Optimal. Leaf size=698 \[ \frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{16 b c (n+1) \sqrt {1-c^2 x^2}}-\frac {15 i d^2 2^{-n-7} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {3 i d^2 2^{-2 n-7} e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i d^2 2^{-n-7} 3^{-n-1} e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {15 i d^2 2^{-n-7} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {3 i d^2 2^{-2 n-7} e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d^2 2^{-n-7} 3^{-n-1} e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 698, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ -\frac {15 i d^2 2^{-n-7} e^{-\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {3 i d^2 2^{-2 n-7} e^{-\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}-\frac {i d^2 2^{-n-7} 3^{-n-1} e^{-\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {15 i d^2 2^{-n-7} e^{\frac {2 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {3 i d^2 2^{-2 n-7} e^{\frac {4 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i d^2 2^{-n-7} 3^{-n-1} e^{\frac {6 i a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{n+1}}{16 b c (n+1) \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2181
Rule 3307
Rule 3312
Rule 4661
Rule 4663
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^n \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos ^6(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {5}{16} (a+b x)^n+\frac {15}{32} (a+b x)^n \cos (2 x)+\frac {3}{16} (a+b x)^n \cos (4 x)+\frac {1}{32} (a+b x)^n \cos (6 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{16 b c (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cos (6 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 \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 )}{16 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 \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 )}{32 c \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{16 b c (1+n) \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-6 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{6 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 \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 )}{32 c \sqrt {1-c^2 x^2}}+\frac {\left (3 d^2 \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 )}{32 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 \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 )}{64 c \sqrt {1-c^2 x^2}}+\frac {\left (15 d^2 \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 )}{64 c \sqrt {1-c^2 x^2}}\\ &=\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^{1+n}}{16 b c (1+n) \sqrt {1-c^2 x^2}}-\frac {15 i 2^{-7-n} d^2 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 {15 i 2^{-7-n} d^2 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 {3 i 2^{-7-2 n} d^2 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 {3 i 2^{-7-2 n} d^2 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 2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 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 {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}+\frac {i 2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 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 {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.76, size = 477, normalized size = 0.68 \[ \frac {d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (9 i 4^{-n} e^{\frac {4 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 6^{-n} e^{\frac {6 i a}{b}} \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \Gamma \left (n+1,\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-9 i 4^{-n} e^{-\frac {4 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \Gamma \left (n+1,-\frac {4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-i 6^{-n} e^{-\frac {6 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \left (\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \Gamma \left (n+1,-\frac {6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-45 i 2^{-n} e^{-\frac {2 i a}{b}} \left (-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+45 i 2^{-n} e^{\frac {2 i a}{b}} \left (\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\frac {120 a}{b n+b}+\frac {120 \sin ^{-1}(c x)}{n+1}\right )}{384 c \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________