Optimal. Leaf size=86 \[ -\frac {e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac {e \log (d+e x)}{a e^2+c d^2}+\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {706, 31, 635, 205, 260} \[ -\frac {e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac {e \log (d+e x)}{a e^2+c d^2}+\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 205
Rule 260
Rule 635
Rule 706
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+c x^2\right )} \, dx &=\frac {\int \frac {c d-c e x}{a+c x^2} \, dx}{c d^2+a e^2}+\frac {e^2 \int \frac {1}{d+e x} \, dx}{c d^2+a e^2}\\ &=\frac {e \log (d+e x)}{c d^2+a e^2}+\frac {(c d) \int \frac {1}{a+c x^2} \, dx}{c d^2+a e^2}-\frac {(c e) \int \frac {x}{a+c x^2} \, dx}{c d^2+a e^2}\\ &=\frac {\sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )}+\frac {e \log (d+e x)}{c d^2+a e^2}-\frac {e \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.73 \[ \frac {\frac {2 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a}}-e \log \left (a+c x^2\right )+2 e \log (d+e x)}{2 a e^2+2 c d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 135, normalized size = 1.57 \[ \left [\frac {d \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}}, \frac {2 \, d \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 79, normalized size = 0.92 \[ \frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}} + \frac {e^{2} \log \left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 77, normalized size = 0.90 \[ \frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {a c}}-\frac {e \ln \left (c \,x^{2}+a \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )}+\frac {e \ln \left (e x +d \right )}{a \,e^{2}+c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 76, normalized size = 0.88 \[ \frac {c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {a c}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, {\left (c d^{2} + a e^{2}\right )}} + \frac {e \log \left (e x + d\right )}{c d^{2} + a e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 230, normalized size = 2.67 \[ \frac {e\,\ln \left (d+e\,x\right )}{c\,d^2+a\,e^2}-\frac {\ln \left (3\,c^2\,e^2\,x+c^2\,d\,e-\frac {c^2\,e\,\left (a\,e-d\,\sqrt {-a\,c}\right )\,\left (-c\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{a^2\,e^2+c\,a\,d^2}\right )\,\left (a\,e-d\,\sqrt {-a\,c}\right )}{2\,\left (a^2\,e^2+c\,a\,d^2\right )}-\frac {\ln \left (3\,c^2\,e^2\,x+c^2\,d\,e-\frac {c^2\,e\,\left (a\,e+d\,\sqrt {-a\,c}\right )\,\left (-c\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{a^2\,e^2+c\,a\,d^2}\right )\,\left (a\,e+d\,\sqrt {-a\,c}\right )}{2\,\left (a^2\,e^2+c\,a\,d^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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