Optimal. Leaf size=167 \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2-b d e+c d^2\right )}-\frac{(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (a e^2-b d e+c d^2\right )}+\frac{\log (x)}{a d} \]
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Rubi [A] time = 0.305797, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1251, 893, 634, 618, 206, 628} \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2-b d e+c d^2\right )}-\frac{(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (a e^2-b d e+c d^2\right )}+\frac{\log (x)}{a d} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 893
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a d x}-\frac{e^3}{d \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{-b c d+b^2 e-a c e-c (c d-b e) x}{a \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\log (x)}{a d}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (c d^2-b d e+a e^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-b c d+b^2 e-a c e-c (c d-b e) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2-b d e+a e^2\right )}\\ &=\frac{\log (x)}{a d}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (c d^2-b d e+a e^2\right )}-\frac{(c d-b e) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2-b d e+a e^2\right )}-\frac{\left (b c d-b^2 e+2 a c e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2-b d e+a e^2\right )}\\ &=\frac{\log (x)}{a d}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (c d^2-b d e+a e^2\right )}-\frac{(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (c d^2-b d e+a e^2\right )}+\frac{\left (b c d-b^2 e+2 a c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a \left (c d^2-b d e+a e^2\right )}\\ &=\frac{\left (b c d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac{\log (x)}{a d}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (c d^2-b d e+a e^2\right )}-\frac{(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.324116, size = 242, normalized size = 1.45 \[ \frac{4 \log (x) \sqrt{b^2-4 a c} \left (e (a e-b d)+c d^2\right )-2 a e^2 \sqrt{b^2-4 a c} \log \left (d+e x^2\right )-d \left (c d \sqrt{b^2-4 a c}-b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )+d \left (-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{4 a d \sqrt{b^2-4 a c} \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 298, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) be}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) a}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) d}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) a}}-{\frac{ec}{a{e}^{2}-deb+c{d}^{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}^{2}e}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bcd}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ) a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{\ln \left ( x \right ) }{ad}}-{\frac{{e}^{2}\ln \left ( e{x}^{2}+d \right ) }{2\,d \left ( a{e}^{2}-deb+c{d}^{2} \right ) }} \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 [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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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 = 1.14009, size = 232, normalized size = 1.39 \begin{align*} -\frac{{\left (c d - b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (a c d^{2} - a b d e + a^{2} e^{2}\right )}} - \frac{e^{3} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}} - \frac{{\left (b c d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{\log \left (x^{2}\right )}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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