Optimal. Leaf size=62 \[ \frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac {b \sqrt {1-c^2 x^4}}{6 c^3} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4842, 12, 266, 43} \[ \frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac {b \sqrt {1-c^2 x^4}}{6 c^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 4842
Rubi steps
\begin {align*} \int x^5 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{6} b \int \frac {2 c x^7}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{3} (b c) \int \frac {x^7}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-c^2 x}} \, dx,x,x^4\right )\\ &=\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{12} (b c) \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \sqrt {1-c^2 x}}-\frac {\sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^4\right )\\ &=\frac {b \sqrt {1-c^2 x^4}}{6 c^3}-\frac {b \left (1-c^2 x^4\right )^{3/2}}{18 c^3}+\frac {1}{6} x^6 \left (a+b \sin ^{-1}\left (c x^2\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 1.13 \[ \frac {a x^6}{6}+\frac {b x^4 \sqrt {1-c^2 x^4}}{18 c}+\frac {b \sqrt {1-c^2 x^4}}{9 c^3}+\frac {1}{6} b x^6 \sin ^{-1}\left (c x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 55, normalized size = 0.89 \[ \frac {3 \, b c^{3} x^{6} \arcsin \left (c x^{2}\right ) + 3 \, a c^{3} x^{6} + {\left (b c^{2} x^{4} + 2 \, b\right )} \sqrt {-c^{2} x^{4} + 1}}{18 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 87, normalized size = 1.40 \[ \frac {3 \, a c x^{6} + {\left (\frac {3 \, {\left (c^{2} x^{4} - 1\right )} x^{2} \arcsin \left (c x^{2}\right )}{c} + \frac {3 \, x^{2} \arcsin \left (c x^{2}\right )}{c} - \frac {{\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{4} + 1}}{c^{2}}\right )} b}{18 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 62, normalized size = 1.00 \[ \frac {x^{6} a}{6}+b \left (\frac {x^{6} \arcsin \left (c \,x^{2}\right )}{6}-\frac {\left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (c^{2} x^{4}+2\right )}{18 c^{3} \sqrt {-c^{2} x^{4}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 59, normalized size = 0.95 \[ \frac {1}{6} \, a x^{6} + \frac {1}{18} \, {\left (3 \, x^{6} \arcsin \left (c x^{2}\right ) - c {\left (\frac {{\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{c^{4}} - \frac {3 \, \sqrt {-c^{2} x^{4} + 1}}{c^{4}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^5\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.52, size = 65, normalized size = 1.05 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {asin}{\left (c x^{2} \right )}}{6} + \frac {b x^{4} \sqrt {- c^{2} x^{4} + 1}}{18 c} + \frac {b \sqrt {- c^{2} x^{4} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\\frac {a x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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