Optimal. Leaf size=106 \[ \frac{6}{35} b c^{7/2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{c} x\right ),-1\right )-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6}{35} b c^{7/2} E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right ) \]
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Rubi [A] time = 0.0742363, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4842, 12, 325, 307, 221, 1199, 424} \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}+\frac{6}{35} b c^{7/2} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )-\frac{6}{35} b c^{7/2} E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 325
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^8} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac{1}{7} b \int \frac{2 c}{x^6 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac{1}{7} (2 b c) \int \frac{1}{x^6 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac{1}{35} \left (6 b c^3\right ) \int \frac{1}{x^2 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}-\frac{1}{35} \left (6 b c^5\right ) \int \frac{x^2}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac{1}{35} \left (6 b c^4\right ) \int \frac{1}{\sqrt{1-c^2 x^4}} \, dx-\frac{1}{35} \left (6 b c^4\right ) \int \frac{1+c x^2}{\sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac{6}{35} b c^{7/2} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )-\frac{1}{35} \left (6 b c^4\right ) \int \frac{\sqrt{1+c x^2}}{\sqrt{1-c x^2}} \, dx\\ &=-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}-\frac{6}{35} b c^{7/2} E\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )+\frac{6}{35} b c^{7/2} F\left (\left .\sin ^{-1}\left (\sqrt{c} x\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.239822, size = 100, normalized size = 0.94 \[ -\frac{-6 i b (-c)^{7/2} x^7 \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-c} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c} x\right ),-1\right )\right )+5 a+2 b x^2 \sqrt{1-c^2 x^4} \left (3 c^3 x^4+c\right )+5 b \sin ^{-1}\left (c x^2\right )}{35 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 118, normalized size = 1.1 \begin{align*} -{\frac{a}{7\,{x}^{7}}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{7\,{x}^{7}}}+{\frac{2\,c}{7} \left ( -{\frac{1}{5\,{x}^{5}}\sqrt{-{c}^{2}{x}^{4}+1}}-{\frac{3\,{c}^{2}}{5\,x}\sqrt{-{c}^{2}{x}^{4}+1}}+{\frac{3}{5}{c}^{{\frac{5}{2}}}\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 ){\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) } \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 (\frac{b \arcsin \left (c x^{2}\right ) + a}{x^{8}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.61101, size = 65, normalized size = 0.61 \begin{align*} - \frac{a}{7 x^{7}} + \frac{b c \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{c^{2} x^{4} e^{2 i \pi }} \right )}}{14 x^{5} \Gamma \left (- \frac{1}{4}\right )} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{7 x^{7}} \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 \frac{b \arcsin \left (c x^{2}\right ) + a}{x^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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