Optimal. Leaf size=58 \[ \frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0836604, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3, 2018, 640, 620, 206} \[ \frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3
Rule 2018
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{2+2 a-2 (1+a)+b x^2+c x^4}} \, dx &=\int \frac{x^3}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{b x+c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=\frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c}\\ &=\frac{\sqrt{b x^2+c x^4}}{2 c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0313152, size = 73, normalized size = 1.26 \[ \frac{x \left (\sqrt{c} x \left (b+c x^2\right )-b \sqrt{b+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b+c x^2}}\right )\right )}{2 c^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 64, normalized size = 1.1 \begin{align*}{\frac{x}{2}\sqrt{c{x}^{2}+b} \left ( x\sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}-b\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{5}{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 = 1.47644, size = 265, normalized size = 4.57 \begin{align*} \left [\frac{b \sqrt{c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} c}{4 \, c^{2}}, \frac{b \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}{c x^{2} + b}\right ) + \sqrt{c x^{4} + b x^{2}} c}{2 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19506, size = 80, normalized size = 1.38 \begin{align*} \frac{b \log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2}}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{4} + b x^{2}}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]