Optimal. Leaf size=128 \[ -\frac {c^3}{2 d^3 (b c-a d) (c+d x)^2}+\frac {c^2 (2 b c-3 a d)}{d^3 (b c-a d)^2 (c+d x)}-\frac {a^3 \log (a+b x)}{b (b c-a d)^3}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^3} \]
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Rubi [A]
time = 0.08, antiderivative size = 128, 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 = {90}
\begin {gather*} -\frac {a^3 \log (a+b x)}{b (b c-a d)^3}+\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^3}-\frac {c^3}{2 d^3 (c+d x)^2 (b c-a d)}+\frac {c^2 (2 b c-3 a d)}{d^3 (c+d x) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x) (c+d x)^3} \, dx &=\int \left (-\frac {a^3}{(b c-a d)^3 (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)^3}-\frac {c^2 (2 b c-3 a d)}{d^2 (-b c+a d)^2 (c+d x)^2}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{d^2 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {c^3}{2 d^3 (b c-a d) (c+d x)^2}+\frac {c^2 (2 b c-3 a d)}{d^3 (b c-a d)^2 (c+d x)}-\frac {a^3 \log (a+b x)}{b (b c-a d)^3}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (c+d x)}{d^3 (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 134, normalized size = 1.05 \begin {gather*} \frac {c^3}{2 d^3 (-b c+a d) (c+d x)^2}+\frac {2 b c^3-3 a c^2 d}{d^3 (-b c+a d)^2 (c+d x)}-\frac {a^3 \log (a+b x)}{b (b c-a d)^3}-\frac {\left (-b^2 c^3+3 a b c^2 d-3 a^2 c d^2\right ) \log (c+d x)}{d^3 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 128, normalized size = 1.00
method | result | size |
default | \(\frac {a^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{3} b}-\frac {c^{2} \left (3 a d -2 b c \right )}{d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}+\frac {c^{3}}{2 d^{3} \left (a d -b c \right ) \left (d x +c \right )^{2}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{3} d^{3}}\) | \(128\) |
norman | \(\frac {\frac {\left (-3 a c d +2 b \,c^{2}\right ) c x}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-5 a c d +3 b \,c^{2}\right ) c^{2}}{2 d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {a^{3} \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(218\) |
risch | \(\frac {-\frac {c^{2} \left (3 a d -2 b c \right ) x}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {c^{3} \left (5 a d -3 b c \right )}{2 d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (d x +c \right )^{2}}+\frac {a^{3} \ln \left (-b x -a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}-\frac {3 c \ln \left (d x +c \right ) a^{2}}{d \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {3 c^{2} \ln \left (d x +c \right ) a b}{d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {c^{3} \ln \left (d x +c \right ) b^{2}}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(308\) |
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. 259 vs.
\(2 (126) = 252\).
time = 0.29, size = 259, normalized size = 2.02 \begin {gather*} -\frac {a^{3} \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac {3 \, b c^{4} - 5 \, a c^{3} d + 2 \, {\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x}{2 \, {\left (b^{2} c^{4} d^{3} - 2 \, a b c^{3} d^{4} + a^{2} c^{2} d^{5} + {\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{4} - 2 \, a b c^{2} d^{5} + a^{2} c d^{6}\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. 369 vs.
\(2 (126) = 252\).
time = 0.87, size = 369, normalized size = 2.88 \begin {gather*} \frac {3 \, b^{3} c^{5} - 8 \, a b^{2} c^{4} d + 5 \, a^{2} b c^{3} d^{2} + 2 \, {\left (2 \, b^{3} c^{4} d - 5 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} x - 2 \, {\left (a^{3} d^{5} x^{2} + 2 \, a^{3} c d^{4} x + a^{3} c^{2} d^{3}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\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}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{4} c^{5} d^{3} - 3 \, a b^{3} c^{4} d^{4} + 3 \, a^{2} b^{2} c^{3} d^{5} - a^{3} b c^{2} d^{6} + {\left (b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d^{4} - 3 \, a b^{3} c^{3} d^{5} + 3 \, a^{2} b^{2} c^{2} d^{6} - a^{3} b c d^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs.
\(2 (116) = 232\).
time = 40.83, size = 653, normalized size = 5.10 \begin {gather*} \frac {a^{3} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )^{3}} - \frac {4 a^{6} c d^{5}}{\left (a d - b c\right )^{3}} + \frac {6 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac {a^{3} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - 3 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b \left (a d - b c\right )^{3}} - \frac {c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- \frac {a^{4} c d^{3} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c^{2} d^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - \frac {6 a^{2} b^{2} c^{3} d \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 3 a^{2} b c^{2} d + \frac {4 a b^{3} c^{4} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} - \frac {b^{4} c^{5} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{3}} + \frac {- 5 a c^{3} d + 3 b c^{4} + x \left (- 6 a c^{2} d^{2} + 4 b c^{3} d\right )}{2 a^{2} c^{2} d^{5} - 4 a b c^{3} d^{4} + 2 b^{2} c^{4} d^{3} + x^{2} \cdot \left (2 a^{2} d^{7} - 4 a b c d^{6} + 2 b^{2} c^{2} d^{5}\right ) + x \left (4 a^{2} c d^{6} - 8 a b c^{2} d^{5} + 4 b^{2} c^{3} d^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 216, normalized size = 1.69 \begin {gather*} -\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac {2 \, {\left (2 \, b^{2} c^{4} - 5 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} x + \frac {3 \, b^{2} c^{5} - 8 \, a b c^{4} d + 5 \, a^{2} c^{3} d^{2}}{d}}{2 \, {\left (b c - a d\right )}^{3} {\left (d x + c\right )}^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 232, normalized size = 1.81 \begin {gather*} \frac {\frac {3\,b\,c^4-5\,a\,c^3\,d}{2\,d^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {c^2\,x\,\left (3\,a\,d-2\,b\,c\right )}{d^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 {a^3\,\ln \left (a+b\,x\right )}{-a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+b^4\,c^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,a^2\,c\,d^2-3\,a\,b\,c^2\,d+b^2\,c^3\right )}{a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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