Optimal. Leaf size=81 \[ -\frac {b}{(b d-a e)^2 (a+b x)}-\frac {e}{(b d-a e)^2 (d+e x)}-\frac {2 b e \log (a+b x)}{(b d-a e)^3}+\frac {2 b e \log (d+e x)}{(b d-a e)^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 46}
\begin {gather*} -\frac {b}{(a+b x) (b d-a e)^2}-\frac {e}{(d+e x) (b d-a e)^2}-\frac {2 b e \log (a+b x)}{(b d-a e)^3}+\frac {2 b e \log (d+e x)}{(b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 46
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x)^2 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^2}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)}+\frac {e^2}{(b d-a e)^2 (d+e x)^2}+\frac {2 b e^2}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {b}{(b d-a e)^2 (a+b x)}-\frac {e}{(b d-a e)^2 (d+e x)}-\frac {2 b e \log (a+b x)}{(b d-a e)^3}+\frac {2 b e \log (d+e x)}{(b d-a e)^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 66, normalized size = 0.81 \begin {gather*} \frac {\frac {b (-b d+a e)}{a+b x}+\frac {e (-b d+a e)}{d+e x}-2 b e \log (a+b x)+2 b e \log (d+e x)}{(b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 82, normalized size = 1.01
method | result | size |
default | \(-\frac {b}{\left (a e -b d \right )^{2} \left (b x +a \right )}+\frac {2 b e \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}-\frac {e}{\left (a e -b d \right )^{2} \left (e x +d \right )}-\frac {2 b e \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}\) | \(82\) |
risch | \(\frac {-\frac {2 b e x}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {a e +b d}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}}{\left (b x +a \right ) \left (e x +d \right )}-\frac {2 b e \ln \left (e x +d \right )}{e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {2 b e \ln \left (-b x -a \right )}{e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}\) | \(177\) |
norman | \(\frac {\frac {-a b \,e^{2}-b^{2} d e}{e b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {2 b e x}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}}{\left (b x +a \right ) \left (e x +d \right )}+\frac {2 b e \ln \left (b x +a \right )}{e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {2 b e \ln \left (e x +d \right )}{e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}\) | \(187\) |
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. 209 vs.
\(2 (90) = 180\).
time = 0.29, size = 209, normalized size = 2.58 \begin {gather*} -\frac {2 \, b e \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e \log \left (x e + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {2 \, b x e + b d + a e}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \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. 236 vs.
\(2 (90) = 180\).
time = 3.00, size = 236, normalized size = 2.91 \begin {gather*} -\frac {2 \, b^{2} d x e + b^{2} d^{2} - {\left (2 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )} \log \left (x e + d\right )}{b^{4} d^{4} x + a b^{3} d^{4} - {\left (a^{3} b x^{2} + a^{4} x\right )} e^{4} + {\left (3 \, a^{2} b^{2} d x^{2} + 2 \, a^{3} b d x - a^{4} d\right )} e^{3} - 3 \, {\left (a b^{3} d^{2} x^{2} - a^{3} b d^{2}\right )} e^{2} + {\left (b^{4} d^{3} x^{2} - 2 \, a b^{3} d^{3} x - 3 \, a^{2} b^{2} d^{3}\right )} e} \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. 406 vs.
\(2 (70) = 140\).
time = 0.64, size = 406, normalized size = 5.01 \begin {gather*} - \frac {2 b e \log {\left (x + \frac {- \frac {2 a^{4} b e^{5}}{\left (a e - b d\right )^{3}} + \frac {8 a^{3} b^{2} d e^{4}}{\left (a e - b d\right )^{3}} - \frac {12 a^{2} b^{3} d^{2} e^{3}}{\left (a e - b d\right )^{3}} + \frac {8 a b^{4} d^{3} e^{2}}{\left (a e - b d\right )^{3}} + 2 a b e^{2} - \frac {2 b^{5} d^{4} e}{\left (a e - b d\right )^{3}} + 2 b^{2} d e}{4 b^{2} e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac {2 b e \log {\left (x + \frac {\frac {2 a^{4} b e^{5}}{\left (a e - b d\right )^{3}} - \frac {8 a^{3} b^{2} d e^{4}}{\left (a e - b d\right )^{3}} + \frac {12 a^{2} b^{3} d^{2} e^{3}}{\left (a e - b d\right )^{3}} - \frac {8 a b^{4} d^{3} e^{2}}{\left (a e - b d\right )^{3}} + 2 a b e^{2} + \frac {2 b^{5} d^{4} e}{\left (a e - b d\right )^{3}} + 2 b^{2} d e}{4 b^{2} e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac {- a e - b d - 2 b e x}{a^{3} d e^{2} - 2 a^{2} b d^{2} e + a b^{2} d^{3} + x^{2} \left (a^{2} b e^{3} - 2 a b^{2} d e^{2} + b^{3} d^{2} e\right ) + x \left (a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e + b^{3} d^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.47, size = 156, normalized size = 1.93 \begin {gather*} -\frac {2 \, b e^{2} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} - \frac {e^{3}}{{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (x e + d\right )}} - \frac {b^{2} e}{{\left (b d - a e\right )}^{3} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 182, normalized size = 2.25 \begin {gather*} \frac {4\,b\,e\,\mathrm {atanh}\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{{\left (a\,e-b\,d\right )}^3}+\frac {2\,b\,e\,x\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3}\right )}{{\left (a\,e-b\,d\right )}^3}-\frac {\frac {a\,e+b\,d}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}+\frac {2\,b\,e\,x}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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