Optimal. Leaf size=137 \[ -\frac {d^3}{2 b^3 x^2}+\frac {3 d^2 (c d-b e)}{b^4 x}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {3 d (c d-b e) (2 c d-b e) \log (x)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712}
\begin {gather*} \frac {3 d \log (x) (c d-b e) (2 c d-b e)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}+\frac {3 d^2 (c d-b e)}{b^4 x}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}-\frac {d^3}{2 b^3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d^3}{b^3 x^3}+\frac {3 d^2 (-c d+b e)}{b^4 x^2}+\frac {3 d (c d-b e) (2 c d-b e)}{b^5 x}+\frac {(-c d+b e)^3}{b^3 (b+c x)^3}-\frac {3 c d (-c d+b e)^2}{b^4 (b+c x)^2}+\frac {3 c d (c d-b e) (-2 c d+b e)}{b^5 (b+c x)}\right ) \, dx\\ &=-\frac {d^3}{2 b^3 x^2}+\frac {3 d^2 (c d-b e)}{b^4 x}+\frac {(c d-b e)^3}{2 b^3 c (b+c x)^2}+\frac {3 d (c d-b e)^2}{b^4 (b+c x)}+\frac {3 d (c d-b e) (2 c d-b e) \log (x)}{b^5}-\frac {3 d (c d-b e) (2 c d-b e) \log (b+c x)}{b^5}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 138, normalized size = 1.01 \begin {gather*} -\frac {\frac {b^2 d^3}{x^2}+\frac {6 b d^2 (-c d+b e)}{x}+\frac {b^2 (-c d+b e)^3}{c (b+c x)^2}-\frac {6 b d (c d-b e)^2}{b+c x}-6 d \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (x)+6 d \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) \log (b+c x)}{2 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 181, normalized size = 1.32
method | result | size |
default | \(-\frac {b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}}{2 b^{3} c \left (c x +b \right )^{2}}-\frac {3 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (c x +b \right )}{b^{5}}+\frac {3 d \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}{b^{4} \left (c x +b \right )}-\frac {d^{3}}{2 b^{3} x^{2}}+\frac {3 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 d^{2} \left (b e -c d \right )}{b^{4} x}\) | \(181\) |
norman | \(\frac {\frac {\left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +18 b \,c^{2} d^{2} e -12 c^{3} d^{3}\right ) x^{3}}{b^{4}}-\frac {d^{3}}{2 b}+\frac {c \left (b^{3} e^{3}-9 b^{2} d \,e^{2} c +27 b \,c^{2} d^{2} e -18 c^{3} d^{3}\right ) x^{4}}{2 b^{5}}-\frac {d^{2} \left (3 b e -2 c d \right ) x}{b^{2}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\) | \(192\) |
risch | \(\frac {\frac {3 c d \left (b^{2} e^{2}-3 b c d e +2 d^{2} c^{2}\right ) x^{3}}{b^{4}}-\frac {\left (b^{3} e^{3}-9 b^{2} d \,e^{2} c +27 b \,c^{2} d^{2} e -18 c^{3} d^{3}\right ) x^{2}}{2 b^{3} c}-\frac {d^{2} \left (3 b e -2 c d \right ) x}{b^{2}}-\frac {d^{3}}{2 b}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d \ln \left (-x \right ) e^{2}}{b^{3}}-\frac {9 d^{2} \ln \left (-x \right ) c e}{b^{4}}+\frac {6 d^{3} \ln \left (-x \right ) c^{2}}{b^{5}}-\frac {3 d \ln \left (c x +b \right ) e^{2}}{b^{3}}+\frac {9 d^{2} \ln \left (c x +b \right ) c e}{b^{4}}-\frac {6 d^{3} \ln \left (c x +b \right ) c^{2}}{b^{5}}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 217, normalized size = 1.58 \begin {gather*} -\frac {b^{3} c d^{3} - 6 \, {\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{2} c^{2} d e^{2}\right )} x^{3} - {\left (18 \, b c^{3} d^{3} - 27 \, b^{2} c^{2} d^{2} e + 9 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{2} d^{3} - 3 \, b^{3} c d^{2} e\right )} x}{2 \, {\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left (x\right )}{b^{5}} \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. 408 vs.
