Optimal. Leaf size=103 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{f x^2}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.178958, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1663, 1657, 634, 618, 206, 628} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{f x^2}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1663
Rule 1657
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x+f x^2}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{f}{c}+\frac{c d-a f+(c e-b f) x}{c \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{f x^2}{2 c}+\frac{\operatorname{Subst}\left (\int \frac{c d-a f+(c e-b f) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c}\\ &=\frac{f x^2}{2 c}+\frac{(c e-b f) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}+\frac{\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{f x^2}{2 c}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}-\frac{\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2}\\ &=\frac{f x^2}{2 c}-\frac{\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}+\frac{(c e-b f) \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.068704, size = 100, normalized size = 0.97 \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )}{\sqrt{4 a c-b^2}}+(c e-b f) \log \left (a+b x^2+c x^4\right )+2 c f x^2}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.005, size = 211, normalized size = 2.1 \begin{align*}{\frac{f{x}^{2}}{2\,c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,c}}-{\frac{af}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{d\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}f}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{2\,c}\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.
[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 = 1.74658, size = 691, normalized size = 6.71 \begin{align*} \left [\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} -{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} \sqrt{b^{2} - 4 \, a c} \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^{2} c - 4 \, a c^{2}\right )} e -{\left (b^{3} - 4 \, a b c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} - 2 \,{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} \sqrt{-b^{2} + 4 \, a c} \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^{2} c - 4 \, a c^{2}\right )} e -{\left (b^{3} - 4 \, a b c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 11.0933, size = 498, normalized size = 4.83 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- a b f - 8 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) + 2 a c e + 2 b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) - b c d}{2 a c f - b^{2} f + b c e - 2 c^{2} d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- a b f - 8 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) + 2 a c e + 2 b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{4 c^{2} \left (4 a c - b^{2}\right )} - \frac{b f - c e}{4 c^{2}}\right ) - b c d}{2 a c f - b^{2} f + b c e - 2 c^{2} d} \right )} + \frac{f x^{2}}{2 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14739, size = 134, normalized size = 1.3 \begin{align*} \frac{f x^{2}}{2 \, c} - \frac{{\left (b f - c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac{{\left (2 \, c^{2} d + b^{2} f - 2 \, a c f - b 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} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]