Optimal. Leaf size=129 \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac {c^3}{2 d^2 (c+d x)^2 (b c-a d)^2}-\frac {c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3} \]
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Rubi [A] time = 0.13, antiderivative size = 129, 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} \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}-\frac {c^2 (b c-3 a d)}{d^2 (c+d x) (b c-a d)^3}+\frac {c^3}{2 d^2 (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (-\frac {a^3}{(b c-a d)^3 (a+b x)^2}+\frac {3 a^2 b c}{(b c-a d)^4 (a+b x)}-\frac {c^3}{d (-b c+a d)^2 (c+d x)^3}-\frac {c^2 (b c-3 a d)}{d (-b c+a d)^3 (c+d x)^2}-\frac {3 a^2 c d}{(-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac {a^3}{b (b c-a d)^3 (a+b x)}+\frac {c^3}{2 d^2 (b c-a d)^2 (c+d x)^2}-\frac {c^2 (b c-3 a d)}{d^2 (b c-a d)^3 (c+d x)}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 130, normalized size = 1.01 \[ \frac {a^3}{b (a+b x) (b c-a d)^3}+\frac {3 a^2 c \log (a+b x)}{(b c-a d)^4}-\frac {3 a^2 c \log (c+d x)}{(b c-a d)^4}+\frac {c^3}{2 d^2 (c+d x)^2 (a d-b c)^2}+\frac {b c^3-3 a c^2 d}{d^2 (c+d x) (a d-b c)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 621, normalized size = 4.81 \[ -\frac {a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} + 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4} + a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - 4 \, a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3} + 4 \, a^{4} c d^{4}\right )} x - 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (a^{2} b^{2} c d^{4} x^{3} + a^{3} b c^{3} d^{2} + {\left (2 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4}\right )} x^{2} + {\left (a^{2} b^{2} c^{3} d^{2} + 2 \, a^{3} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{5} c^{6} d^{2} - 4 \, a^{2} b^{4} c^{5} d^{3} + 6 \, a^{3} b^{3} c^{4} d^{4} - 4 \, a^{4} b^{2} c^{3} d^{5} + a^{5} b c^{2} d^{6} + {\left (b^{6} c^{4} d^{4} - 4 \, a b^{5} c^{3} d^{5} + 6 \, a^{2} b^{4} c^{2} d^{6} - 4 \, a^{3} b^{3} c d^{7} + a^{4} b^{2} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{5} d^{3} - 7 \, a b^{5} c^{4} d^{4} + 8 \, a^{2} b^{4} c^{3} d^{5} - 2 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7} + a^{5} b d^{8}\right )} x^{2} + {\left (b^{6} c^{6} d^{2} - 2 \, a b^{5} c^{5} d^{3} - 2 \, a^{2} b^{4} c^{4} d^{4} + 8 \, a^{3} b^{3} c^{3} d^{5} - 7 \, a^{4} b^{2} c^{2} d^{6} + 2 \, a^{5} b c d^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 230, normalized size = 1.78 \[ -\frac {3 \, a^{2} b c \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {a^{3} b^{2}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} {\left (b x + a\right )}} + \frac {b^{2} c^{3} - 6 \, a b c^{2} d - \frac {6 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 147, normalized size = 1.14 \[ \frac {3 a^{2} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4}}-\frac {3 a^{2} c \ln \left (d x +c \right )}{\left (a d -b c \right )^{4}}-\frac {a^{3}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b}-\frac {3 a \,c^{2}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d}+\frac {b \,c^{3}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{2}}+\frac {c^{3}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.28, size = 463, normalized size = 3.59 \[ \frac {3 \, a^{2} c \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {3 \, a^{2} c \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {a b^{2} c^{4} - 5 \, a^{2} b c^{3} d - 2 \, a^{3} c^{2} d^{2} + 2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - a^{3} d^{4}\right )} x^{2} + {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d - 6 \, a^{2} b c^{2} d^{2} - 4 \, a^{3} c d^{3}\right )} x}{2 \, {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5} + {\left (b^{5} c^{3} d^{4} - 3 \, a b^{4} c^{2} d^{5} + 3 \, a^{2} b^{3} c d^{6} - a^{3} b^{2} d^{7}\right )} x^{3} + {\left (2 \, b^{5} c^{4} d^{3} - 5 \, a b^{4} c^{3} d^{4} + 3 \, a^{2} b^{3} c^{2} d^{5} + a^{3} b^{2} c d^{6} - a^{4} b d^{7}\right )} x^{2} + {\left (b^{5} c^{5} d^{2} - a b^{4} c^{4} d^{3} - 3 \, a^{2} b^{3} c^{3} d^{4} + 5 \, a^{3} b^{2} c^{2} d^{5} - 2 \, a^{4} b c d^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 396, normalized size = 3.07 \[ \frac {6\,a^2\,c\,\mathrm {atanh}\left (\frac {a^4\,d^4-2\,a^3\,b\,c\,d^3+2\,a\,b^3\,c^3\,d-b^4\,c^4}{{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {\frac {x^2\,\left (a^3\,d^3+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^2\,d^2+5\,a\,b\,c\,d-b^2\,c^2\right )}{2\,b\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.09, size = 717, normalized size = 5.56 \[ - \frac {3 a^{2} c \log {\left (x + \frac {- \frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} + \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} - \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} + \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} - \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d + \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {3 a^{2} c \log {\left (x + \frac {\frac {3 a^{7} c d^{5}}{\left (a d - b c\right )^{4}} - \frac {15 a^{6} b c^{2} d^{4}}{\left (a d - b c\right )^{4}} + \frac {30 a^{5} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{4}} - \frac {30 a^{4} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{4}} + \frac {15 a^{3} b^{4} c^{5} d}{\left (a d - b c\right )^{4}} + 3 a^{3} c d - \frac {3 a^{2} b^{5} c^{6}}{\left (a d - b c\right )^{4}} + 3 a^{2} b c^{2}}{6 a^{2} b c d} \right )}}{\left (a d - b c\right )^{4}} + \frac {- 2 a^{3} c^{2} d^{2} - 5 a^{2} b c^{3} d + a b^{2} c^{4} + x^{2} \left (- 2 a^{3} d^{4} - 6 a b^{2} c^{2} d^{2} + 2 b^{3} c^{3} d\right ) + x \left (- 4 a^{3} c d^{3} - 6 a^{2} b c^{2} d^{2} - 3 a b^{2} c^{3} d + b^{3} c^{4}\right )}{2 a^{4} b c^{2} d^{5} - 6 a^{3} b^{2} c^{3} d^{4} + 6 a^{2} b^{3} c^{4} d^{3} - 2 a b^{4} c^{5} d^{2} + x^{3} \left (2 a^{3} b^{2} d^{7} - 6 a^{2} b^{3} c d^{6} + 6 a b^{4} c^{2} d^{5} - 2 b^{5} c^{3} d^{4}\right ) + x^{2} \left (2 a^{4} b d^{7} - 2 a^{3} b^{2} c d^{6} - 6 a^{2} b^{3} c^{2} d^{5} + 10 a b^{4} c^{3} d^{4} - 4 b^{5} c^{4} d^{3}\right ) + x \left (4 a^{4} b c d^{6} - 10 a^{3} b^{2} c^{2} d^{5} + 6 a^{2} b^{3} c^{3} d^{4} + 2 a b^{4} c^{4} d^{3} - 2 b^{5} c^{5} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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