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

Optimal. Leaf size=157 \[ -\frac {b (2 a d+b c)}{(a+b x) (b c-a d)^4}+\frac {a b}{2 (a+b x)^2 (b c-a d)^3}-\frac {d (a d+2 b c)}{(c+d x) (b c-a d)^4}-\frac {c d}{2 (c+d x)^2 (b c-a d)^3}-\frac {3 b d (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {3 b d (a d+b c) \log (c+d x)}{(b c-a d)^5} \]

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Rubi [A]  time = 0.15, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} -\frac {b (2 a d+b c)}{(a+b x) (b c-a d)^4}+\frac {a b}{2 (a+b x)^2 (b c-a d)^3}-\frac {d (a d+2 b c)}{(c+d x) (b c-a d)^4}-\frac {c d}{2 (c+d x)^2 (b c-a d)^3}-\frac {3 b d (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac {3 b d (a d+b c) \log (c+d x)}{(b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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]))))

Rubi steps

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

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Mathematica [A]  time = 0.21, size = 142, normalized size = 0.90 \begin {gather*} \frac {\frac {a b (b c-a d)^2}{(a+b x)^2}-\frac {c d (b c-a d)^2}{(c+d x)^2}-\frac {2 b (2 a d+b c) (b c-a d)}{a+b x}+\frac {2 d (a d-b c) (a d+2 b c)}{c+d x}-6 b d (a d+b c) \log (a+b x)+6 b d (a d+b c) \log (c+d x)}{2 (b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*b*(b*c - a*d)^2)/(a + b*x)^2 - (2*b*(b*c - a*d)*(b*c + 2*a*d))/(a + b*x) - (c*d*(b*c - a*d)^2)/(c + d*x)^2
+ (2*d*(-(b*c) + a*d)*(2*b*c + a*d))/(c + d*x) - 6*b*d*(b*c + a*d)*Log[a + b*x] + 6*b*d*(b*c + a*d)*Log[c + d
*x])/(2*(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}{(a+b x)^3 (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.57, size = 891, normalized size = 5.68 \begin {gather*} -\frac {a b^{3} c^{4} + 9 \, a^{2} b^{2} c^{3} d - 9 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 9 \, {\left (b^{4} c^{3} d + a b^{3} c^{2} d^{2} - a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} + 7 \, a b^{3} c^{3} d - 7 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x + 6 \, {\left (a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} + {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{3} + {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} d + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 6 \, {\left (a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} + {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{3} + {\left (b^{4} c^{3} d + 5 \, a b^{3} c^{2} d^{2} + 5 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{3} d + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} + {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, 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} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, 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}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.95, size = 388, normalized size = 2.47 \begin {gather*} -\frac {3 \, {\left (b^{3} c d + a b^{2} d^{2}\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 (b^{2} c d^{2} + a b d^{3}\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 \, b^{3} c d^{2} x^{3} + 6 \, a b^{2} d^{3} x^{3} + 9 \, b^{3} c^{2} d x^{2} + 18 \, a b^{2} c d^{2} x^{2} + 9 \, a^{2} b d^{3} x^{2} + 2 \, b^{3} c^{3} x + 16 \, a b^{2} c^{2} d x + 16 \, a^{2} b c d^{2} x + 2 \, a^{3} d^{3} x + a b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2}}{2 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (b d x^{2} + b c x + a d x + a c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(b^3*c*d + a*b^2*d^2)*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*(b^2*c*d^2 + a*b*d^3)*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*b^3*c*d^2*x^3 + 6*a*b^2*d^3*x^3 + 9*b^
3*c^2*d*x^2 + 18*a*b^2*c*d^2*x^2 + 9*a^2*b*d^3*x^2 + 2*b^3*c^3*x + 16*a*b^2*c^2*d*x + 16*a^2*b*c*d^2*x + 2*a^3
*d^3*x + a*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3
+ a^4*d^4)*(b*d*x^2 + b*c*x + a*d*x + a*c)^2)

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maple [A]  time = 0.01, size = 226, normalized size = 1.44 \begin {gather*} \frac {3 a b \,d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {3 a b \,d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {3 b^{2} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}-\frac {3 b^{2} c d \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {2 a b d}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {a \,d^{2}}{\left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {b^{2} c}{\left (a d -b c \right )^{4} \left (b x +a \right )}-\frac {2 b c d}{\left (a d -b c \right )^{4} \left (d x +c \right )}-\frac {a b}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {c d}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.00, size = 627, normalized size = 3.99 \begin {gather*} -\frac {3 \, {\left (b^{2} c d + a b d^{2}\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 (b^{2} c d + a b d^{2}\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 b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} + 8 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{2 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.66, size = 495, normalized size = 3.15 \begin {gather*} \frac {6\,b\,d\,\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 {a^3\,c\,d^2+10\,a^2\,b\,c^2\,d+a\,b^2\,c^3}{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 {9\,x^2\,\left (a^2\,b\,d^3+2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}{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 {x\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+7\,a\,b\,c\,d+b^2\,c^2\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 {3\,b^2\,d^2\,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}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((a + b*x)^3*(c + d*x)^3),x)

[Out]

