Optimal. Leaf size=287 \[ \frac {2 b x \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac {1}{3} x^3 \left (a+b \text {ArcSin}\left (c+d x^2\right )\right )+\frac {8 b \sqrt {1-c} c (1+c) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{9 d^{3/2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}-\frac {2 b \sqrt {1-c} (1+c) (1+3 c) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{9 d^{3/2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \]
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Rubi [A]
time = 0.23, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4926, 12, 1136,
1216, 538, 435, 430} \begin {gather*} \frac {1}{3} x^3 \left (a+b \text {ArcSin}\left (c+d x^2\right )\right )-\frac {2 b \sqrt {1-c} (c+1) (3 c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{9 d^{3/2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}+\frac {8 b \sqrt {1-c} c (c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{9 d^{3/2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}+\frac {2 b x \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 430
Rule 435
Rule 538
Rule 1136
Rule 1216
Rule 4926
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{3} b \int \frac {2 d x^4}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {1}{3} (2 b d) \int \frac {x^4}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=\frac {2 b x \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {(2 b) \int \frac {1-c^2-4 c d x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx}{9 d}\\ &=\frac {2 b x \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )-\frac {\left (2 b \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {1-c^2-4 c d x^2}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ &=\frac {2 b x \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {\left (8 b c (1+c) \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}}}{\sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}-\frac {\left (2 b (1+c) (1+3 c) \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {1}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{9 d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ &=\frac {2 b x \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{9 d}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c+d x^2\right )\right )+\frac {8 b \sqrt {1-c} c (1+c) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{9 d^{3/2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}-\frac {2 b \sqrt {1-c} (1+c) (1+3 c) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{9 d^{3/2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.36, size = 307, normalized size = 1.07 \begin {gather*} \frac {\sqrt {\frac {d}{1+c}} x \left (3 a d x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}-2 b \left (-1+c^2+2 c d x^2+d^2 x^4\right )+3 b d x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4} \text {ArcSin}\left (c+d x^2\right )\right )-8 i b (-1+c) c \sqrt {\frac {-1+c+d x^2}{-1+c}} \sqrt {\frac {1+c+d x^2}{1+c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{1+c}} x\right )|\frac {1+c}{-1+c}\right )+2 i b \left (1-4 c+3 c^2\right ) \sqrt {\frac {-1+c+d x^2}{-1+c}} \sqrt {\frac {1+c+d x^2}{1+c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{1+c}} x\right )|\frac {1+c}{-1+c}\right )}{9 d \sqrt {\frac {d}{1+c}} \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 295, normalized size = 1.03
method | result | size |
default | \(\frac {x^{3} a}{3}+b \left (\frac {x^{3} \arcsin \left (d \,x^{2}+c \right )}{3}-\frac {2 d \left (-\frac {x \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{3 d^{2}}+\frac {\left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \EllipticF \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )}{3 d^{2} \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}+\frac {8 c \left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )\right )}{3 d \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}\, \left (-2 d c +2 d \right )}\right )}{3}\right )\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 72, normalized size = 0.25 \begin {gather*} \frac {3 \, b d^{2} x^{4} \arcsin \left (d x^{2} + c\right ) + 3 \, a d^{2} x^{4} + 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (b d x^{2} - 4 \, b c\right )}}{9 \, d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {asin}{\left (c + d x^{2} \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asin}\left (d\,x^2+c\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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