3.3.100 \(\int \frac {x^2}{(a+b x)^3 (c+d x)^3} \, dx\)

Optimal. Leaf size=180 \[ \frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac {a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac {a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac {c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]

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Rubi [A]  time = 0.19, antiderivative size = 180, 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 {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac {\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac {a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac {a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac {c (2 a d+b c)}{(c+d x) (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^3*(c + d*x)^3),x]

[Out]

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

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Mathematica [A]  time = 0.24, size = 168, normalized size = 0.93 \begin {gather*} \frac {2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)-2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)-\frac {a^2 (b c-a d)^2}{(a+b x)^2}+\frac {c^2 (b c-a d)^2}{(c+d x)^2}+\frac {2 a (a d+2 b c) (b c-a d)}{a+b x}+\frac {2 c (2 a d+b c) (b c-a d)}{c+d x}}{2 (b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/((a + b*x)^3*(c + d*x)^3),x]

[Out]

(-((a^2*(b*c - a*d)^2)/(a + b*x)^2) + (2*a*(b*c - a*d)*(2*b*c + a*d))/(a + b*x) + (c^2*(b*c - a*d)^2)/(c + d*x
)^2 + (2*c*(b*c - a*d)*(b*c + 2*a*d))/(c + d*x) + 2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*Log[a + b*x] - 2*(b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*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^2}{(a+b x)^3 (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2/((a + b*x)^3*(c + d*x)^3),x]

[Out]

IntegrateAlgebraic[x^2/((a + b*x)^3*(c + d*x)^3), x]

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fricas [B]  time = 1.70, size = 990, normalized size = 5.50 \begin {gather*} \frac {6 \, a^{2} b^{2} c^{4} - 6 \, a^{4} c^{2} d^{2} + 2 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2} + 2 \, {\left (5 \, a b^{3} c^{4} + 3 \, a^{2} b^{2} c^{3} d - 3 \, a^{3} b c^{2} d^{2} - 5 \, a^{4} c d^{3}\right )} x + 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\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^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (a^{2} b^{2} c^{4} + 4 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{4} + 2 \, {\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^{3} + {\left (b^{4} c^{4} + 8 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 8 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{4} + 5 \, a^{2} b^{2} c^{3} d + 5 \, a^{3} b c^{2} d^{2} + a^{4} 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^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(6*a^2*b^2*c^4 - 6*a^4*c^2*d^2 + 2*(b^4*c^3*d + 3*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^3 + 3*(b^
4*c^4 + 4*a*b^3*c^3*d - 4*a^3*b*c*d^3 - a^4*d^4)*x^2 + 2*(5*a*b^3*c^4 + 3*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 - 5*
a^4*c*d^3)*x + 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4
+ 2*(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^3 + (b^4*c^4 + 8*a*b^3*c^3*d + 18*a^2*b^2*c^
2*d^2 + 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^4 + 5*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 + a^4*c*d^3)*x)*log(b*
x + a) - 2*(a^2*b^2*c^4 + 4*a^3*b*c^3*d + a^4*c^2*d^2 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^4 + 2*(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^3 + (b^4*c^4 + 8*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2
+ 8*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3*c^4 + 5*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 + a^4*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 = 0.78, size = 412, normalized size = 2.29 \begin {gather*} \frac {{\left (b^{3} c^{2} + 4 \, a b^{2} c d + a^{2} b 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 {{\left (b^{2} c^{2} d + 4 \, a b c d^{2} + a^{2} 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 {2 \, b^{3} c^{2} d x^{3} + 8 \, a b^{2} c d^{2} x^{3} + 2 \, a^{2} b d^{3} x^{3} + 3 \, b^{3} c^{3} x^{2} + 15 \, a b^{2} c^{2} d x^{2} + 15 \, a^{2} b c d^{2} x^{2} + 3 \, a^{3} d^{3} x^{2} + 10 \, a b^{2} c^{3} x + 16 \, a^{2} b c^{2} d x + 10 \, a^{3} c d^{2} x + 6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d}{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^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")

[Out]

(b^3*c^2 + 4*a*b^2*c*d + a^2*b*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) - (b^2*c^2*d + 4*a*b*c*d^2 + a^2*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*(2*b^3*c^2*d*x^3 +
8*a*b^2*c*d^2*x^3 + 2*a^2*b*d^3*x^3 + 3*b^3*c^3*x^2 + 15*a*b^2*c^2*d*x^2 + 15*a^2*b*c*d^2*x^2 + 3*a^3*d^3*x^2
+ 10*a*b^2*c^3*x + 16*a^2*b*c^2*d*x + 10*a^3*c*d^2*x + 6*a^2*b*c^3 + 6*a^3*c^2*d)/((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 = 0.02, size = 272, normalized size = 1.51 \begin {gather*} -\frac {a^{2} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {a^{2} d^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {4 a b c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {4 a b c d \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}-\frac {b^{2} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {b^{2} c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {a^{2} d}{\left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {2 a b c}{\left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {2 a c d}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {b \,c^{2}}{\left (a d -b c \right )^{4} \left (d x +c \right )}+\frac {a^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}-\frac {c^{2}}{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^2/(b*x+a)^3/(d*x+c)^3,x)

[Out]

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

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maxima [B]  time = 1.08, size = 646, normalized size = 3.59 \begin {gather*} \frac {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} 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 {{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} 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 {6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d + 2 \, {\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \, {\left (b^{3} c^{3} + 5 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c^{3} + 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\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^2/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

