Optimal. Leaf size=83 \[ \frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {6 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}-\frac {6 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}+\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c} \]
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Rubi [A] time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4842, 12, 321, 307, 221, 1199, 424} \[ \frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c}+\frac {6 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}-\frac {6 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 307
Rule 321
Rule 424
Rule 1199
Rule 4842
Rubi steps
\begin {align*} \int x^4 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{5} b \int \frac {2 c x^6}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{5} (2 b c) \int \frac {x^6}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c}+\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {(6 b) \int \frac {x^2}{\sqrt {1-c^2 x^4}} \, dx}{25 c}\\ &=\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c}+\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {(6 b) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx}{25 c^2}-\frac {(6 b) \int \frac {1+c x^2}{\sqrt {1-c^2 x^4}} \, dx}{25 c^2}\\ &=\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c}+\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )+\frac {6 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}-\frac {(6 b) \int \frac {\sqrt {1+c x^2}}{\sqrt {1-c x^2}} \, dx}{25 c^2}\\ &=\frac {2 b x^3 \sqrt {1-c^2 x^4}}{25 c}+\frac {1}{5} x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {6 b E\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}+\frac {6 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{25 c^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 93, normalized size = 1.12 \[ \frac {1}{25} \left (5 a x^5+\frac {2 b x^3 \sqrt {1-c^2 x^4}}{c}+5 b x^5 \sin ^{-1}\left (c x^2\right )+\frac {6 i b \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )\right )}{(-c)^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{4} \arcsin \left (c x^{2}\right ) + a x^{4}, x\right ) \]
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 \[ \int {\left (b \arcsin \left (c x^{2}\right ) + a\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 1.22 \[ \frac {a \,x^{5}}{5}+b \left (\frac {x^{5} \arcsin \left (c \,x^{2}\right )}{5}-\frac {2 c \left (-\frac {x^{3} \sqrt {-c^{2} x^{4}+1}}{5 c^{2}}-\frac {3 \sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \left (\EllipticF \left (x \sqrt {c}, i\right )-\EllipticE \left (x \sqrt {c}, i\right )\right )}{5 c^{\frac {7}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{5}\right ) \]
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 \[ \frac {1}{5} \, a x^{5} + \frac {1}{5} \, {\left (x^{5} \arctan \left (c x^{2}, \sqrt {c x^{2} + 1} \sqrt {-c x^{2} + 1}\right ) + 10 \, c \int \frac {x^{6} e^{\left (\frac {1}{2} \, \log \left (c x^{2} + 1\right ) + \frac {1}{2} \, \log \left (-c x^{2} + 1\right )\right )}}{5 \, {\left (c^{4} x^{8} - c^{2} x^{4} + {\left (c^{2} x^{4} - 1\right )} e^{\left (\log \left (c x^{2} + 1\right ) + \log \left (-c x^{2} + 1\right )\right )}\right )}}\,{d x}\right )} b \]
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 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.09, size = 58, normalized size = 0.70 \[ \frac {a x^{5}}{5} - \frac {b c x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{10 \Gamma \left (\frac {11}{4}\right )} + \frac {b x^{5} \operatorname {asin}{\left (c x^{2} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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