Optimal. Leaf size=75 \[ -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {626, 44} \begin {gather*} -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 626
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {1}{(a e+c d x)^2 (d+e x)} \, dx\\ &=\int \left (\frac {c d}{\left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac {e^2}{\left (c d^2-a e^2\right )^2 (d+e x)}\right ) \, dx\\ &=-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac {e \log (d+e x)}{\left (c d^2-a e^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 74, normalized size = 0.99 \begin {gather*} \frac {1}{\left (a e^2-c d^2\right ) (a e+c d x)}-\frac {e \log (a e+c d x)}{\left (a e^2-c d^2\right )^2}+\frac {e \log (d+e x)}{\left (a e^2-c d^2\right )^2} \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 {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 116, normalized size = 1.55 \begin {gather*} -\frac {c d^{2} - a e^{2} + {\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right ) - {\left (c d e x + a e^{2}\right )} \log \left (e x + d\right )}{a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} + {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 194, normalized size = 2.59 \begin {gather*} -\frac {2 \, {\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac {c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 75, normalized size = 1.00 \begin {gather*} \frac {e \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}-\frac {e \ln \left (c d x +a e \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {1}{\left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 113, normalized size = 1.51 \begin {gather*} -\frac {e \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {e \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {1}{a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 96, normalized size = 1.28 \begin {gather*} \frac {1}{\left (a\,e+c\,d\,x\right )\,\left (a\,e^2-c\,d^2\right )}-\frac {2\,e\,\mathrm {atanh}\left (\frac {a^2\,e^4-c^2\,d^4}{{\left (a\,e^2-c\,d^2\right )}^2}+\frac {2\,c\,d\,e\,x}{a\,e^2-c\,d^2}\right )}{{\left (a\,e^2-c\,d^2\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.79, size = 287, normalized size = 3.83 \begin {gather*} \frac {e \log {\left (x + \frac {- \frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} + \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {e \log {\left (x + \frac {\frac {a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} - \frac {c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {1}{a^{2} e^{3} - a c d^{2} e + x \left (a c d e^{2} - c^{2} d^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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