Optimal. Leaf size=81 \[ \frac{418}{105} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+\frac{1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{315} x \left (669 x^2+1087\right ) \sqrt{-x^4+x^2+2}+\frac{4432}{315} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0554957, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1176, 1180, 524, 424, 419} \[ \frac{1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{315} x \left (669 x^2+1087\right ) \sqrt{-x^4+x^2+2}+\frac{418}{105} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{4432}{315} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx &=\frac{1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{1}{21} \int \left (-262-223 x^2\right ) \sqrt{2+x^2-x^4} \, dx\\ &=\frac{1}{315} x \left (1087+669 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}+\frac{1}{315} \int \frac{5686+4432 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{1}{315} x \left (1087+669 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}+\frac{2}{315} \int \frac{5686+4432 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{315} x \left (1087+669 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}+\frac{836}{105} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{4432}{315} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=\frac{1}{315} x \left (1087+669 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{63} x \left (48+35 x^2\right ) \left (2+x^2-x^4\right )^{3/2}+\frac{4432}{315} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{418}{105} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.006, size = 176, normalized size = 2.2 \begin{align*} -{\frac{5\,{x}^{7}}{9}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{13\,{x}^{5}}{63}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{1259\,{x}^{3}}{315}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{1567\,x}{315}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2843\,\sqrt{2}}{315}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{2216\,\sqrt{2}}{315}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (5 \, x^{6} + 2 \, x^{4} - 17 \, x^{2} - 14\right )} \sqrt{-x^{4} + x^{2} + 2}, x\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 \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]