\(2 (140) = 280\).
time = 1.63, size = 408, normalized size = 2.98 \begin {gather*} \frac {12 \, b c^{4} d^{3} x^{3} + 18 \, b^{2} c^{3} d^{3} x^{2} + 4 \, b^{3} c^{2} d^{3} x - b^{4} c d^{3} - b^{5} x^{2} e^{3} + 3 \, {\left (2 \, b^{3} c^{2} d x^{3} + 3 \, b^{4} c d x^{2}\right )} e^{2} - 3 \, {\left (6 \, b^{2} c^{3} d^{2} x^{3} + 9 \, b^{3} c^{2} d^{2} x^{2} + 2 \, b^{4} c d^{2} x\right )} e - 6 \, {\left (2 \, c^{5} d^{3} x^{4} + 4 \, b c^{4} d^{3} x^{3} + 2 \, b^{2} c^{3} d^{3} x^{2} + {\left (b^{2} c^{3} d x^{4} + 2 \, b^{3} c^{2} d x^{3} + b^{4} c d x^{2}\right )} e^{2} - 3 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + b^{3} c^{2} d^{2} x^{2}\right )} e\right )} \log \left (c x + b\right ) + 6 \, {\left (2 \, c^{5} d^{3} x^{4} + 4 \, b c^{4} d^{3} x^{3} + 2 \, b^{2} c^{3} d^{3} x^{2} + {\left (b^{2} c^{3} d x^{4} + 2 \, b^{3} c^{2} d x^{3} + b^{4} c d x^{2}\right )} e^{2} - 3 \, {\left (b c^{4} d^{2} x^{4} + 2 \, b^{2} c^{3} d^{2} x^{3} + b^{3} c^{2} d^{2} x^{2}\right )} e\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\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. 371 vs.
\(2 (124) = 248\).
time = 0.99, size = 371, normalized size = 2.71 \begin {gather*} \frac {- b^{3} c d^{3} + x^{3} \cdot \left (6 b^{2} c^{2} d e^{2} - 18 b c^{3} d^{2} e + 12 c^{4} d^{3}\right ) + x^{2} \left (- b^{4} e^{3} + 9 b^{3} c d e^{2} - 27 b^{2} c^{2} d^{2} e + 18 b c^{3} d^{3}\right ) + x \left (- 6 b^{3} c d^{2} e + 4 b^{2} c^{2} d^{3}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac {3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (x + \frac {3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} - 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} - \frac {3 d \left (b e - 2 c d\right ) \left (b e - c d\right ) \log {\left (x + \frac {3 b^{3} d e^{2} - 9 b^{2} c d^{2} e + 6 b c^{2} d^{3} + 3 b d \left (b e - 2 c d\right ) \left (b e - c d\right )}{6 b^{2} c d e^{2} - 18 b c^{2} d^{2} e + 12 c^{3} d^{3}} \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 219, normalized size = 1.60 \begin {gather*} \frac {3 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{4} d^{3} x^{3} - 18 \, b c^{3} d^{2} x^{3} e + 18 \, b c^{3} d^{3} x^{2} + 6 \, b^{2} c^{2} d x^{3} e^{2} - 27 \, b^{2} c^{2} d^{2} x^{2} e + 4 \, b^{2} c^{2} d^{3} x + 9 \, b^{3} c d x^{2} e^{2} - 6 \, b^{3} c d^{2} x e - b^{3} c d^{3} - b^{4} x^{2} e^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 211, normalized size = 1.54 \begin {gather*} -\frac {\frac {d^3}{2\,b}+\frac {d^2\,x\,\left (3\,b\,e-2\,c\,d\right )}{b^2}+\frac {x^2\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+27\,b\,c^2\,d^2\,e-18\,c^3\,d^3\right )}{2\,b^3\,c}-\frac {3\,c\,d\,x^3\,\left (b^2\,e^2-3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^4}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,d\,\mathrm {atanh}\left (\frac {3\,d\,\left (b\,e-c\,d\right )\,\left (b\,e-2\,c\,d\right )\,\left (b+2\,c\,x\right )}{b\,\left (3\,b^2\,d\,e^2-9\,b\,c\,d^2\,e+6\,c^2\,d^3\right )}\right )\,\left (b\,e-c\,d\right )\,\left (b\,e-2\,c\,d\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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