Optimal. Leaf size=466 \[ \frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt {1-c^2 x^2}}-\frac {i d 2^{-n-3} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i d 2^{-2 (n+3)} e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d 2^{-n-3} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d 2^{-2 (n+3)} e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.41, antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4663, 4661, 3312, 3307, 2181} \[ -\frac {i d 2^{-n-3} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i d 2^{-2 (n+3)} e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d 2^{-n-3} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d 2^{-2 (n+3)} e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{8 b c (n+1) \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rule 4661
Rule 4663
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos ^4(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{8} (a+b x)^n+\frac {1}{2} (a+b x)^n \cos (2 x)+\frac {1}{8} (a+b x)^n \cos (4 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (4 x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {1-c^2 x^2}}\\ &=\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{4 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{2 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {1-c^2 x^2}}\\ &=\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{8 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {i 2^{-3-n} d e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-3-n} d e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i 4^{-3-n} d e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 4^{-3-n} d e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 1.99, size = 326, normalized size = 0.70 \[ \frac {d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (i 4^{-n} e^{-\frac {4 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (e^{\frac {8 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )-\frac {8 \left (a+b \sin ^{-1}(c x)\right )}{b n+b}+8 \left (\frac {4 a+4 b \sin ^{-1}(c x)}{b n+b}-i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )\right )}{64 c \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx \]
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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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