Optimal. Leaf size=61 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac {2}{15} b c^{5/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {2 b c \sqrt {1-c^2 x^4}}{15 x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 325, 221} \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}-\frac {2 b c \sqrt {1-c^2 x^4}}{15 x^3}+\frac {2}{15} b c^{5/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 325
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c x^2\right )}{x^6} \, dx &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} b \int \frac {2 c}{x^4 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{5} (2 b c) \int \frac {1}{x^4 \sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{15 x^3}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac {1}{15} \left (2 b c^3\right ) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx\\ &=-\frac {2 b c \sqrt {1-c^2 x^4}}{15 x^3}-\frac {a+b \sin ^{-1}\left (c x^2\right )}{5 x^5}+\frac {2}{15} b c^{5/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.15, size = 72, normalized size = 1.18 \[ -\frac {3 a+2 b c x^2 \sqrt {1-c^2 x^4}-2 i b (-c)^{5/2} x^5 F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )+3 b \sin ^{-1}\left (c x^2\right )}{15 x^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x^{2}\right ) + a}{x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 1.43 \[ -\frac {a}{5 x^{5}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{5 x^{5}}+\frac {2 c \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{3 x^{3}}+\frac {c^{\frac {3}{2}} \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{3 \sqrt {-c^{2} x^{4}+1}}\right )}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (2 \, c x^{5} \int \frac {e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{c^{4} x^{12} - c^{2} x^{8} - {\left (c^{2} x^{8} - x^{4}\right )} {\left (c x^{2} + 1\right )} {\left (c x^{2} - 1\right )}}\,{d x} + \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right )\right )} b}{5 \, x^{5}} - \frac {a}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 61, normalized size = 1.00 \[ - \frac {a}{5 x^{5}} + \frac {b c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{10 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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