Optimal. Leaf size=41 \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{4 x^4}-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2} \]
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Rubi [A] time = 0.026372, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4842, 12, 264} \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{4 x^4}-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 4842
Rule 12
Rule 264
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^5} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{4 x^4}+\frac{1}{4} b \int \frac{2 c}{x^3 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{4 x^4}+\frac{1}{2} (b c) \int \frac{1}{x^3 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0163505, size = 46, normalized size = 1.12 \[ -\frac{a}{4 x^4}-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2}-\frac{b \sin ^{-1}\left (c x^2\right )}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 54, normalized size = 1.3 \begin{align*} -{\frac{a}{4\,{x}^{4}}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{4\,{x}^{4}}}+{\frac{c \left ( c{x}^{2}-1 \right ) \left ( c{x}^{2}+1 \right ) }{4\,{x}^{2}}{\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41391, size = 51, normalized size = 1.24 \begin{align*} -\frac{1}{4} \, b{\left (\frac{\sqrt{-c^{2} x^{4} + 1} c}{x^{2}} + \frac{\arcsin \left (c x^{2}\right )}{x^{4}}\right )} - \frac{a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.50562, size = 92, normalized size = 2.24 \begin{align*} \frac{a x^{4} - \sqrt{-c^{2} x^{4} + 1} b c x^{2} - b \arcsin \left (c x^{2}\right ) - a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.38465, size = 70, normalized size = 1.71 \begin{align*} - \frac{a}{4 x^{4}} + \frac{b c \left (\begin{cases} - \frac{i \sqrt{c^{2} x^{4} - 1}}{2 x^{2}} & \text{for}\: \left |{c^{2} x^{4}}\right | > 1 \\- \frac{\sqrt{- c^{2} x^{4} + 1}}{2 x^{2}} & \text{otherwise} \end{cases}\right )}{2} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23014, size = 238, normalized size = 5.8 \begin{align*} -\frac{\frac{b c^{5} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac{a c^{5} x^{4}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac{2 \, b c^{4} x^{2}}{\sqrt{-c^{2} x^{4} + 1} + 1} + 2 \, b c^{3} \arcsin \left (c x^{2}\right ) + 2 \, a c^{3} + \frac{2 \, b c^{2}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac{b c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac{a c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}}}{16 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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