(6*b*d*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 - ((a*b^2*c^3 + a^3*c*d^2 + 10*a^2*b*c^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)) + (9*x^2*(a^2*b*d^3 + b^3*c^2*d + 2*a*b^2*c*d^2))/(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)) + (x*(a*d + b*c)*(a^2*d^2 + b^2*c^2 + 7*a*b
*c*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*b^2*d^2*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))/(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 = 3.04, size = 1052, normalized size = 6.70 \begin {gather*} - \frac {3 b d \left (a d + b c\right ) \log {\left (x + \frac {- \frac {3 a^{6} b d^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {18 a^{5} b^{2} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{4} b^{3} c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {60 a^{3} b^{4} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {45 a^{2} b^{5} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{2} b d^{3} + \frac {18 a b^{6} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a b^{2} c d^{2} - \frac {3 b^{7} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 b^{3} c^{2} d}{6 a b^{2} d^{3} + 6 b^{3} c d^{2}} \right )}}{\left (a d - b c\right )^{5}} + \frac {3 b d \left (a d + b c\right ) \log {\left (x + \frac {\frac {3 a^{6} b d^{7} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {18 a^{5} b^{2} c d^{6} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{4} b^{3} c^{2} d^{5} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} - \frac {60 a^{3} b^{4} c^{3} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + \frac {45 a^{2} b^{5} c^{4} d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 a^{2} b d^{3} - \frac {18 a b^{6} c^{5} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 6 a b^{2} c d^{2} + \frac {3 b^{7} c^{6} d \left (a d + b c\right )}{\left (a d - b c\right )^{5}} + 3 b^{3} c^{2} d}{6 a b^{2} d^{3} + 6 b^{3} c d^{2}} \right )}}{\left (a d - b c\right )^{5}} + \frac {- a^{3} c d^{2} - 10 a^{2} b c^{2} d - a b^{2} c^{3} + x^{3} \left (- 6 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x^{2} \left (- 9 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right ) + x \left (- 2 a^{3} d^{3} - 16 a^{2} b c d^{2} - 16 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{2 a^{6} c^{2} d^{4} - 8 a^{5} b c^{3} d^{3} + 12 a^{4} b^{2} c^{4} d^{2} - 8 a^{3} b^{3} c^{5} d + 2 a^{2} b^{4} c^{6} + x^{4} \left (2 a^{4} b^{2} d^{6} - 8 a^{3} b^{3} c d^{5} + 12 a^{2} b^{4} c^{2} d^{4} - 8 a b^{5} c^{3} d^{3} + 2 b^{6} c^{4} d^{2}\right ) + x^{3} \left (4 a^{5} b d^{6} - 12 a^{4} b^{2} c d^{5} + 8 a^{3} b^{3} c^{2} d^{4} + 8 a^{2} b^{4} c^{3} d^{3} - 12 a b^{5} c^{4} d^{2} + 4 b^{6} c^{5} d\right ) + x^{2} \left (2 a^{6} d^{6} - 18 a^{4} b^{2} c^{2} d^{4} + 32 a^{3} b^{3} c^{3} d^{3} - 18 a^{2} b^{4} c^{4} d^{2} + 2 b^{6} c^{6}\right ) + x \left (4 a^{6} c d^{5} - 12 a^{5} b c^{2} d^{4} + 8 a^{4} b^{2} c^{3} d^{3} + 8 a^{3} b^{3} c^{4} d^{2} - 12 a^{2} b^{4} c^{5} d + 4 a b^{5} c^{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*d*(a*d + b*c)*log(x + (-3*a**6*b*d**7*(a*d + b*c)/(a*d - b*c)**5 + 18*a**5*b**2*c*d**6*(a*d + b*c)/(a*d -
b*c)**5 - 45*a**4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 + 60*a**3*b**4*c**3*d**4*(a*d + b*c)/(a*d - b*c)*
*5 - 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 + 18*a*b**6*c**5*d**2*(a*d + b*c)/(a*d
- b*c)**5 + 6*a*b**2*c*d**2 - 3*b**7*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2*d)/(6*a*b**2*d**3 + 6*b**
3*c*d**2))/(a*d - b*c)**5 + 3*b*d*(a*d + b*c)*log(x + (3*a**6*b*d**7*(a*d + b*c)/(a*d - b*c)**5 - 18*a**5*b**2
*c*d**6*(a*d + b*c)/(a*d - b*c)**5 + 45*a**4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 - 60*a**3*b**4*c**3*d**
4*(a*d + b*c)/(a*d - b*c)**5 + 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 - 18*a*b**6*c
**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 6*a*b**2*c*d**2 + 3*b**7*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2
*d)/(6*a*b**2*d**3 + 6*b**3*c*d**2))/(a*d - b*c)**5 + (-a**3*c*d**2 - 10*a**2*b*c**2*d - a*b**2*c**3 + x**3*(-
6*a*b**2*d**3 - 6*b**3*c*d**2) + x**2*(-9*a**2*b*d**3 - 18*a*b**2*c*d**2 - 9*b**3*c**2*d) + x*(-2*a**3*d**3 -
16*a**2*b*c*d**2 - 16*a*b**2*c**2*d - 2*b**3*c**3))/(2*a**6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4
*d**2 - 8*a**3*b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**
2*d**4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4*a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2
*d**4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*d**2 + 4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 18*a**4*b**2*c**2*d
**4 + 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 2*b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4
+ 8*a**4*b**2*c**3*d**3 + 8*a**3*b**3*c**4*d**2 - 12*a**2*b**4*c**5*d + 4*a*b**5*c**6))

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