Optimal. Leaf size=595 \[ -\frac {d e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (1+n,\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.34, antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4809, 4491,
3389, 2212} \begin {gather*} -\frac {d e^{-\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{-\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{-\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {i (a+b \text {ArcSin}(c x))}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d 3^{-n} e^{\frac {3 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {d 5^{-n-1} e^{\frac {5 i a}{b}} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2212
Rule 3389
Rule 4491
Rule 4809
Rubi steps
\begin {align*} \int x \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 x \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 ) \text {Subst}\left (\int (a+b x)^n \cos ^4(x) \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{8} (a+b x)^n \sin (x)+\frac {3}{16} (a+b x)^n \sin (3 x)+\frac {1}{16} (a+b x)^n \sin (5 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sin (5 x) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sin (3 x) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}\\ &=\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{5 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (3 i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (3 i d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{3 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {d e^{-\frac {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 {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {d e^{\frac {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 {i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{-\frac {3 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 {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {3^{-n} d e^{\frac {3 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 {3 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{-\frac {5 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 {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}-\frac {5^{-1-n} d e^{\frac {5 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 {5 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.39, size = 464, normalized size = 0.78 \begin {gather*} -\frac {15^{-1-n} d^2 e^{-\frac {5 i a}{b}} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^n \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^{-3 n} \left (2\ 15^{1+n} e^{\frac {4 i a}{b}} \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^n \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^{2 n} \text {Gamma}\left (1+n,-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+\left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^n \left (2\ 15^{1+n} e^{\frac {6 i a}{b}} \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^{2 n} \text {Gamma}\left (1+n,\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+3 \left (5^{1+n} e^{\frac {2 i a}{b}} \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{2 n} \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^n \text {Gamma}\left (1+n,-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+5^{1+n} e^{\frac {8 i a}{b}} \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^{2 n} \text {Gamma}\left (1+n,\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+3^n \left (\left (-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^n \left (\frac {i (a+b \text {ArcSin}(c x))}{b}\right )^{3 n} \text {Gamma}\left (1+n,-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )+e^{\frac {10 i a}{b}} \left (\frac {(a+b \text {ArcSin}(c x))^2}{b^2}\right )^{2 n} \text {Gamma}\left (1+n,\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )\right )\right )\right )\right )}{32 c^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________