Optimal. Leaf size=140 \[ \frac {a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac {c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}+\frac {c^4}{2 d^4 (c+d x)^2 (b c-a d)}-\frac {c^3 (3 b c-4 a d)}{d^4 (c+d x) (b c-a d)^2}+\frac {x}{b d^3} \]
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Rubi [A] time = 0.12, antiderivative size = 140, 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} \[ -\frac {c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}+\frac {a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac {c^3 (3 b c-4 a d)}{d^4 (c+d x) (b c-a d)^2}+\frac {c^4}{2 d^4 (c+d x)^2 (b c-a d)}+\frac {x}{b d^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x) (c+d x)^3} \, dx &=\int \left (\frac {1}{b d^3}+\frac {a^4}{b (b c-a d)^3 (a+b x)}+\frac {c^4}{d^3 (-b c+a d) (c+d x)^3}+\frac {c^3 (3 b c-4 a d)}{d^3 (-b c+a d)^2 (c+d x)^2}+\frac {c^2 \left (3 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{d^3 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=\frac {x}{b d^3}+\frac {c^4}{2 d^4 (b c-a d) (c+d x)^2}-\frac {c^3 (3 b c-4 a d)}{d^4 (b c-a d)^2 (c+d x)}+\frac {a^4 \log (a+b x)}{b^2 (b c-a d)^3}-\frac {c^2 \left (3 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 138, normalized size = 0.99 \[ \frac {a^4 \log (a+b x)}{b^2 (b c-a d)^3}+\frac {c^2 \left (6 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (a d-b c)^3}-\frac {c^4}{2 d^4 (c+d x)^2 (a d-b c)}+\frac {c^3 (4 a d-3 b c)}{d^4 (c+d x) (b c-a d)^2}+\frac {x}{b d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 501, normalized size = 3.58 \[ -\frac {5 \, b^{4} c^{6} - 12 \, a b^{3} c^{5} d + 7 \, a^{2} b^{2} c^{4} d^{2} - 2 \, {\left (b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}\right )} x^{3} - 4 \, {\left (b^{4} c^{4} d^{2} - 3 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} - a^{3} b c d^{5}\right )} x^{2} + 2 \, {\left (2 \, b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + a^{2} b^{2} c^{3} d^{3} + a^{3} b c^{2} d^{4}\right )} x - 2 \, {\left (a^{4} d^{6} x^{2} + 2 \, a^{4} c d^{5} x + a^{4} c^{2} d^{4}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{4} c^{6} - 8 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} + {\left (3 \, b^{4} c^{4} d^{2} - 8 \, a b^{3} c^{3} d^{3} + 6 \, a^{2} b^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (3 \, b^{4} c^{5} d - 8 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{5} c^{5} d^{4} - 3 \, a b^{4} c^{4} d^{5} + 3 \, a^{2} b^{3} c^{3} d^{6} - a^{3} b^{2} c^{2} d^{7} + {\left (b^{5} c^{3} d^{6} - 3 \, a b^{4} c^{2} d^{7} + 3 \, a^{2} b^{3} c d^{8} - a^{3} b^{2} d^{9}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d^{5} - 3 \, a b^{4} c^{3} d^{6} + 3 \, a^{2} b^{3} c^{2} d^{7} - a^{3} b^{2} c d^{8}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 227, normalized size = 1.62 \[ \frac {a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}} - \frac {{\left (3 \, b^{2} c^{4} - 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} + \frac {x}{b d^{3}} - \frac {5 \, b^{2} c^{6} - 12 \, a b c^{5} d + 7 \, a^{2} c^{4} d^{2} + 2 \, {\left (3 \, b^{2} c^{5} d - 7 \, a b c^{4} d^{2} + 4 \, a^{2} c^{3} d^{3}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 191, normalized size = 1.36 \[ -\frac {a^{4} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} b^{2}}+\frac {6 a^{2} c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{2}}-\frac {8 a b \,c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{3}}+\frac {3 b^{2} c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{4}}+\frac {4 a \,c^{3}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{3}}-\frac {3 b \,c^{4}}{\left (a d -b c \right )^{2} \left (d x +c \right ) d^{4}}-\frac {c^{4}}{2 \left (a d -b c \right ) \left (d x +c \right )^{2} d^{4}}+\frac {x}{b \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 272, normalized size = 1.94 \[ \frac {a^{4} \log \left (b x + a\right )}{b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}} - \frac {{\left (3 \, b^{2} c^{4} - 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}} - \frac {5 \, b c^{5} - 7 \, a c^{4} d + 2 \, {\left (3 \, b c^{4} d - 4 \, a c^{3} d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d^{4} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6} + {\left (b^{2} c^{2} d^{6} - 2 \, a b c d^{7} + a^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{5} - 2 \, a b c^{2} d^{6} + a^{2} c d^{7}\right )} x\right )}} + \frac {x}{b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 258, normalized size = 1.84 \[ \frac {x}{b\,d^3}-\frac {\frac {5\,b^2\,c^5-7\,a\,b\,c^4\,d}{2\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {c^2\,x\,\left (3\,b^2\,c^2-4\,a\,b\,c\,d\right )}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{b\,c^2\,d^3+2\,b\,c\,d^4\,x+b\,d^5\,x^2}+\frac {a^4\,\ln \left (a+b\,x\right )}{-a^3\,b^2\,d^3+3\,a^2\,b^3\,c\,d^2-3\,a\,b^4\,c^2\,d+b^5\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (6\,a^2\,c^2\,d^2-8\,a\,b\,c^3\,d+3\,b^2\,c^4\right )}{a^3\,d^7-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^2\,d^5-b^3\,c^3\,d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.84, size = 719, normalized size = 5.14 \[ - \frac {a^{4} \log {\left (x + \frac {\frac {a^{8} d^{7}}{b \left (a d - b c\right )^{3}} - \frac {4 a^{7} c d^{6}}{\left (a d - b c\right )^{3}} + \frac {6 a^{6} b c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac {4 a^{5} b^{2} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + \frac {a^{4} b^{3} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + a^{4} c d^{3} + 6 a^{3} b c^{2} d^{2} - 8 a^{2} b^{2} c^{3} d + 3 a b^{3} c^{4}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} - 8 a b^{3} c^{3} d + 3 b^{4} c^{4}} \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac {c^{2} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right ) \log {\left (x + \frac {- \frac {a^{4} b c^{2} d^{3} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + a^{4} c d^{3} + \frac {4 a^{3} b^{2} c^{3} d^{2} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 6 a^{3} b c^{2} d^{2} - \frac {6 a^{2} b^{3} c^{4} d \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 8 a^{2} b^{2} c^{3} d + \frac {4 a b^{4} c^{5} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 3 a b^{3} c^{4} - \frac {b^{5} c^{6} \left (6 a^{2} d^{2} - 8 a b c d + 3 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} - 8 a b^{3} c^{3} d + 3 b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )^{3}} + \frac {7 a c^{4} d - 5 b c^{5} + x \left (8 a c^{3} d^{2} - 6 b c^{4} d\right )}{2 a^{2} c^{2} d^{6} - 4 a b c^{3} d^{5} + 2 b^{2} c^{4} d^{4} + x^{2} \left (2 a^{2} d^{8} - 4 a b c d^{7} + 2 b^{2} c^{2} d^{6}\right ) + x \left (4 a^{2} c d^{7} - 8 a b c^{2} d^{6} + 4 b^{2} c^{3} d^{5}\right )} + \frac {x}{b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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