Optimal. Leaf size=137 \[ \frac {2 (f x)^{5/2} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )}{5 f \sqrt {d-c^2 d x^2}}-\frac {4 b c (f x)^{7/2} \sqrt {1-c^2 x^2} \text {HypergeometricPFQ}\left (\left \{1,\frac {7}{4},\frac {7}{4}\right \},\left \{\frac {9}{4},\frac {11}{4}\right \},c^2 x^2\right )}{35 f^2 \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {4805}
\begin {gather*} \frac {2 \sqrt {1-c^2 x^2} (f x)^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{5 f \sqrt {d-c^2 d x^2}}-\frac {4 b c \sqrt {1-c^2 x^2} (f x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4805
Rubi steps
\begin {align*} \int \frac {(f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 (f x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right )}{5 f \sqrt {d-c^2 d x^2}}-\frac {4 b c (f x)^{7/2} \sqrt {1-c^2 x^2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{35 f^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 97, normalized size = 0.71 \begin {gather*} -\frac {2 x (f x)^{3/2} \sqrt {1-c^2 x^2} \left (-7 (a+b \text {ArcSin}(c x)) \text {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {9}{4},c^2 x^2\right )+2 b c x \text {HypergeometricPFQ}\left (\left \{1,\frac {7}{4},\frac {7}{4}\right \},\left \{\frac {9}{4},\frac {11}{4}\right \},c^2 x^2\right )\right )}{35 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.50, size = 0, normalized size = 0.00 \[\int \frac {\left (f x \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\sqrt {-c^{2} d \,x^{2}+d}}\, dx\]
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]
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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (f\,x\right )}^{3/2}}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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