(b^2*c^2 + 4*a*b*c*d + a^2*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) - (b^2*c^2 + 4*a*b*c*d + a^2*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*(6*a^2*b*c^3 + 6*a^3*c^2*d + 2*(b^3*c^2*d +
4*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 3*(b^3*c^3 + 5*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3)*x^2 + 2*(5*a*b^2*c^3 +
8*a^2*b*c^2*d + 5*a^3*c*d^2)*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 = 0.68, size = 569, normalized size = 3.16 \begin {gather*} \frac {\frac {3\,\left (d\,a^3\,c^2+b\,a^2\,c^3\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\,x^2\,\left (a\,d+b\,c\right )\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\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 {b\,d\,x^3\,\left (a^2\,d^2+4\,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 {a\,c\,x\,\left (5\,a^2\,d^2+8\,a\,b\,c\,d+5\,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}}{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 {2\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\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^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x)^3*(c + d*x)^3),x)

[Out]

((3*(a^2*b*c^3 + a^3*c^2*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*x^2*
(a*d + b*c)*(a^2*d^2 + b^2*c^2 + 4*a*b*c*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)) + (b*d*x^3*(a^2*d^2 + b^2*c^2 + 4*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) + (a*c*x*(5*a^2*d^2 + 5*b^2*c^2 + 8*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))/(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) - (2*atanh((a^5*d^5 + b^5*c^5 + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*
a*b^4*c^4*d - 3*a^4*b*c*d^4)/(a*d - b*c)^5 + (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^2*d^2 + b^2*c^2 + 4*a*b*c*d))/(a*d - b*c)^5

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sympy [B]  time = 3.85, size = 1299, normalized size = 7.22 \begin {gather*} \frac {6 a^{3} c^{2} d + 6 a^{2} b c^{3} + x^{3} \left (2 a^{2} b d^{3} + 8 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \left (3 a^{3} d^{3} + 15 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 3 b^{3} c^{3}\right ) + x \left (10 a^{3} c d^{2} + 16 a^{2} b c^{2} d + 10 a b^{2} 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 )} + \frac {\left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- \frac {a^{6} d^{6} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + \frac {6 a^{5} b c d^{5} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} - \frac {15 a^{4} b^{2} c^{2} d^{4} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + \frac {20 a^{3} b^{3} c^{3} d^{3} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + a^{3} d^{3} - \frac {15 a^{2} b^{4} c^{4} d^{2} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + 5 a^{2} b c d^{2} + \frac {6 a b^{5} c^{5} d \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + 5 a b^{2} c^{2} d - \frac {b^{6} c^{6} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + b^{3} c^{3}}{2 a^{2} b d^{3} + 8 a b^{2} c d^{2} + 2 b^{3} c^{2} d} \right )}}{\left (a d - b c\right )^{5}} - \frac {\left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {\frac {a^{6} d^{6} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} - \frac {6 a^{5} b c d^{5} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + \frac {15 a^{4} b^{2} c^{2} d^{4} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} - \frac {20 a^{3} b^{3} c^{3} d^{3} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + a^{3} d^{3} + \frac {15 a^{2} b^{4} c^{4} d^{2} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + 5 a^{2} b c d^{2} - \frac {6 a b^{5} c^{5} d \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + 5 a b^{2} c^{2} d + \frac {b^{6} c^{6} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{5}} + b^{3} c^{3}}{2 a^{2} b d^{3} + 8 a b^{2} c d^{2} + 2 b^{3} c^{2} d} \right )}}{\left (a d - b c\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**3/(d*x+c)**3,x)

[Out]

(6*a**3*c**2*d + 6*a**2*b*c**3 + x**3*(2*a**2*b*d**3 + 8*a*b**2*c*d**2 + 2*b**3*c**2*d) + x**2*(3*a**3*d**3 +
15*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 3*b**3*c**3) + x*(10*a**3*c*d**2 + 16*a**2*b*c**2*d + 10*a*b**2*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)) + (a**2*d**2 + 4*a*b*c*d + b**2*c**2)*log(x + (-a**6*d**6*(a**2*d**2 + 4*a*
b*c*d + b**2*c**2)/(a*d - b*c)**5 + 6*a**5*b*c*d**5*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 15*a*
*4*b**2*c**2*d**4*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 20*a**3*b**3*c**3*d**3*(a**2*d**2 + 4*a
*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a**3*d**3 - 15*a**2*b**4*c**4*d**2*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a
*d - b*c)**5 + 5*a**2*b*c*d**2 + 6*a*b**5*c**5*d*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a*b**2
*c**2*d - b**6*c**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + b**3*c**3)/(2*a**2*b*d**3 + 8*a*b**2*
c*d**2 + 2*b**3*c**2*d))/(a*d - b*c)**5 - (a**2*d**2 + 4*a*b*c*d + b**2*c**2)*log(x + (a**6*d**6*(a**2*d**2 +
4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 6*a**5*b*c*d**5*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 1
5*a**4*b**2*c**2*d**4*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 - 20*a**3*b**3*c**3*d**3*(a**2*d**2 +
4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + a**3*d**3 + 15*a**2*b**4*c**4*d**2*(a**2*d**2 + 4*a*b*c*d + b**2*c**2
)/(a*d - b*c)**5 + 5*a**2*b*c*d**2 - 6*a*b**5*c**5*d*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + 5*a*
b**2*c**2*d + b**6*c**6*(a**2*d**2 + 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**5 + b**3*c**3)/(2*a**2*b*d**3 + 8*a*b
**2*c*d**2 + 2*b**3*c**2*d))/(a*d - b*c)**5

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