - 5*a^4*b^2*c*d^5 + 2
*a^5*b*d^6)*x^3 + (3*b^6*c^6 - a*b^5*c^5*d - 20*a^2*b^4*c^4*d^2 - 20*a^4*b^2*c^2*d^4 - a^5*b*c*d^5 + 3*a^6*d^6
)*x^2 + 2*(3*a*b^5*c^6 - 7*a^2*b^4*c^5*d - 5*a^3*b^3*c^4*d^2 - 5*a^4*b^2*c^3*d^3 - 7*a^5*b*c^2*d^4 + 3*a^6*c*d
^5)*x)/(a^2*b^7*c^6*d^3 - 4*a^3*b^6*c^5*d^4 + 6*a^4*b^5*c^4*d^5 - 4*a^5*b^4*c^3*d^6 + a^6*b^3*c^2*d^7 + (b^9*c
^4*d^5 - 4*a*b^8*c^3*d^6 + 6*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^4 + 2*(b^9*c^5*d^4 - 3*a*b^8*c
^4*d^5 + 2*a^2*b^7*c^3*d^6 + 2*a^3*b^6*c^2*d^7 - 3*a^4*b^5*c*d^8 + a^5*b^4*d^9)*x^3 + (b^9*c^6*d^3 - 9*a^2*b^7
*c^4*d^5 + 16*a^3*b^6*c^3*d^6 - 9*a^4*b^5*c^2*d^7 + a^6*b^3*d^9)*x^2 + 2*(a*b^8*c^6*d^3 - 3*a^2*b^7*c^5*d^4 +
2*a^3*b^6*c^4*d^5 + 2*a^4*b^5*c^3*d^6 - 3*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8)*x)

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mupad [B]  time = 1.37, size = 759, normalized size = 3.56 \begin {gather*} \frac {\frac {3\,a^2\,c^2\,\left (a^4\,d^4-3\,a^3\,b\,c\,d^3-3\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^2\,\left (-3\,a^6\,d^6+a^5\,b\,c\,d^5+20\,a^4\,b^2\,c^2\,d^4+20\,a^2\,b^4\,c^4\,d^2+a\,b^5\,c^5\,d-3\,b^6\,c^6\right )}{2\,b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {x^3\,\left (a\,d+b\,c\right )\,\left (2\,a^4\,d^4-7\,a^3\,b\,c\,d^3+7\,a^2\,b^2\,c^2\,d^2-7\,a\,b^3\,c^3\,d+2\,b^4\,c^4\right )}{b^2\,d^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a\,c\,x\,\left (a\,d+b\,c\right )\,\left (3\,a^4\,d^4-10\,a^3\,b\,c\,d^3+5\,a^2\,b^2\,c^2\,d^2-10\,a\,b^3\,c^3\,d+3\,b^4\,c^4\right )}{b^3\,d^3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^2-5\,a^4\,b\,c\,d+10\,a^3\,b^2\,c^2\right )}{-a^5\,b^3\,d^5+5\,a^4\,b^4\,c\,d^4-10\,a^3\,b^5\,c^2\,d^3+10\,a^2\,b^6\,c^3\,d^2-5\,a\,b^7\,c^4\,d+b^8\,c^5}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-5\,a\,b\,c^4\,d+b^2\,c^5\right )}{a^5\,d^8-5\,a^4\,b\,c\,d^7+10\,a^3\,b^2\,c^2\,d^6-10\,a^2\,b^3\,c^3\,d^5+5\,a\,b^4\,c^4\,d^4-b^5\,c^5\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((a + b*x)^3*(c + d*x)^3),x)

[Out]

((3*a^2*c^2*(a^4*d^4 + b^4*c^4 - 3*a*b^3*c^3*d - 3*a^3*b*c*d^3))/(2*b^3*d^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2
*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (x^2*(20*a^2*b^4*c^4*d^2 - 3*b^6*c^6 - 3*a^6*d^6 + 20*a^4*b^2*c^2*d^4
+ a*b^5*c^5*d + a^5*b*c*d^5))/(2*b^3*d^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d
^3)) + (x^3*(a*d + b*c)*(2*a^4*d^4 + 2*b^4*c^4 + 7*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 7*a^3*b*c*d^3))/(b^2*d^2*
(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (a*c*x*(a*d + b*c)*(3*a^4*d^4 + 3*b
^4*c^4 + 5*a^2*b^2*c^2*d^2 - 10*a*b^3*c^3*d - 10*a^3*b*c*d^3))/(b^3*d^3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2
- 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)))/(x*(2*a*b*c^2 + 2*a^2*c*d) + x^2*(a^2*d^2 + b^2*c^2 + 4*a*b*c*d) + x^3*(2*
a*b*d^2 + 2*b^2*c*d) + a^2*c^2 + b^2*d^2*x^4) - (log(a + b*x)*(a^5*d^2 + 10*a^3*b^2*c^2 - 5*a^4*b*c*d))/(b^8*c
^5 - a^5*b^3*d^5 + 5*a^4*b^4*c*d^4 + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 - 5*a*b^7*c^4*d) - (log(c + d*x)*
(b^2*c^5 + 10*a^2*c^3*d^2 - 5*a*b*c^4*d))/(a^5*d^8 - b^5*c^5*d^3 + 5*a*b^4*c^4*d^4 - 10*a^2*b^3*c^3*d^5 + 10*a
^3*b^2*c^2*d^6 - 5*a^4*b*c*d^7)

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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(x**5/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

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