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

Optimal. Leaf size=155 \[ \frac {a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac {3 a^2 c}{(a+b x) (b c-a d)^4}-\frac {c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac {3 a c^2}{(c+d x) (b c-a d)^4}-\frac {3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 155, 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} \[ \frac {a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac {3 a^2 c}{(a+b x) (b c-a d)^4}-\frac {3 a c^2}{(c+d x) (b c-a d)^4}-\frac {c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac {3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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Mathematica [A]  time = 0.28, size = 153, normalized size = 0.99 \[ \frac {1}{2} \left (\frac {a^3}{b (a+b x)^2 (b c-a d)^3}-\frac {6 a^2 c}{(a+b x) (b c-a d)^4}+\frac {c^3}{d (c+d x)^2 (a d-b c)^3}-\frac {6 a c^2}{(c+d x) (b c-a d)^4}-\frac {6 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {6 a c (a d+b c) \log (c+d x)}{(b c-a d)^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3/(b*(b*c - a*d)^3*(a + b*x)^2) - (6*a^2*c)/((b*c - a*d)^4*(a + b*x)) + c^3/(d*(-(b*c) + a*d)^3*(c + d*x)^2
) - (6*a*c^2)/((b*c - a*d)^4*(c + d*x)) - (6*a*c*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (6*a*c*(b*c + a*d)*
Log[c + d*x])/(b*c - a*d)^5)/2

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fricas [B]  time = 0.97, size = 991, normalized size = 6.39 \[ -\frac {a^{2} b^{3} c^{5} + 9 \, a^{3} b^{2} c^{4} d - 9 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + 6 \, {\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x^{3} + {\left (b^{5} c^{5} + 4 \, a b^{4} c^{4} d + 19 \, a^{2} b^{3} c^{3} d^{2} - 19 \, a^{3} b^{2} c^{2} d^{3} - 4 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x^{2} + 2 \, {\left (a b^{4} c^{5} + 7 \, a^{2} b^{3} c^{4} d - 7 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x + 6 \, {\left (a^{3} b^{2} c^{4} d + a^{4} b c^{3} d^{2} + {\left (a b^{4} c^{2} d^{3} + a^{2} b^{3} c d^{4}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} d^{2} + 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x^{3} + {\left (a b^{4} c^{4} d + 5 \, a^{2} b^{3} c^{3} d^{2} + 5 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4}\right )} x^{2} + 2 \, {\left (a^{2} b^{3} c^{4} d + 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (a^{3} b^{2} c^{4} d + a^{4} b c^{3} d^{2} + {\left (a b^{4} c^{2} d^{3} + a^{2} b^{3} c d^{4}\right )} x^{4} + 2 \, {\left (a b^{4} c^{3} d^{2} + 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x^{3} + {\left (a b^{4} c^{4} d + 5 \, a^{2} b^{3} c^{3} d^{2} + 5 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4}\right )} x^{2} + 2 \, {\left (a^{2} b^{3} c^{4} d + 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{6} c^{7} d - 5 \, a^{3} b^{5} c^{6} d^{2} + 10 \, a^{4} b^{4} c^{5} d^{3} - 10 \, a^{5} b^{3} c^{4} d^{4} + 5 \, a^{6} b^{2} c^{3} d^{5} - a^{7} b c^{2} d^{6} + {\left (b^{8} c^{5} d^{3} - 5 \, a b^{7} c^{4} d^{4} + 10 \, a^{2} b^{6} c^{3} d^{5} - 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} - a^{5} b^{3} d^{8}\right )} x^{4} + 2 \, {\left (b^{8} c^{6} d^{2} - 4 \, a b^{7} c^{5} d^{3} + 5 \, a^{2} b^{6} c^{4} d^{4} - 5 \, a^{4} b^{4} c^{2} d^{6} + 4 \, a^{5} b^{3} c d^{7} - a^{6} b^{2} d^{8}\right )} x^{3} + {\left (b^{8} c^{7} d - a b^{7} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{5} d^{3} + 25 \, a^{3} b^{5} c^{4} d^{4} - 25 \, a^{4} b^{4} c^{3} d^{5} + 9 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7} - a^{7} b d^{8}\right )} x^{2} + 2 \, {\left (a b^{7} c^{7} d - 4 \, a^{2} b^{6} c^{6} d^{2} + 5 \, a^{3} b^{5} c^{5} d^{3} - 5 \, a^{5} b^{3} c^{3} d^{5} + 4 \, a^{6} b^{2} c^{2} d^{6} - a^{7} b c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.23, size = 438, normalized size = 2.83 \[ -\frac {3 \, {\left (a b^{2} c^{2} + a^{2} b c d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} + \frac {3 \, {\left (a b c^{2} d + a^{2} c d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {6 \, a b^{3} c^{2} d^{2} x^{3} + 6 \, a^{2} b^{2} c d^{3} x^{3} + b^{4} c^{4} x^{2} + 5 \, a b^{3} c^{3} d x^{2} + 24 \, a^{2} b^{2} c^{2} d^{2} x^{2} + 5 \, a^{3} b c d^{3} x^{2} + a^{4} d^{4} x^{2} + 2 \, a b^{3} c^{4} x + 16 \, a^{2} b^{2} c^{3} d x + 16 \, a^{3} b c^{2} d^{2} x + 2 \, a^{4} c d^{3} x + a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}}{2 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(a*b^2*c^2 + a^2*b*c*d)*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^
3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5) + 3*(a*b*c^2*d + a^2*c*d^2)*log(abs(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 +
10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5*a^4*b*c*d^5 - a^5*d^6) - 1/2*(6*a*b^3*c^2*d^2*x^3 + 6*a^2*b^2*c*d^
3*x^3 + b^4*c^4*x^2 + 5*a*b^3*c^3*d*x^2 + 24*a^2*b^2*c^2*d^2*x^2 + 5*a^3*b*c*d^3*x^2 + a^4*d^4*x^2 + 2*a*b^3*c
^4*x + 16*a^2*b^2*c^3*d*x + 16*a^3*b*c^2*d^2*x + 2*a^4*c*d^3*x + a^2*b^2*c^4 + 10*a^3*b*c^3*d + a^4*c^2*d^2)/(
(b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*(b*d*x^2 + b*c*x + a*d*x + a*c
)^2)

