Optimal. Leaf size=221 \[ \frac {\log (x)}{a^3 c^3}+\frac {b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac {d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}-\frac {b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac {d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac {d^3}{2 c (c+d x)^2 (b c-a d)^3} \]
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Rubi [A] time = 0.24, antiderivative size = 221, 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} \[ -\frac {b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac {b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac {\log (x)}{a^3 c^3}+\frac {b^3}{2 a (a+b x)^2 (b c-a d)^3}-\frac {d^3 (4 b c-a d)}{c^2 (c+d x) (b c-a d)^4}-\frac {d^3}{2 c (c+d x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x)^3 (c+d x)^3} \, dx &=\int \left (\frac {1}{a^3 c^3 x}+\frac {b^4}{a (-b c+a d)^3 (a+b x)^3}+\frac {b^4 (-b c+4 a d)}{a^2 (-b c+a d)^4 (a+b x)^2}+\frac {b^4 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right )}{a^3 (-b c+a d)^5 (a+b x)}+\frac {d^4}{c (b c-a d)^3 (c+d x)^3}+\frac {d^4 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^5 (c+d x)}\right ) \, dx\\ &=\frac {b^3}{2 a (b c-a d)^3 (a+b x)^2}+\frac {b^3 (b c-4 a d)}{a^2 (b c-a d)^4 (a+b x)}-\frac {d^3}{2 c (b c-a d)^3 (c+d x)^2}-\frac {d^3 (4 b c-a d)}{c^2 (b c-a d)^4 (c+d x)}+\frac {\log (x)}{a^3 c^3}-\frac {b^3 \left (b^2 c^2-5 a b c d+10 a^2 d^2\right ) \log (a+b x)}{a^3 (b c-a d)^5}+\frac {d^3 \left (10 b^2 c^2-5 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 218, normalized size = 0.99 \[ \frac {\log (x)}{a^3 c^3}+\frac {b^3 (b c-4 a d)}{a^2 (a+b x) (b c-a d)^4}+\frac {d^3 \left (a^2 d^2-5 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^5}+\frac {b^3 \left (10 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{a^3 (a d-b c)^5}-\frac {b^3}{2 a (a+b x)^2 (a d-b c)^3}+\frac {d^3 (a d-4 b c)}{c^2 (c+d x) (b c-a d)^4}-\frac {d^3}{2 c (c+d x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 97.66, size = 1630, normalized size = 7.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.95, size = 504, normalized size = 2.28 \[ -\frac {{\left (b^{6} c^{2} - 5 \, a b^{5} c d + 10 \, a^{2} b^{4} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}} + \frac {{\left (10 \, b^{2} c^{2} d^{4} - 5 \, a b c d^{5} + a^{2} d^{6}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{8} d - 5 \, a b^{4} c^{7} d^{2} + 10 \, a^{2} b^{3} c^{6} d^{3} - 10 \, a^{3} b^{2} c^{5} d^{4} + 5 \, a^{4} b c^{4} d^{5} - a^{5} c^{3} d^{6}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3} c^{3}} + \frac {3 \, a^{2} b^{4} c^{6} - 9 \, a^{3} b^{3} c^{5} d - 9 \, a^{5} b c^{3} d^{3} + 3 \, a^{6} c^{2} d^{4} + 2 \, {\left (a b^{5} c^{4} d^{2} - 4 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x^{3} + {\left (4 \, a b^{5} c^{5} d - 13 \, a^{2} b^{4} c^{4} d^{2} - 18 \, a^{3} b^{3} c^{3} d^{3} - 13 \, a^{4} b^{2} c^{2} d^{4} + 4 \, a^{5} b c d^{5}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - a^{2} b^{4} c^{5} d - 9 \, a^{3} b^{3} c^{4} d^{2} - 9 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 322, normalized size = 1.46 \[ -\frac {a^{2} d^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c^{3}}+\frac {5 a b \,d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c^{2}}+\frac {10 b^{3} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a}-\frac {5 b^{4} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a^{2}}+\frac {b^{5} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} a^{3}}-\frac {10 b^{2} d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} c}+\frac {a \,d^{4}}{\left (a d -b c \right )^{4} \left (d x +c \right ) c^{2}}-\frac {4 b^{3} d}{\left (a d -b c \right )^{4} \left (b x +a \right ) a}+\frac {b^{4} c}{\left (a d -b c \right )^{4} \left (b x +a \right ) a^{2}}-\frac {4 b \,d^{3}}{\left (a d -b c \right )^{4} \left (d x +c \right ) c}-\frac {b^{3}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} a}+\frac {d^{3}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} c}+\frac {\ln \relax (x )}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 804, normalized size = 3.64 \[ -\frac {{\left (b^{5} c^{2} - 5 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} \log \left (b x + a\right )}{a^{3} b^{5} c^{5} - 5 \, a^{4} b^{4} c^{4} d + 10 \, a^{5} b^{3} c^{3} d^{2} - 10 \, a^{6} b^{2} c^{2} d^{3} + 5 \, a^{7} b c d^{4} - a^{8} d^{5}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 5 \, a b c d^{4} + a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{5} c^{8} - 5 \, a b^{4} c^{7} d + 10 \, a^{2} b^{3} c^{6} d^{2} - 10 \, a^{3} b^{2} c^{5} d^{3} + 5 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}} + \frac {3 \, a b^{4} c^{5} - 9 \, a^{2} b^{3} c^{4} d - 9 \, a^{4} b c^{2} d^{3} + 3 \, a^{5} c d^{4} + 2 \, {\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} - 4 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{3} + {\left (4 \, b^{5} c^{4} d - 13 \, a b^{4} c^{3} d^{2} - 18 \, a^{2} b^{3} c^{2} d^{3} - 13 \, a^{3} b^{2} c d^{4} + 4 \, a^{4} b d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{5} - a b^{4} c^{4} d - 9 \, a^{2} b^{3} c^{3} d^{2} - 9 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4} + a^{5} d^{5}\right )} x}{2 \, {\left (a^{4} b^{4} c^{8} - 4 \, a^{5} b^{3} c^{7} d + 6 \, a^{6} b^{2} c^{6} d^{2} - 4 \, a^{7} b c^{5} d^{3} + a^{8} c^{4} d^{4} + {\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6}\right )} x^{4} + 2 \, {\left (a^{2} b^{6} c^{7} d - 3 \, a^{3} b^{5} c^{6} d^{2} + 2 \, a^{4} b^{4} c^{5} d^{3} + 2 \, a^{5} b^{3} c^{4} d^{4} - 3 \, a^{6} b^{2} c^{3} d^{5} + a^{7} b c^{2} d^{6}\right )} x^{3} + {\left (a^{2} b^{6} c^{8} - 9 \, a^{4} b^{4} c^{6} d^{2} + 16 \, a^{5} b^{3} c^{5} d^{3} - 9 \, a^{6} b^{2} c^{4} d^{4} + a^{8} c^{2} d^{6}\right )} x^{2} + 2 \, {\left (a^{3} b^{5} c^{8} - 3 \, a^{4} b^{4} c^{7} d + 2 \, a^{5} b^{3} c^{6} d^{2} + 2 \, a^{6} b^{2} c^{5} d^{3} - 3 \, a^{7} b c^{4} d^{4} + a^{8} c^{3} d^{5}\right )} x\right )}} + \frac {\log \relax (x)}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 624, normalized size = 2.82 \[ \frac {\frac {3\,\left (a^4\,d^4-3\,a^3\,b\,c\,d^3-3\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,c\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x^2\,\left (-4\,a^4\,b\,d^5+13\,a^3\,b^2\,c\,d^4+18\,a^2\,b^3\,c^2\,d^3+13\,a\,b^4\,c^3\,d^2-4\,b^5\,c^4\,d\right )}{2\,a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {x\,\left (-a^5\,d^5+a^4\,b\,c\,d^4+9\,a^3\,b^2\,c^2\,d^3+9\,a^2\,b^3\,c^3\,d^2+a\,b^4\,c^4\,d-b^5\,c^5\right )}{a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b\,d\,x^3\,\left (a^3\,b\,d^4-4\,a^2\,b^2\,c\,d^3-4\,a\,b^3\,c^2\,d^2+b^4\,c^3\,d\right )}{a^2\,c^2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}}{x\,\left (2\,d\,a^2\,c+2\,b\,a\,c^2\right )+x^2\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )+x^3\,\left (2\,c\,b^2\,d+2\,a\,b\,d^2\right )+a^2\,c^2+b^2\,d^2\,x^4}+\frac {\ln \relax (x)}{a^3\,c^3}+\frac {b^3\,\ln \left (a+b\,x\right )\,\left (10\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2\right )}{a^3\,{\left (a\,d-b\,c\right )}^5}-\frac {d^3\,\ln \left (c+d\,x\right )\,\left (a^2\,d^2-5\,a\,b\,c\,d+10\,b^2\,c^2\right )}{c^3\,{\left (a\,d-b\,c\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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