Optimal. Leaf size=134 \[ -\frac {d}{2 c (b c-a d) (c+d x)^2}-\frac {d (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a c^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 134, 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 = {84}
\begin {gather*} \frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}-\frac {d (2 b c-a d)}{c^2 (c+d x) (b c-a d)^2}-\frac {d}{2 c (c+d x)^2 (b c-a d)}+\frac {\log (x)}{a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {1}{x (a+b x) (c+d x)^3} \, dx &=\int \left (\frac {1}{a c^3 x}+\frac {b^4}{a (-b c+a d)^3 (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)^3}+\frac {d^2 (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)^2}+\frac {d^2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{c^3 (b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {d}{2 c (b c-a d) (c+d x)^2}-\frac {d (2 b c-a d)}{c^2 (b c-a d)^2 (c+d x)}+\frac {\log (x)}{a c^3}-\frac {b^3 \log (a+b x)}{a (b c-a d)^3}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)}{c^3 (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 116, normalized size = 0.87 \begin {gather*} \frac {\log (x)}{a c^3}+\frac {\frac {2 b^3 \log (a+b x)}{a}+\frac {d \left (\frac {c (b c-a d) (-a d (3 c+2 d x)+b c (5 c+4 d x))}{(c+d x)^2}-2 \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \log (c+d x)\right )}{c^3}}{2 (-b c+a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 131, normalized size = 0.98
method | result | size |
default | \(\frac {b^{3} \ln \left (b x +a \right )}{a \left (a d -b c \right )^{3}}+\frac {d}{2 c \left (a d -b c \right ) \left (d x +c \right )^{2}}+\frac {d \left (a d -2 b c \right )}{c^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a d -b c \right )^{3}}+\frac {\ln \left (x \right )}{a \,c^{3}}\) | \(131\) |
norman | \(\frac {\frac {\left (-2 a \,d^{2}+3 b c d \right ) d x}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-3 a \,d^{2}+5 b c d \right ) d^{2} x^{2}}{2 c^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {\ln \left (x \right )}{a \,c^{3}}+\frac {b^{3} \ln \left (b x +a \right )}{a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(230\) |
risch | \(\frac {\frac {d^{2} \left (a d -2 b c \right ) x}{c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \left (3 a d -5 b c \right )}{2 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {b^{3} \ln \left (b x +a \right )}{a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\ln \left (-x \right )}{a \,c^{3}}-\frac {d^{3} \ln \left (-d x -c \right ) a^{2}}{c^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 d^{2} \ln \left (-d x -c \right ) a b}{c^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {3 d \ln \left (-d x -c \right ) b^{2}}{c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(321\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs.
\(2 (132) = 264\).
time = 0.30, size = 266, normalized size = 1.99 \begin {gather*} -\frac {b^{3} \log \left (b x + a\right )}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}} - \frac {5 \, b c^{2} d - 3 \, a c d^{2} + 2 \, {\left (2 \, b c d^{2} - a d^{3}\right )} x}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x\right )}} + \frac {\log \left (x\right )}{a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 506 vs.
\(2 (132) = 264\).
time = 7.71, size = 506, normalized size = 3.78 \begin {gather*} -\frac {5 \, a b^{2} c^{4} d - 8 \, a^{2} b c^{3} d^{2} + 3 \, a^{3} c^{2} d^{3} + 2 \, {\left (2 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x + 2 \, {\left (b^{3} c^{3} d^{2} x^{2} + 2 \, b^{3} c^{4} d x + b^{3} c^{5}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, a b^{2} c^{4} d - 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (3 \, a b^{2} c^{2} d^{3} - 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} x\right )} \log \left (d x + c\right ) - 2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x\right )} \log \left (x\right )}{2 \, {\left (a b^{3} c^{8} - 3 \, a^{2} b^{2} c^{7} d + 3 \, a^{3} b c^{6} d^{2} - a^{4} c^{5} d^{3} + {\left (a b^{3} c^{6} d^{2} - 3 \, a^{2} b^{2} c^{5} d^{3} + 3 \, a^{3} b c^{4} d^{4} - a^{4} c^{3} d^{5}\right )} x^{2} + 2 \, {\left (a b^{3} c^{7} d - 3 \, a^{2} b^{2} c^{6} d^{2} + 3 \, a^{3} b c^{5} d^{3} - a^{4} c^{4} d^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.22, size = 234, normalized size = 1.75 \begin {gather*} -\frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}} + \frac {{\left (3 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3} + a^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}} + \frac {\log \left ({\left | x \right |}\right )}{a c^{3}} - \frac {5 \, b^{2} c^{4} d - 8 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3} + 2 \, {\left (2 \, b^{2} c^{3} d^{2} - 3 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 235, normalized size = 1.75 \begin {gather*} \frac {\frac {3\,a\,d^2-5\,b\,c\,d}{2\,c\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,x\,\left (a\,d-2\,b\,c\right )}{c^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{c^2+2\,c\,d\,x+d^2\,x^2}+\frac {b^3\,\ln \left (a+b\,x\right )}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^3-3\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{-a^3\,c^3\,d^3+3\,a^2\,b\,c^4\,d^2-3\,a\,b^2\,c^5\,d+b^3\,c^6}+\frac {\ln \left (x\right )}{a\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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