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maple [A]  time = 0.02, size = 190, normalized size = 1.23 \[ \frac {3 a^{2} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {3 a^{2} c d \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {3 a b \,c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {3 a b \,c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {3 a^{2} c}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {3 a \,c^{2}}{\left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {a^{3}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} b}+\frac {c^{3}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.07, size = 682, normalized size = 4.40 \[ -\frac {3 \, {\left (a b c^{2} + a^{2} c d\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {3 \, {\left (a b c^{2} + a^{2} c d\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + 6 \, {\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x^{3} + {\left (b^{4} c^{4} + 5 \, a b^{3} c^{3} d + 24 \, a^{2} b^{2} c^{2} d^{2} + 5 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 8 \, a^{2} b^{2} c^{3} d + 8 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \, {\left (a^{2} b^{5} c^{6} d - 4 \, a^{3} b^{4} c^{5} d^{2} + 6 \, a^{4} b^{3} c^{4} d^{3} - 4 \, a^{5} b^{2} c^{3} d^{4} + a^{6} b c^{2} d^{5} + {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{5} d^{2} - 3 \, a b^{6} c^{4} d^{3} + 2 \, a^{2} b^{5} c^{3} d^{4} + 2 \, a^{3} b^{4} c^{2} d^{5} - 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{3} + {\left (b^{7} c^{6} d - 9 \, a^{2} b^{5} c^{4} d^{3} + 16 \, a^{3} b^{4} c^{3} d^{4} - 9 \, a^{4} b^{3} c^{2} d^{5} + a^{6} b d^{7}\right )} x^{2} + 2 \, {\left (a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2} + 2 \, a^{3} b^{4} c^{4} d^{3} + 2 \, a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(a*b*c^2 + a^2*c*d)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4
*b*c*d^4 - a^5*d^5) + 3*(a*b*c^2 + a^2*c*d)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^
3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 1/2*(a^2*b^2*c^4 + 10*a^3*b*c^3*d + a^4*c^2*d^2 + 6*(a*b^3*c^2*d^2
+ a^2*b^2*c*d^3)*x^3 + (b^4*c^4 + 5*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3
*c^4 + 8*a^2*b^2*c^3*d + 8*a^3*b*c^2*d^2 + a^4*c*d^3)*x)/(a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^
3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5 + (b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 +
a^4*b^3*d^7)*x^4 + 2*(b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6
+ a^5*b^2*d^7)*x^3 + (b^7*c^6*d - 9*a^2*b^5*c^4*d^3 + 16*a^3*b^4*c^3*d^4 - 9*a^4*b^3*c^2*d^5 + a^6*b*d^7)*x^2
+ 2*(a*b^6*c^6*d - 3*a^2*b^5*c^5*d^2 + 2*a^3*b^4*c^4*d^3 + 2*a^4*b^3*c^3*d^4 - 3*a^5*b^2*c^2*d^5 + a^6*b*c*d^
6)*x)

