Optimal. Leaf size=144 \[ -\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 144, 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 {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac {2 \log (x) (b e+c d)}{b^3 d^3}-\frac {c^3}{b^2 (b+c x) (c d-b e)^2}-\frac {1}{b^2 d^2 x}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac {e^3}{d^2 (d+e x) (c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {1}{b^2 d^2 x^2}-\frac {2 (c d+b e)}{b^3 d^3 x}+\frac {c^4}{b^2 (-c d+b e)^2 (b+c x)^2}+\frac {2 c^4 (-c d+2 b e)}{b^3 (-c d+b e)^3 (b+c x)}+\frac {e^4}{d^2 (c d-b e)^2 (d+e x)^2}+\frac {2 e^4 (2 c d-b e)}{d^3 (c d-b e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 145, normalized size = 1.01 \begin {gather*} -\frac {1}{b^2 d^2 x}-\frac {c^3}{b^2 (c d-b e)^2 (b+c x)}-\frac {e^3}{d^2 (c d-b e)^2 (d+e x)}-\frac {2 (c d+b e) \log (x)}{b^3 d^3}+\frac {2 c^3 (-c d+2 b e) \log (b+c x)}{b^3 (-c d+b e)^3}+\frac {2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 146, normalized size = 1.01
method | result | size |
default | \(-\frac {c^{3}}{\left (b e -c d \right )^{2} b^{2} \left (c x +b \right )}+\frac {2 c^{3} \left (2 b e -c d \right ) \ln \left (c x +b \right )}{\left (b e -c d \right )^{3} b^{3}}-\frac {1}{x \,b^{2} d^{2}}+\frac {\left (-2 b e -2 c d \right ) \ln \left (x \right )}{b^{3} d^{3}}-\frac {e^{3}}{\left (b e -c d \right )^{2} d^{2} \left (e x +d \right )}+\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (b e -c d \right )^{3} d^{3}}\) | \(146\) |
norman | \(\frac {\frac {\left (2 b^{4} e^{4}-b^{3} c d \,e^{3}-b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x^{2}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {\left (2 b^{3} e^{3}-b^{2} d \,e^{2} c -b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) c e \,x^{3}}{d^{3} b^{3} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}-\frac {1}{b d}}{x \left (c x +b \right ) \left (e x +d \right )}-\frac {2 \left (b e +c d \right ) \ln \left (x \right )}{b^{3} d^{3}}+\frac {2 c^{3} \left (2 b e -c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {2 e^{3} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}\) | \(309\) |
risch | \(\frac {-\frac {2 c e \left (b^{2} e^{2}-b c d e +d^{2} c^{2}\right ) x^{2}}{b^{2} d^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}-\frac {\left (2 b^{3} e^{3}-b^{2} d \,e^{2} c -b \,c^{2} d^{2} e +2 c^{3} d^{3}\right ) x}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}-\frac {1}{b d}}{x \left (c x +b \right ) \left (e x +d \right )}+\frac {2 e^{4} \ln \left (-e x -d \right ) b}{d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {4 e^{3} \ln \left (-e x -d \right ) c}{d^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {4 c^{3} \ln \left (c x +b \right ) e}{b^{2} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {2 c^{4} \ln \left (c x +b \right ) d}{b^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {2 \ln \left (-x \right ) e}{b^{2} d^{3}}-\frac {2 \ln \left (-x \right ) c}{b^{3} d^{2}}\) | \(399\) |
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. 368 vs.
\(2 (151) = 302\).
time = 0.29, size = 368, normalized size = 2.56 \begin {gather*} \frac {2 \, {\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac {2 \, {\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (x e + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} + {\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} + {\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} + {\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac {2 \, {\left (c d + b e\right )} \log \left (x\right )}{b^{3} d^{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. 663 vs.
