Optimal. Leaf size=49 \[ \frac{2 b \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )}{\sqrt{c}}+a x+b x \sin ^{-1}\left (c x^2\right )-\frac{2 b E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}} \]
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Rubi [A] time = 0.0426012, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4840, 12, 307, 221, 1199, 424} \[ a x+b x \sin ^{-1}\left (c x^2\right )+\frac{2 b F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}}-\frac{2 b E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 4840
Rule 12
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-b \int \frac{2 c x^2}{\sqrt{1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-(2 b c) \int \frac{x^2}{\sqrt{1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )+(2 b) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx-(2 b) \int \frac{1+c x^2}{\sqrt{1-c^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )+\frac{2 b F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}}-(2 b) \int \frac{\sqrt{1+c x^2}}{\sqrt{1-c x^2}} \, dx\\ &=a x+b x \sin ^{-1}\left (c x^2\right )-\frac{2 b E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}}+\frac{2 b F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [C] time = 0.0051358, size = 39, normalized size = 0.8 \[ -\frac{2}{3} b c x^3 \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},c^2 x^4\right )+a x+b x \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 71, normalized size = 1.5 \begin{align*} ax+b \left ( x\arcsin \left ( c{x}^{2} \right ) +2\,{\frac{\sqrt{-c{x}^{2}+1}\sqrt{c{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{c},i \right ) -{\it EllipticE} \left ( x\sqrt{c},i \right ) \right ) }{\sqrt{c}\sqrt{-{c}^{2}{x}^{4}+1}}} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b \arcsin \left (c x^{2}\right ) + a, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13843, size = 49, normalized size = 1. \begin{align*} a x + b \left (- \frac{c x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{c^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} + x \operatorname{asin}{\left (c x^{2} \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int b \arcsin \left (c x^{2}\right ) + a\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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