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mupad [B]  time = 0.67, size = 532, normalized size = 3.43 \[ \frac {6\,a\,c\,\mathrm {atanh}\left (\frac {\left (a\,d+b\,c+2\,b\,d\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^5}-\frac {\frac {x^2\,\left (a^4\,d^4+5\,a^3\,b\,c\,d^3+24\,a^2\,b^2\,c^2\,d^2+5\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {a^2\,c^2\,\left (a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )}{2\,b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {3\,a\,b\,c\,d\,x^3\,\left (a\,d+b\,c\right )}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}+\frac {a\,c\,x\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\right )}{b\,d\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*x)^3*(c + d*x)^3),x)

[Out]

(6*a*c*atanh(((a*d + b*c + 2*b*d*x)*(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*d - b*c)^5)*(a*d + b*c))/(a*d - b*c)^5 - ((x^2*(a^4*d^4 + b^4*c^4 + 24*a^2*b^2*c^2*d^2 + 5*a*b^3*c^3*d + 5*a
^3*b*c*d^3))/(2*b*d*(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^2*c^2*(a^2*d
^2 + b^2*c^2 + 10*a*b*c*d))/(2*b*d*(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)) +
(3*a*b*c*d*x^3*(a*d + b*c))/(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)*(a^2*d^2 + b^2*c^2 + 7*a*b*c*d))/(b*d*(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)

