Optimal. Leaf size=83 \[ \frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {c d}{a^2 x}+\frac {c e \log \left (a+c x^2\right )}{2 a^2}-\frac {c e \log (x)}{a^2}-\frac {d}{3 a x^3}-\frac {e}{2 a x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 83, 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^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {c d}{a^2 x}+\frac {c e \log \left (a+c x^2\right )}{2 a^2}-\frac {c e \log (x)}{a^2}-\frac {d}{3 a x^3}-\frac {e}{2 a x^2} \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^4 \left (a+c x^2\right )} \, dx &=\int \left (\frac {d}{a x^4}+\frac {e}{a x^3}-\frac {c d}{a^2 x^2}-\frac {c e}{a^2 x}+\frac {c^2 (d+e x)}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac {d}{3 a x^3}-\frac {e}{2 a x^2}+\frac {c d}{a^2 x}-\frac {c e \log (x)}{a^2}+\frac {c^2 \int \frac {d+e x}{a+c x^2} \, dx}{a^2}\\ &=-\frac {d}{3 a x^3}-\frac {e}{2 a x^2}+\frac {c d}{a^2 x}-\frac {c e \log (x)}{a^2}+\frac {\left (c^2 d\right ) \int \frac {1}{a+c x^2} \, dx}{a^2}+\frac {\left (c^2 e\right ) \int \frac {x}{a+c x^2} \, dx}{a^2}\\ &=-\frac {d}{3 a x^3}-\frac {e}{2 a x^2}+\frac {c d}{a^2 x}+\frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {c e \log (x)}{a^2}+\frac {c e \log \left (a+c x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.93 \begin {gather*} \frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {-3 c e x^3 \log \left (a+c x^2\right )+2 a d+3 a e x-6 c d x^2+6 c e x^3 \log (x)}{6 a^2 x^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 {d+e x}{x^4 \left (a+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 170, normalized size = 2.05 \begin {gather*} \left [\frac {3 \, c d x^{3} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} + 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \relax (x) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}, \frac {6 \, c d x^{3} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \relax (x) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 76, normalized size = 0.92 \begin {gather*} \frac {c^{2} d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} a^{2}} + \frac {c e \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c e \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {6 \, c d x^{2} - 3 \, a x e - 2 \, a d}{6 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 75, normalized size = 0.90 \begin {gather*} \frac {c^{2} d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, a^{2}}-\frac {c e \ln \relax (x )}{a^{2}}+\frac {c e \ln \left (c \,x^{2}+a \right )}{2 a^{2}}+\frac {c d}{a^{2} x}-\frac {e}{2 a \,x^{2}}-\frac {d}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 72, normalized size = 0.87 \begin {gather*} \frac {c^{2} d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} a^{2}} + \frac {c e \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c e \log \relax (x)}{a^{2}} + \frac {6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 177, normalized size = 2.13 \begin {gather*} \frac {\ln \left (d\,\sqrt {-a^5\,c^3}-3\,e\,x\,\sqrt {-a^5\,c^3}+3\,a^3\,c\,e+a^2\,c^2\,d\,x\right )\,\left (d\,\sqrt {-a^5\,c^3}+a^3\,c\,e\right )}{2\,a^5}-\frac {\frac {d}{3\,a}+\frac {e\,x}{2\,a}-\frac {c\,d\,x^2}{a^2}}{x^3}-\frac {\ln \left (3\,e\,x\,\sqrt {-a^5\,c^3}-d\,\sqrt {-a^5\,c^3}+3\,a^3\,c\,e+a^2\,c^2\,d\,x\right )\,\left (d\,\sqrt {-a^5\,c^3}-a^3\,c\,e\right )}{2\,a^5}-\frac {c\,e\,\ln \relax (x)}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.77, size = 408, normalized size = 4.92 \begin {gather*} \left (\frac {c e}{2 a^{2}} - \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) \log {\left (x + \frac {12 a^{6} e \left (\frac {c e}{2 a^{2}} - \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac {c e}{2 a^{2}} - \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac {c e}{2 a^{2}} - \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} + \left (\frac {c e}{2 a^{2}} + \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) \log {\left (x + \frac {12 a^{6} e \left (\frac {c e}{2 a^{2}} + \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac {c e}{2 a^{2}} + \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac {c e}{2 a^{2}} + \frac {d \sqrt {- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} - \frac {c e \log {\relax (x )}}{a^{2}} + \frac {- 2 a d - 3 a e x + 6 c d x^{2}}{6 a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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