Optimal. Leaf size=73 \[ -\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {c d \log \left (a+c x^2\right )}{2 a^2}-\frac {c d \log (x)}{a^2}-\frac {d}{2 a x^2}-\frac {e}{a x} \]
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Rubi [A] time = 0.06, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \begin {gather*} \frac {c d \log \left (a+c x^2\right )}{2 a^2}-\frac {c d \log (x)}{a^2}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {d}{2 a x^2}-\frac {e}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rubi steps
\begin {align*} \int \frac {d+e x}{x^3 \left (a+c x^2\right )} \, dx &=\int \left (\frac {d}{a x^3}+\frac {e}{a x^2}-\frac {c d}{a^2 x}-\frac {c (a e-c d x)}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {c d \log (x)}{a^2}-\frac {c \int \frac {a e-c d x}{a+c x^2} \, dx}{a^2}\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {c d \log (x)}{a^2}+\frac {\left (c^2 d\right ) \int \frac {x}{a+c x^2} \, dx}{a^2}-\frac {(c e) \int \frac {1}{a+c x^2} \, dx}{a}\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c d \log (x)}{a^2}+\frac {c d \log \left (a+c x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {c d \log \left (a+c x^2\right )}{2 a^2}-\frac {c d \log (x)}{a^2}-\frac {d}{2 a x^2}-\frac {e}{a x} \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}{x^3 \left (a+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 153, normalized size = 2.10 \begin {gather*} \left [\frac {a e x^{2} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + c d x^{2} \log \left (c x^{2} + a\right ) - 2 \, c d x^{2} \log \relax (x) - 2 \, a e x - a d}{2 \, a^{2} x^{2}}, -\frac {2 \, a e x^{2} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) - c d x^{2} \log \left (c x^{2} + a\right ) + 2 \, c d x^{2} \log \relax (x) + 2 \, a e x + a d}{2 \, a^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 66, normalized size = 0.90 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{\sqrt {a c} a} + \frac {c d \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {2 \, a x e + a d}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 65, normalized size = 0.89 \begin {gather*} -\frac {c e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, a}-\frac {c d \ln \relax (x )}{a^{2}}+\frac {c d \ln \left (c \,x^{2}+a \right )}{2 a^{2}}-\frac {e}{a x}-\frac {d}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 60, normalized size = 0.82 \begin {gather*} -\frac {c e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} a} + \frac {c d \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c d \log \relax (x)}{a^{2}} - \frac {2 \, e x + d}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 154, normalized size = 2.11 \begin {gather*} \frac {\ln \left (a\,e\,\sqrt {-a^5\,c}+3\,a^3\,c\,d-a^3\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}+a^2\,c\,d\right )}{2\,a^4}-\frac {\ln \left (a\,e\,\sqrt {-a^5\,c}-3\,a^3\,c\,d+a^3\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}-a^2\,c\,d\right )}{2\,a^4}-\frac {\frac {d}{2\,a}+\frac {e\,x}{a}}{x^2}-\frac {c\,d\,\ln \relax (x)}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.70, size = 360, normalized size = 4.93 \begin {gather*} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) \log {\left (x + \frac {- 12 a^{4} d \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} + \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) \log {\left (x + \frac {- 12 a^{4} d \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} + \frac {- d - 2 e x}{2 a x^{2}} - \frac {c d \log {\relax (x )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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