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sympy [B]  time = 3.48, size = 1117, normalized size = 7.21 \[ - \frac {3 a c \left (a d + b c\right ) \log {\left (x + \frac {- \frac {3 a^{7} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {18 a^{6} b c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{5} b^{2} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {60 a^{4} b^{3} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{3} b^{4} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{3} c d^{2} + \frac {18 a^{2} b^{5} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{2} d - \frac {3 a b^{6} c^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a b^{2} c^{3}}{6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d} \right )}}{\left (a d - b c\right )^{5}} + \frac {3 a c \left (a d + b c\right ) \log {\left (x + \frac {\frac {3 a^{7} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {18 a^{6} b c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{5} b^{2} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {60 a^{4} b^{3} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{3} b^{4} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{3} c d^{2} - \frac {18 a^{2} b^{5} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a^{2} b c^{2} d + \frac {3 a b^{6} c^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a b^{2} c^{3}}{6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{4} c^{2} d^{2} - 10 a^{3} b c^{3} d - a^{2} b^{2} c^{4} + x^{3} \left (- 6 a^{2} b^{2} c d^{3} - 6 a b^{3} c^{2} d^{2}\right ) + x^{2} \left (- a^{4} d^{4} - 5 a^{3} b c d^{3} - 24 a^{2} b^{2} c^{2} d^{2} - 5 a b^{3} c^{3} d - b^{4} c^{4}\right ) + x \left (- 2 a^{4} c d^{3} - 16 a^{3} b c^{2} d^{2} - 16 a^{2} b^{2} c^{3} d - 2 a b^{3} c^{4}\right )}{2 a^{6} b c^{2} d^{5} - 8 a^{5} b^{2} c^{3} d^{4} + 12 a^{4} b^{3} c^{4} d^{3} - 8 a^{3} b^{4} c^{5} d^{2} + 2 a^{2} b^{5} c^{6} d + x^{4} \left (2 a^{4} b^{3} d^{7} - 8 a^{3} b^{4} c d^{6} + 12 a^{2} b^{5} c^{2} d^{5} - 8 a b^{6} c^{3} d^{4} + 2 b^{7} c^{4} d^{3}\right ) + x^{3} \left (4 a^{5} b^{2} d^{7} - 12 a^{4} b^{3} c d^{6} + 8 a^{3} b^{4} c^{2} d^{5} + 8 a^{2} b^{5} c^{3} d^{4} - 12 a b^{6} c^{4} d^{3} + 4 b^{7} c^{5} d^{2}\right ) + x^{2} \left (2 a^{6} b d^{7} - 18 a^{4} b^{3} c^{2} d^{5} + 32 a^{3} b^{4} c^{3} d^{4} - 18 a^{2} b^{5} c^{4} d^{3} + 2 b^{7} c^{6} d\right ) + x \left (4 a^{6} b c d^{6} - 12 a^{5} b^{2} c^{2} d^{5} + 8 a^{4} b^{3} c^{3} d^{4} + 8 a^{3} b^{4} c^{4} d^{3} - 12 a^{2} b^{5} c^{5} d^{2} + 4 a b^{6} c^{6} d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a*c*(a*d + b*c)*log(x + (-3*a**7*c*d**6*(a*d + b*c)/(a*d - b*c)**5 + 18*a**6*b*c**2*d**5*(a*d + b*c)/(a*d -
b*c)**5 - 45*a**5*b**2*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 + 60*a**4*b**3*c**4*d**3*(a*d + b*c)/(a*d - b*c)*
*5 - 45*a**3*b**4*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 3*a**3*c*d**2 + 18*a**2*b**5*c**6*d*(a*d + b*c)/(a*d
- b*c)**5 + 6*a**2*b*c**2*d - 3*a*b**6*c**7*(a*d + b*c)/(a*d - b*c)**5 + 3*a*b**2*c**3)/(6*a**2*b*c*d**2 + 6*a
*b**2*c**2*d))/(a*d - b*c)**5 + 3*a*c*(a*d + b*c)*log(x + (3*a**7*c*d**6*(a*d + b*c)/(a*d - b*c)**5 - 18*a**6*
b*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 + 45*a**5*b**2*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 - 60*a**4*b**3*c**4
*d**3*(a*d + b*c)/(a*d - b*c)**5 + 45*a**3*b**4*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 3*a**3*c*d**2 - 18*a**2
*b**5*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 6*a**2*b*c**2*d + 3*a*b**6*c**7*(a*d + b*c)/(a*d - b*c)**5 + 3*a*b**
2*c**3)/(6*a**2*b*c*d**2 + 6*a*b**2*c**2*d))/(a*d - b*c)**5 + (-a**4*c**2*d**2 - 10*a**3*b*c**3*d - a**2*b**2*
c**4 + x**3*(-6*a**2*b**2*c*d**3 - 6*a*b**3*c**2*d**2) + x**2*(-a**4*d**4 - 5*a**3*b*c*d**3 - 24*a**2*b**2*c**
2*d**2 - 5*a*b**3*c**3*d - b**4*c**4) + x*(-2*a**4*c*d**3 - 16*a**3*b*c**2*d**2 - 16*a**2*b**2*c**3*d - 2*a*b*
*3*c**4))/(2*a**6*b*c**2*d**5 - 8*a**5*b**2*c**3*d**4 + 12*a**4*b**3*c**4*d**3 - 8*a**3*b**4*c**5*d**2 + 2*a**
2*b**5*c**6*d + x**4*(2*a**4*b**3*d**7 - 8*a**3*b**4*c*d**6 + 12*a**2*b**5*c**2*d**5 - 8*a*b**6*c**3*d**4 + 2*
b**7*c**4*d**3) + x**3*(4*a**5*b**2*d**7 - 12*a**4*b**3*c*d**6 + 8*a**3*b**4*c**2*d**5 + 8*a**2*b**5*c**3*d**4
- 12*a*b**6*c**4*d**3 + 4*b**7*c**5*d**2) + x**2*(2*a**6*b*d**7 - 18*a**4*b**3*c**2*d**5 + 32*a**3*b**4*c**3*
d**4 - 18*a**2*b**5*c**4*d**3 + 2*b**7*c**6*d) + x*(4*a**6*b*c*d**6 - 12*a**5*b**2*c**2*d**5 + 8*a**4*b**3*c**
3*d**4 + 8*a**3*b**4*c**4*d**3 - 12*a**2*b**5*c**5*d**2 + 4*a*b**6*c**6*d))

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