Optimal. Leaf size=399 \[ -\frac {3 b c d x^{2+m} \sqrt {d-c^2 d x^2}}{(2+m)^2 (4+m) \sqrt {1-c^2 x^2}}-\frac {b c d x^{2+m} \sqrt {d-c^2 d x^2}}{\left (8+6 m+m^2\right ) \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^{4+m} \sqrt {d-c^2 d x^2}}{(4+m)^2 \sqrt {1-c^2 x^2}}+\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{8+6 m+m^2}+\frac {x^{1+m} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{4+m}+\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{\left (8+14 m+7 m^2+m^3\right ) \sqrt {1-c^2 x^2}}-\frac {3 b c d x^{2+m} \sqrt {d-c^2 d x^2} \text {HypergeometricPFQ}\left (\left \{1,1+\frac {m}{2},1+\frac {m}{2}\right \},\left \{\frac {3}{2}+\frac {m}{2},2+\frac {m}{2}\right \},c^2 x^2\right )}{(1+m) (2+m)^2 (4+m) \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4787, 4783,
4805, 30, 14} \begin {gather*} -\frac {3 b c d x^{m+2} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,\frac {m}{2}+1,\frac {m}{2}+1;\frac {m}{2}+\frac {3}{2},\frac {m}{2}+2;c^2 x^2\right )}{(m+1) (m+2)^2 (m+4) \sqrt {1-c^2 x^2}}+\frac {3 d x^{m+1} \sqrt {d-c^2 d x^2} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{\left (m^3+7 m^2+14 m+8\right ) \sqrt {1-c^2 x^2}}+\frac {3 d x^{m+1} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{m^2+6 m+8}+\frac {x^{m+1} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{m+4}-\frac {b c d x^{m+2} \sqrt {d-c^2 d x^2}}{\left (m^2+6 m+8\right ) \sqrt {1-c^2 x^2}}-\frac {3 b c d x^{m+2} \sqrt {d-c^2 d x^2}}{(m+2)^2 (m+4) \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^{m+4} \sqrt {d-c^2 d x^2}}{(m+4)^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 4783
Rule 4787
Rule 4805
Rubi steps
\begin {align*} \int x^m \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {x^{1+m} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4+m}+\frac {(3 d) \int x^m \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4+m}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^{1+m} \left (1-c^2 x^2\right ) \, dx}{(4+m) \sqrt {1-c^2 x^2}}\\ &=\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8+6 m+m^2}+\frac {x^{1+m} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4+m}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (x^{1+m}-c^2 x^{3+m}\right ) \, dx}{(4+m) \sqrt {1-c^2 x^2}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^m \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{(2+m) (4+m) \sqrt {1-c^2 x^2}}-\frac {\left (3 b c d \sqrt {d-c^2 d x^2}\right ) \int x^{1+m} \, dx}{(2+m) (4+m) \sqrt {1-c^2 x^2}}\\ &=-\frac {3 b c d x^{2+m} \sqrt {d-c^2 d x^2}}{(2+m)^2 (4+m) \sqrt {1-c^2 x^2}}-\frac {b c d x^{2+m} \sqrt {d-c^2 d x^2}}{\left (8+6 m+m^2\right ) \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^{4+m} \sqrt {d-c^2 d x^2}}{(4+m)^2 \sqrt {1-c^2 x^2}}+\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{8+6 m+m^2}+\frac {x^{1+m} \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{4+m}+\frac {3 d x^{1+m} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{\left (8+14 m+7 m^2+m^3\right ) \sqrt {1-c^2 x^2}}-\frac {3 b c d x^{2+m} \sqrt {d-c^2 d x^2} \, _3F_2\left (1,1+\frac {m}{2},1+\frac {m}{2};\frac {3}{2}+\frac {m}{2},2+\frac {m}{2};c^2 x^2\right )}{(1+m) (2+m)^2 (4+m) \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 237, normalized size = 0.59 \begin {gather*} \frac {d x^{1+m} \sqrt {d-c^2 d x^2} \left (-\frac {b c x \left (4+m-c^2 (2+m) x^2\right )}{(2+m) (4+m) \sqrt {1-c^2 x^2}}+\left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))-\frac {3 \left (b c (1+m) x-(1+m) (2+m) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))-(2+m) (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )+b c x \text {HypergeometricPFQ}\left (\left \{1,1+\frac {m}{2},1+\frac {m}{2}\right \},\left \{\frac {3}{2}+\frac {m}{2},2+\frac {m}{2}\right \},c^2 x^2\right )\right )}{(1+m) (2+m)^2 \sqrt {1-c^2 x^2}}\right )}{4+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.16, size = 0, normalized size = 0.00 \[\int x^{m} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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