\(2 (151) = 302\).
time = 17.40, size = 663, normalized size = 4.60 \begin {gather*} -\frac {2 \, b c^{4} d^{5} x + b^{2} c^{3} d^{5} - 2 \, {\left (b^{4} c d x^{2} + b^{5} d x\right )} e^{4} + {\left (4 \, b^{3} c^{2} d^{2} x^{2} + 3 \, b^{4} c d^{2} x - b^{5} d^{2}\right )} e^{3} - {\left (4 \, b^{2} c^{3} d^{3} x^{2} - 3 \, b^{4} c d^{3}\right )} e^{2} + {\left (2 \, b c^{4} d^{4} x^{2} - 3 \, b^{2} c^{3} d^{4} x - 3 \, b^{3} c^{2} d^{4}\right )} e - 2 \, {\left (c^{5} d^{5} x^{2} + b c^{4} d^{5} x - 2 \, {\left (b c^{4} d^{3} x^{3} + b^{2} c^{3} d^{3} x^{2}\right )} e^{2} + {\left (c^{5} d^{4} x^{3} - b c^{4} d^{4} x^{2} - 2 \, b^{2} c^{3} d^{4} x\right )} e\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (b^{4} c x^{3} + b^{5} x^{2}\right )} e^{5} - {\left (2 \, b^{3} c^{2} d x^{3} + b^{4} c d x^{2} - b^{5} d x\right )} e^{4} - 2 \, {\left (b^{3} c^{2} d^{2} x^{2} + b^{4} c d^{2} x\right )} e^{3}\right )} \log \left (x e + d\right ) + 2 \, {\left (c^{5} d^{5} x^{2} + b c^{4} d^{5} x - {\left (b^{4} c x^{3} + b^{5} x^{2}\right )} e^{5} + {\left (2 \, b^{3} c^{2} d x^{3} + b^{4} c d x^{2} - b^{5} d x\right )} e^{4} + 2 \, {\left (b^{3} c^{2} d^{2} x^{2} + b^{4} c d^{2} x\right )} e^{3} - 2 \, {\left (b c^{4} d^{3} x^{3} + b^{2} c^{3} d^{3} x^{2}\right )} e^{2} + {\left (c^{5} d^{4} x^{3} - b c^{4} d^{4} x^{2} - 2 \, b^{2} c^{3} d^{4} x\right )} e\right )} \log \left (x\right )}{b^{3} c^{4} d^{7} x^{2} + b^{4} c^{3} d^{7} x - {\left (b^{6} c d^{3} x^{3} + b^{7} d^{3} x^{2}\right )} e^{4} + {\left (3 \, b^{5} c^{2} d^{4} x^{3} + 2 \, b^{6} c d^{4} x^{2} - b^{7} d^{4} x\right )} e^{3} - 3 \, {\left (b^{4} c^{3} d^{5} x^{3} - b^{6} c d^{5} x\right )} e^{2} + {\left (b^{3} c^{4} d^{6} x^{3} - 2 \, b^{4} c^{3} d^{6} x^{2} - 3 \, b^{5} c^{2} d^{6} x\right )} e} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 555 vs.
\(2 (151) = 302\).
time = 1.17, size = 555, normalized size = 3.85 \begin {gather*} \frac {{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} e^{\left (-2\right )} \log \left (\frac {{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} - {\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} + {\left | b \right |} e^{2} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left | b \right |}} - \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left ({\left | -c + \frac {2 \, c d}{x e + d} - \frac {c d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac {e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} {\left (x e + d\right )}} - \frac {\frac {2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac {{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )} {\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 304, normalized size = 2.11 \begin {gather*} -\frac {\frac {1}{b\,d}+\frac {2\,x^2\,\left (b^2\,c\,e^3-b\,c^2\,d\,e^2+c^3\,d^2\,e\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}+\frac {x\,\left (b\,e+c\,d\right )\,\left (2\,b^2\,e^2-3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^2\,d^2\,\left (b^2\,e^2-2\,b\,c\,d\,e+c^2\,d^2\right )}}{c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+b\,d\,x}-\frac {\ln \left (b+c\,x\right )\,\left (2\,c^4\,d-4\,b\,c^3\,e\right )}{b^6\,e^3-3\,b^5\,c\,d\,e^2+3\,b^4\,c^2\,d^2\,e-b^3\,c^3\,d^3}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b\,e^4-4\,c\,d\,e^3\right )}{-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}-\frac {2\,\ln \left (x\right )\,\left (b\,e+c\,d\right )}{b^3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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