Optimal. Leaf size=756 \[ \frac {6 b^2 c^2 d^2 x^{3+m}}{(3+m)^2 (5+m)^2}+\frac {2 b^2 c^2 d^2 x^{3+m}}{(3+m) (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m}}{(3+m)^3 (5+m)}-\frac {2 b^2 c^4 d^2 x^{5+m}}{(5+m)^3}-\frac {6 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{(3+m) (5+m)^2}-\frac {8 b c d^2 x^{2+m} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{(3+m)^2 (5+m)}-\frac {2 b c d^2 x^{2+m} \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{(5+m)^2}+\frac {8 d^2 x^{1+m} (a+b \text {ArcSin}(c x))^2}{(5+m) \left (3+4 m+m^2\right )}+\frac {4 d^2 x^{1+m} \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{15+8 m+m^2}+\frac {d^2 x^{1+m} \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2}{5+m}-\frac {8 b c d^2 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)}-\frac {6 b c d^2 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m)^2 \left (6+5 m+m^2\right )}-\frac {16 b c d^2 x^{2+m} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{(5+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {6 b^2 c^2 d^2 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(2+m) (3+m)^2 (5+m)^2}+\frac {8 b^2 c^2 d^2 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(2+m) (3+m)^3 (5+m)}+\frac {16 b^2 c^2 d^2 x^{3+m} \text {HypergeometricPFQ}\left (\left \{1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2}\right \},\left \{2+\frac {m}{2},\frac {5}{2}+\frac {m}{2}\right \},c^2 x^2\right )}{(3+m)^2 (5+m) \left (2+3 m+m^2\right )} \]
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Rubi [A]
time = 0.61, antiderivative size = 756, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4787, 4723,
4805, 4783, 30, 14} \begin {gather*} \frac {16 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+3)^2 (m+5) \left (m^2+3 m+2\right )}+\frac {8 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^3 (m+5)}+\frac {6 b^2 c^2 d^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^2 (m+5)^2}-\frac {6 b c d^2 x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{(m+5)^2 \left (m^2+5 m+6\right )}-\frac {16 b c d^2 x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{(m+5) \left (m^3+6 m^2+11 m+6\right )}-\frac {8 b c d^2 x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{(m+2) (m+3)^2 (m+5)}+\frac {4 d^2 \left (1-c^2 x^2\right ) x^{m+1} (a+b \text {ArcSin}(c x))^2}{m^2+8 m+15}+\frac {d^2 \left (1-c^2 x^2\right )^2 x^{m+1} (a+b \text {ArcSin}(c x))^2}{m+5}-\frac {2 b c d^2 \left (1-c^2 x^2\right )^{3/2} x^{m+2} (a+b \text {ArcSin}(c x))}{(m+5)^2}-\frac {8 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \text {ArcSin}(c x))}{(m+3)^2 (m+5)}-\frac {6 b c d^2 \sqrt {1-c^2 x^2} x^{m+2} (a+b \text {ArcSin}(c x))}{(m+3) (m+5)^2}+\frac {8 d^2 x^{m+1} (a+b \text {ArcSin}(c x))^2}{(m+5) \left (m^2+4 m+3\right )}-\frac {2 b^2 c^4 d^2 x^{m+5}}{(m+5)^3}+\frac {8 b^2 c^2 d^2 x^{m+3}}{(m+3)^3 (m+5)}+\frac {2 b^2 c^2 d^2 x^{m+3}}{(m+3) (m+5)^2}+\frac {6 b^2 c^2 d^2 x^{m+3}}{(m+3)^2 (m+5)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4723
Rule 4783
Rule 4787
Rule 4805
Rubi steps
\begin {align*} \int x^m \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}
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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (d-c^2 d x^2\right )^2 (a+b \text {ArcSin}(c x))^2 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 4.67, size = 0, normalized size = 0.00 \[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{2} \left (\int a^{2} x^{m}\, dx + \int b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{m} \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 2 a^{2} c^{2} x^{2} x^{m}\right )\, dx + \int a^{2} c^{4} x^{4} x^{m}\, dx + \int \left (- 2 b^{2} c^{2} x^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{4} x^{4} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 4 a b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{4} x^{4} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \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^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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