Optimal. Leaf size=195 \[ \frac {b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {b^4}{a^2 (a+b x) (b c-a d)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac {1}{a^2 c^3 x}-\frac {2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac {d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.23, antiderivative size = 195, 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} \[ \frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac {b^4}{a^2 (a+b x) (b c-a d)^3}+\frac {b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac {1}{a^2 c^3 x}-\frac {2 d^3 (2 b c-a d)}{c^3 (c+d x) (b c-a d)^3}-\frac {d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac {1}{a^2 c^3 x^2}+\frac {-2 b c-3 a d}{a^3 c^4 x}-\frac {b^5}{a^2 (-b c+a d)^3 (a+b x)^2}-\frac {b^5 (-2 b c+5 a d)}{a^3 (-b c+a d)^4 (a+b x)}+\frac {d^4}{c^2 (b c-a d)^2 (c+d x)^3}+\frac {2 d^4 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)^2}+\frac {d^4 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )}{c^4 (b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac {1}{a^2 c^3 x}-\frac {b^4}{a^2 (b c-a d)^3 (a+b x)}-\frac {d^3}{2 c^2 (b c-a d)^2 (c+d x)^2}-\frac {2 d^3 (2 b c-a d)}{c^3 (b c-a d)^3 (c+d x)}-\frac {(2 b c+3 a d) \log (x)}{a^3 c^4}+\frac {b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 193, normalized size = 0.99 \[ \frac {b^4 (2 b c-5 a d) \log (a+b x)}{a^3 (b c-a d)^4}-\frac {\log (x) (3 a d+2 b c)}{a^3 c^4}+\frac {b^4}{a^2 (a+b x) (a d-b c)^3}+\frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log (c+d x)}{c^4 (b c-a d)^4}-\frac {1}{a^2 c^3 x}+\frac {2 d^3 (a d-2 b c)}{c^3 (c+d x) (b c-a d)^3}-\frac {d^3}{2 c^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 79.60, size = 1213, normalized size = 6.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 517, normalized size = 2.65 \[ -\frac {b^{9}}{{\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )} {\left (b x + a\right )}} + \frac {{\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}} - \frac {{\left (2 \, b^{2} c + 3 \, a b d\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} b c^{4}} + \frac {2 \, b^{5} c^{5} d^{2} - 8 \, a b^{4} c^{4} d^{3} + 12 \, a^{2} b^{3} c^{3} d^{4} - 17 \, a^{3} b^{2} c^{2} d^{5} + 6 \, a^{4} b c d^{6} + \frac {4 \, b^{7} c^{6} d - 20 \, a b^{6} c^{5} d^{2} + 40 \, a^{2} b^{5} c^{4} d^{3} - 50 \, a^{3} b^{4} c^{3} d^{4} + 43 \, a^{4} b^{3} c^{2} d^{5} - 12 \, a^{5} b^{2} c d^{6}}{{\left (b x + a\right )} b} + \frac {2 \, {\left (b^{9} c^{7} - 6 \, a b^{8} c^{6} d + 15 \, a^{2} b^{7} c^{5} d^{2} - 20 \, a^{3} b^{6} c^{4} d^{3} + 20 \, a^{4} b^{5} c^{3} d^{4} - 13 \, a^{5} b^{4} c^{2} d^{5} + 3 \, a^{6} b^{3} c d^{6}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{2 \, {\left (b c - a d\right )}^{4} a^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} c^{4} {\left (\frac {a}{b x + a} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 266, normalized size = 1.36 \[ \frac {3 a^{2} d^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} c^{4}}-\frac {10 a b \,d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} c^{3}}-\frac {5 b^{4} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} a^{2}}+\frac {2 b^{5} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} a^{3}}+\frac {10 b^{2} d^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} c^{2}}-\frac {2 a \,d^{4}}{\left (a d -b c \right )^{3} \left (d x +c \right ) c^{3}}+\frac {b^{4}}{\left (a d -b c \right )^{3} \left (b x +a \right ) a^{2}}+\frac {4 b \,d^{3}}{\left (a d -b c \right )^{3} \left (d x +c \right ) c^{2}}-\frac {d^{3}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} c^{2}}-\frac {3 d \ln \relax (x )}{a^{2} c^{4}}-\frac {2 b \ln \relax (x )}{a^{3} c^{3}}-\frac {1}{a^{2} c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 639, normalized size = 3.28 \[ \frac {{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x + a\right )}{a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}} + \frac {{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x + c\right )}{b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}} - \frac {2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{3} + {\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{2} + {\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x}{2 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{4} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{3} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{2} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x\right )}} - \frac {{\left (2 \, b c + 3 \, a d\right )} \log \relax (x)}{a^{3} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 535, normalized size = 2.74 \[ \frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,d^5-10\,a\,b\,c\,d^4+10\,b^2\,c^2\,d^3\right )}{a^4\,c^4\,d^4-4\,a^3\,b\,c^5\,d^3+6\,a^2\,b^2\,c^6\,d^2-4\,a\,b^3\,c^7\,d+b^4\,c^8}-\frac {\frac {1}{a\,c}+\frac {x\,\left (9\,a^4\,d^4-19\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d-4\,b^4\,c^4\right )}{2\,a^2\,c^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {x^2\,\left (-6\,a^4\,d^5+5\,a^3\,b\,c\,d^4+15\,a^2\,b^2\,c^2\,d^3-10\,a\,b^3\,c^3\,d^2+8\,b^4\,c^4\,d\right )}{2\,a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,d^2\,x^3\,\left (3\,a^3\,d^3-7\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{a^2\,c^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^2\,\left (b\,c^2+2\,a\,d\,c\right )+x^3\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^4+a\,c^2\,x}+\frac {\ln \left (a+b\,x\right )\,\left (2\,b^5\,c-5\,a\,b^4\,d\right )}{a^7\,d^4-4\,a^6\,b\,c\,d^3+6\,a^5\,b^2\,c^2\,d^2-4\,a^4\,b^3\,c^3\,d+a^3\,b^4\,c^4}-\frac {\ln \relax (x)\,\left (3\,a\,d+2\,b\,c\right )}{a^3\,c^4} \]
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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