Optimal. Leaf size=106 \[ -\frac {a+b \sin ^{-1}\left (c x^2\right )}{7 x^7}+\frac {6}{35} b c^{7/2} F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {6}{35} b c^{7/2} E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )-\frac {2 b c \sqrt {1-c^2 x^4}}{35 x^5}-\frac {6 b c^3 \sqrt {1-c^2 x^4}}{35 x} \]
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Rubi [A] time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 12
Rule 221
Rule 307
Rule 325
Rule 424
Rule 1199
Rule 4842
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*}
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Mathematica [C] time = 0.25, size = 100, normalized size = 0.94 \[ -\frac {5 a+2 b x^2 \sqrt {1-c^2 x^4} \left (3 c^3 x^4+c\right )-6 i b (-c)^{7/2} x^7 \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )\right )+5 b \sin ^{-1}\left (c x^2\right )}{35 x^7} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (c x^{2}\right ) + a}{x^{8}}, x\right ) \]
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 \[ \int \frac {b \arcsin \left (c x^{2}\right ) + a}{x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 1.11 \[ -\frac {a}{7 x^{7}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{7 x^{7}}+\frac {2 c \left (-\frac {\sqrt {-c^{2} x^{4}+1}}{5 x^{5}}-\frac {3 c^{2} \sqrt {-c^{2} x^{4}+1}}{5 x}+\frac {3 c^{\frac {5}{2}} \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c}, i\right )-\EllipticE \left (x \sqrt {c}, i\right )\right )}{5 \sqrt {-c^{2} x^{4}+1}}\right )}{7}\right ) \]
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 \[ -\frac {{\left (2 \, c x^{7} \int \frac {e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{c^{4} x^{14} - c^{2} x^{10} - {\left (c^{2} x^{10} - x^{6}\right )} {\left (c x^{2} + 1\right )} {\left (c x^{2} - 1\right )}}\,{d x} + \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right )\right )} b}{7 \, x^{7}} - \frac {a}{7 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^8} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.08, size = 65, normalized size = 0.61 \[ - \frac {a}{7 x^{7}} + \frac {b c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{14 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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