Optimal. Leaf size=161 \[ \frac {d}{2 \left (b^2 c^2-a^2 d^2\right ) (c+d x)^2}+\frac {2 b^2 c d}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)}-\frac {b^2 \log (a-b x)}{2 a (b c+a d)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3}-\frac {b^2 d \left (3 b^2 c^2+a^2 d^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {84}
\begin {gather*} \frac {2 b^2 c d}{(c+d x) \left (b^2 c^2-a^2 d^2\right )^2}+\frac {d}{2 (c+d x)^2 \left (b^2 c^2-a^2 d^2\right )}-\frac {b^2 d \left (a^2 d^2+3 b^2 c^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}-\frac {b^2 \log (a-b x)}{2 a (a d+b c)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {1}{(a-b x) (a+b x) (c+d x)^3} \, dx &=\int \left (\frac {b^3}{2 a (b c+a d)^3 (a-b x)}-\frac {b^3}{2 a (-b c+a d)^3 (a+b x)}-\frac {d^2}{\left (b^2 c^2-a^2 d^2\right ) (c+d x)^3}-\frac {2 b^2 c d^2}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)^2}-\frac {d^2 \left (3 b^4 c^2+a^2 b^2 d^2\right )}{\left (b^2 c^2-a^2 d^2\right )^3 (c+d x)}\right ) \, dx\\ &=\frac {d}{2 \left (b^2 c^2-a^2 d^2\right ) (c+d x)^2}+\frac {2 b^2 c d}{\left (b^2 c^2-a^2 d^2\right )^2 (c+d x)}-\frac {b^2 \log (a-b x)}{2 a (b c+a d)^3}+\frac {b^2 \log (a+b x)}{2 a (b c-a d)^3}-\frac {b^2 d \left (3 b^2 c^2+a^2 d^2\right ) \log (c+d x)}{\left (b^2 c^2-a^2 d^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 147, normalized size = 0.91 \begin {gather*} \frac {1}{2} \left (-\frac {b^2 \log (a-b x)}{a (b c+a d)^3}-\frac {b^2 \log (a+b x)}{a (-b c+a d)^3}+\frac {d \left (\frac {\left (b^2 c^2-a^2 d^2\right ) \left (-a^2 d^2+b^2 c (5 c+4 d x)\right )}{(c+d x)^2}-2 \left (3 b^4 c^2+a^2 b^2 d^2\right ) \log (c+d x)\right )}{\left (b^2 c^2-a^2 d^2\right )^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 158, normalized size = 0.98
method | result | size |
default | \(-\frac {b^{2} \ln \left (-b x +a \right )}{2 a \left (a d +b c \right )^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{2 a \left (a d -b c \right )^{3}}-\frac {d}{2 \left (a d +b c \right ) \left (a d -b c \right ) \left (d x +c \right )^{2}}+\frac {d \,b^{2} \left (a^{2} d^{2}+3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d +b c \right )^{3} \left (a d -b c \right )^{3}}+\frac {2 d \,b^{2} c}{\left (a d +b c \right )^{2} \left (a d -b c \right )^{2} \left (d x +c \right )}\) | \(158\) |
norman | \(\frac {\frac {-a^{2} d^{5}+5 b^{2} c^{2} d^{3}}{2 d^{2} \left (a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}\right )}+\frac {2 c \,d^{2} b^{2} x}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}}{\left (d x +c \right )^{2}}+\frac {b^{2} d \left (a^{2} d^{2}+3 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}-\frac {b^{2} \ln \left (b x +a \right )}{2 a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} \ln \left (-b x +a \right )}{2 \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) a}\) | \(285\) |
risch | \(\frac {\frac {2 c \,d^{2} b^{2} x}{a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}}-\frac {\left (a^{2} d^{2}-5 b^{2} c^{2}\right ) d}{2 \left (a^{4} d^{4}-2 a^{2} b^{2} c^{2} d^{2}+b^{4} c^{4}\right )}}{\left (d x +c \right )^{2}}+\frac {b^{2} d^{3} \ln \left (d x +c \right ) a^{2}}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}+\frac {3 b^{4} d \ln \left (d x +c \right ) c^{2}}{a^{6} d^{6}-3 a^{4} b^{2} c^{2} d^{4}+3 a^{2} b^{4} c^{4} d^{2}-b^{6} c^{6}}-\frac {b^{2} \ln \left (b x +a \right )}{2 a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} \ln \left (-b x +a \right )}{2 \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) a}\) | \(329\) |
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. 314 vs.
