Optimal. Leaf size=81 \[ -\frac {b}{(a+b x) (b d-a e)^2}-\frac {e}{(d+e x) (b d-a e)^2}-\frac {2 b e \log (a+b x)}{(b d-a e)^3}+\frac {2 b e \log (d+e x)}{(b d-a e)^3} \]
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Rubi [A] time = 0.05, 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, 44} \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 44
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 = 0.07, size = 66, normalized size = 0.81 \begin {gather*} \frac {\frac {b (a e-b d)}{a+b x}+\frac {e (a e-b d)}{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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 241, normalized size = 2.98 \begin {gather*} -\frac {b^{2} d^{2} - a^{2} e^{2} + 2 \, {\left (b^{2} d e - a b e^{2}\right )} x + 2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} + {\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, 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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maple [A] time = 0.06, size = 82, normalized size = 1.01 \begin {gather*} \frac {2 b e \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}-\frac {2 b e \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}-\frac {b}{\left (a e -b d \right )^{2} \left (b x +a \right )}-\frac {e}{\left (a e -b d \right )^{2} \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 208, normalized size = 2.57 \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 (e x + 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 e x + 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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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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sympy [B] time = 1.15, 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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