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 = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 12
Rule 264
Rule 4842
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.48, size = 42, normalized size = 1.02 \[ \frac {a x^{4} - \sqrt {-c^{2} x^{4} + 1} b c x^{2} - b \arcsin \left (c x^{2}\right ) - a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 176, normalized size = 4.29 \[ -\frac {\frac {b c^{5} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {a c^{5} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac {2 \, b c^{4} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} + 2 \, b c^{3} \arcsin \left (c x^{2}\right ) + 2 \, a c^{3} + \frac {2 \, b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac {a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}}}{16 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 1.32 \[ -\frac {a}{4 x^{4}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{4 x^{4}}+\frac {c \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right )}{4 x^{2} \sqrt {-c^{2} x^{4}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 38, normalized size = 0.93 \[ -\frac {1}{4} \, b {\left (\frac {\sqrt {-c^{2} x^{4} + 1} c}{x^{2}} + \frac {\arcsin \left (c x^{2}\right )}{x^{4}}\right )} - \frac {a}{4 \, x^{4}} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 70, normalized size = 1.71 \[ - \frac {a}{4 x^{4}} + \frac {b c \left (\begin {cases} - \frac {i \sqrt {c^{2} x^{4} - 1}}{2 x^{2}} & \text {for}\: \left |{c^{2} x^{4}}\right | > 1 \\- \frac {\sqrt {- c^{2} x^{4} + 1}}{2 x^{2}} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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