\(2 (155) = 310\).
time = 0.30, size = 314, normalized size = 1.95 \begin {gather*} \frac {b^{2} \log \left (b x + a\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} - \frac {b^{2} \log \left (b x - a\right )}{2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d + a^{2} b^{2} d^{3}\right )} \log \left (d x + c\right )}{b^{6} c^{6} - 3 \, a^{2} b^{4} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{2} d^{4} - a^{6} d^{6}} + \frac {4 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - a^{2} d^{3}}{2 \, {\left (b^{4} c^{6} - 2 \, a^{2} b^{2} c^{4} d^{2} + a^{4} c^{2} d^{4} + {\left (b^{4} c^{4} d^{2} - 2 \, a^{2} b^{2} c^{2} d^{4} + a^{4} d^{6}\right )} x^{2} + 2 \, {\left (b^{4} c^{5} d - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{4} c d^{5}\right )} x\right )}} \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. 605 vs.
\(2 (155) = 310\).
time = 6.16, size = 605, normalized size = 3.76 \begin {gather*} \frac {5 \, a b^{4} c^{4} d - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{5} d^{5} + 4 \, {\left (a b^{4} c^{3} d^{2} - a^{3} b^{2} c d^{4}\right )} x + {\left (b^{5} c^{5} + 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} + a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} + 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d + 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x + a\right ) - {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (b x - a\right ) - 2 \, {\left (3 \, a b^{4} c^{4} d + a^{3} b^{2} c^{2} d^{3} + {\left (3 \, a b^{4} c^{2} d^{3} + a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (3 \, a b^{4} c^{3} d^{2} + a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{8} - 3 \, a^{3} b^{4} c^{6} d^{2} + 3 \, a^{5} b^{2} c^{4} d^{4} - a^{7} c^{2} d^{6} + {\left (a b^{6} c^{6} d^{2} - 3 \, a^{3} b^{4} c^{4} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + 2 \, {\left (a b^{6} c^{7} d - 3 \, a^{3} b^{4} c^{5} d^{3} + 3 \, a^{5} b^{2} c^{3} d^{5} - a^{7} c d^{7}\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 = 1.35, size = 277, normalized size = 1.72 \begin {gather*} \frac {b^{3} \log \left ({\left | b x + a \right |}\right )}{2 \, {\left (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}\right )}} - \frac {b^{3} \log \left ({\left | b x - a \right |}\right )}{2 \, {\left (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}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d^{2} + a^{2} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{6} c^{6} d - 3 \, a^{2} b^{4} c^{4} d^{3} + 3 \, a^{4} b^{2} c^{2} d^{5} - a^{6} d^{7}} + \frac {5 \, b^{4} c^{4} d - 6 \, a^{2} b^{2} c^{2} d^{3} + a^{4} d^{5} + 4 \, {\left (b^{4} c^{3} d^{2} - a^{2} b^{2} c d^{4}\right )} x}{2 \, {\left (b c + a d\right )}^{3} {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.71, size = 294, normalized size = 1.83 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (a^2\,b^2\,d^3+3\,b^4\,c^2\,d\right )}{a^6\,d^6-3\,a^4\,b^2\,c^2\,d^4+3\,a^2\,b^4\,c^4\,d^2-b^6\,c^6}-\frac {b^2\,\ln \left (a+b\,x\right )}{2\,\left (a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3\right )}-\frac {b^2\,\ln \left (a-b\,x\right )}{2\,\left (a^4\,d^3+3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d+a\,b^3\,c^3\right )}-\frac {\frac {a^2\,d^3-5\,b^2\,c^2\,d}{2\,\left (a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4\right )}-\frac {2\,b^2\,c\,d^2\,x}{a^4\,d^4-2\,a^2\,b^2\,c^2\,d^2+b^4\,c^4}}{c^2+2\,c\,d\,x+d^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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