Optimal. Leaf size=174 \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt{b^2-4 a c}}+\frac{\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac{b d-a e}{2 a^2 x^2}-\frac{d}{4 a x^4} \]
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Rubi [A] time = 0.40734, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1663, 1628, 634, 618, 206, 628} \[ -\frac{\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt{b^2-4 a c}}+\frac{\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac{b d-a e}{2 a^2 x^2}-\frac{d}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1628
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x+f x^2}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d}{a x^3}+\frac{-b d+a e}{a^2 x^2}+\frac{b^2 d-a b e-a (c d-a f)}{a^3 x}+\frac{-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{d}{4 a x^4}+\frac{b d-a e}{2 a^2 x^2}+\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}+\frac{\operatorname{Subst}\left (\int \frac{-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3}\\ &=-\frac{d}{4 a x^4}+\frac{b d-a e}{2 a^2 x^2}+\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}-\frac{\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}\\ &=-\frac{d}{4 a x^4}+\frac{b d-a e}{2 a^2 x^2}+\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac{\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^3}\\ &=-\frac{d}{4 a x^4}+\frac{b d-a e}{2 a^2 x^2}+\frac{\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \sqrt{b^2-4 a c}}+\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac{\left (b^2 d-a b e-a (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.35299, size = 314, normalized size = 1.8 \[ -\frac{\frac{a^2 d}{x^4}+\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a b \left (-e \sqrt{b^2-4 a c}+a f-3 c d\right )+a \left (-c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+2 a c e\right )+b^2 \left (d \sqrt{b^2-4 a c}-a e\right )+b^3 d\right )}{\sqrt{b^2-4 a c}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-a b \left (e \sqrt{b^2-4 a c}+a f-3 c d\right )+a \left (a f \sqrt{b^2-4 a c}-c \left (d \sqrt{b^2-4 a c}+2 a e\right )\right )+b^2 \left (d \sqrt{b^2-4 a c}+a e\right )+b^3 (-d)\right )}{\sqrt{b^2-4 a c}}-4 \log (x) \left (-a b e+a (a f-c d)+b^2 d\right )+\frac{2 a (a e-b d)}{x^2}}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 356, normalized size = 2.1 \begin{align*} -{\frac{d}{4\,a{x}^{4}}}-{\frac{e}{2\,a{x}^{2}}}+{\frac{bd}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ) f}{a}}-{\frac{\ln \left ( x \right ) be}{{a}^{2}}}-{\frac{\ln \left ( x \right ) cd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}d}{{a}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) f}{4\,a}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) be}{4\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{a}^{3}}}-{\frac{bf}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{ce}{a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{e{b}^{2}}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,bcd}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}d}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 5.26681, size = 1283, normalized size = 7.37 \begin{align*} \left [\frac{{\left (a^{2} b f +{\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt{b^{2} - 4 \, a c} x^{4} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c +{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (x\right ) + 2 \,{\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}, \frac{2 \,{\left (a^{2} b f +{\left (b^{3} - 3 \, a b c\right )} d -{\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt{-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d -{\left (a b^{3} - 4 \, a^{2} b c\right )} e +{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (x\right ) + 2 \,{\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} -{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \,{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13726, size = 286, normalized size = 1.64 \begin{align*} -\frac{{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac{{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{{\left (b^{3} d - 3 \, a b c d + a^{2} b f - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{3}} - \frac{3 \, b^{2} d x^{4} - 3 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 3 \, a b x^{4} e - 2 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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