Optimal. Leaf size=133 \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.123315, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1247, 705, 31, 634, 618, 206, 628} \[ -\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1247
Rule 705
Rule 31
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{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}{(d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{c d-b e-c e x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac{e \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}+\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}-\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}+\frac{e \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}-\frac{e \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0688062, size = 112, normalized size = 0.84 \[ \frac{e \sqrt{4 a c-b^2} \left (\log \left (a+b x^2+c x^4\right )-2 \log \left (d+e x^2\right )\right )+(2 b e-4 c d) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{4 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 176, normalized size = 1.3 \begin{align*} -{\frac{e\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a{e}^{2}-4\,deb+4\,c{d}^{2}}}-{\frac{be}{2\,a{e}^{2}-2\,deb+2\,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{cd}{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{e\ln \left ( e{x}^{2}+d \right ) }{2\,a{e}^{2}-2\,deb+2\,c{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 78.3602, size = 718, normalized size = 5.4 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} e \log \left (e x^{2} + d\right ) + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c d - b e\right )} \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 )}{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, -\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{4} + b x^{2} + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} e \log \left (e x^{2} + d\right ) + 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c d - b e\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1654, size = 181, normalized size = 1.36 \begin{align*} -\frac{e \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (c d^{2} - b d e + a e^{2}\right )}} + \frac{e^{2} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]