Optimal. Leaf size=86 \[ -\frac{10 b \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )}{147 c^{7/2}}+\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{10 b x \sqrt{1-c^2 x^4}}{147 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0496963, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 321, 221} \[ \frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{10 b x \sqrt{1-c^2 x^4}}{147 c^3}-\frac{10 b F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{147 c^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4842
Rule 12
Rule 321
Rule 221
Rubi steps
\begin{align*} \int x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{7} b \int \frac{2 c x^8}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{1}{7} (2 b c) \int \frac{x^8}{\sqrt{1-c^2 x^4}} \, dx\\ &=\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{(10 b) \int \frac{x^4}{\sqrt{1-c^2 x^4}} \, dx}{49 c}\\ &=\frac{10 b x \sqrt{1-c^2 x^4}}{147 c^3}+\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{(10 b) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx}{147 c^3}\\ &=\frac{10 b x \sqrt{1-c^2 x^4}}{147 c^3}+\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{1}{7} x^7 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac{10 b F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )}{147 c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.200106, size = 82, normalized size = 0.95 \[ \frac{1}{147} \left (-\frac{10 i b \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c} x\right ),-1\right )}{(-c)^{7/2}}+21 a x^7+\frac{2 b x \sqrt{1-c^2 x^4} \left (3 c^2 x^4+5\right )}{c^3}+21 b x^7 \sin ^{-1}\left (c x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 108, normalized size = 1.3 \begin{align*}{\frac{{x}^{7}a}{7}}+b \left ({\frac{{x}^{7}\arcsin \left ( c{x}^{2} \right ) }{7}}-{\frac{2\,c}{7} \left ( -{\frac{{x}^{5}}{7\,{c}^{2}}\sqrt{-{c}^{2}{x}^{4}+1}}-{\frac{5\,x}{21\,{c}^{4}}\sqrt{-{c}^{2}{x}^{4}+1}}+{\frac{5}{21}\sqrt{-c{x}^{2}+1}\sqrt{c{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{c},i \right ){c}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) } \right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{6} \arcsin \left (c x^{2}\right ) + a x^{6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.85858, size = 58, normalized size = 0.67 \begin{align*} \frac{a x^{7}}{7} - \frac{b c x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{c^{2} x^{4} e^{2 i \pi }} \right )}}{14 \Gamma \left (\frac{13}{4}\right )} + \frac{b x^{7} \operatorname{asin}{\left (c x^{2} \right )}}{7} \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 (b \arcsin \left (c x^{2}\right ) + a\right )} x^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]