Optimal. Leaf size=73 \[ \frac {1}{(d+e x) \left (c d^2-a e^2\right )}+\frac {c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac {c d \log (d+e x)}{\left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 44} \begin {gather*} \frac {1}{(d+e x) \left (c d^2-a e^2\right )}+\frac {c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac {c d \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 {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^2} \, dx\\ &=\int \left (\frac {c^2 d^2}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {e}{\left (c d^2-a e^2\right ) (d+e x)^2}-\frac {c d e}{\left (c d^2-a e^2\right )^2 (d+e x)}\right ) \, dx\\ &=\frac {1}{\left (c d^2-a e^2\right ) (d+e x)}+\frac {c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac {c d \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 = 66, normalized size = 0.90 \begin {gather*} \frac {c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left (c d^2-a e^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 {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 109, normalized size = 1.49 \begin {gather*} \frac {c d^{2} - a e^{2} + {\left (c d e x + c d^{2}\right )} \log \left (c d x + a e\right ) - {\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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 = 1.03 \begin {gather*} -\frac {c d \ln \left (e x +d \right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {c d \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 (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 107, normalized size = 1.47 \begin {gather*} \frac {c d \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {c d \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {1}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 95, normalized size = 1.30 \begin {gather*} \frac {2\,c\,d\,\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}-\frac {1}{\left (a\,e^2-c\,d^2\right )\,\left (d+e\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.79, size = 301, normalized size = 4.12 \begin {gather*} - \frac {c d \log {\left (x + \frac {- \frac {a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} + \frac {c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {c d \log {\left (x + \frac {\frac {a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac {3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} - \frac {c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac {1}{a d e^{2} - c d^{3} + x \left (a e^{3} - c d^{2} e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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