Optimal. Leaf size=906 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.959616, antiderivative size = 906, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {4725, 4723, 4406, 3307, 2181} \[ -\frac{i 2^{-n-7} d^2 e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{2 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}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-2 (n+4)} d^2 e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text{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}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-n-7} 3^{-n-1} d^2 e^{-\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{6 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}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-3 n-11} d^2 e^{-\frac{8 i a}{b}} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^n \text{Gamma}\left (n+1,-\frac{8 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}}{c^3 \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}}{128 b c^3 (n+1) \sqrt{1-c^2 x^2}}+\frac{i 2^{-n-7} 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} \text{Gamma}\left (n+1,\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-2 (n+4)} 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} \text{Gamma}\left (n+1,\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-n-7} 3^{-n-1} 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} \text{Gamma}\left (n+1,\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-3 n-11} d^2 e^{\frac{8 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{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4725
Rule 4723
Rule 4406
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^2 \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 x^2 \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) \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \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}{128} (a+b x)^n+\frac{1}{32} (a+b x)^n \cos (2 x)-\frac{1}{32} (a+b x)^n \cos (4 x)-\frac{1}{32} (a+b x)^n \cos (6 x)-\frac{1}{128} (a+b x)^n \cos (8 x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \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}}{128 b c^3 (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 (8 x) \, dx,x,\sin ^{-1}(c x)\right )}{128 c^3 \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 (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3 \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 (4 x) \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3 \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^3 \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}}{128 b c^3 (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^{-8 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{256 c^3 \sqrt{1-c^2 x^2}}-\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int e^{8 i x} (a+b x)^n \, dx,x,\sin ^{-1}(c x)\right )}{256 c^3 \sqrt{1-c^2 x^2}}+\frac{\left (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^3 \sqrt{1-c^2 x^2}}+\frac{\left (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^3 \sqrt{1-c^2 x^2}}-\frac{\left (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 )}{64 c^3 \sqrt{1-c^2 x^2}}-\frac{\left (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 )}{64 c^3 \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^3 \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^3 \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}}{128 b c^3 (1+n) \sqrt{1-c^2 x^2}}-\frac{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^3 \sqrt{1-c^2 x^2}}+\frac{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^3 \sqrt{1-c^2 x^2}}+\frac{i 4^{-4-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^3 \sqrt{1-c^2 x^2}}-\frac{i 4^{-4-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^3 \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^3 \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^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-11-3 n} d^2 e^{-\frac{8 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{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-11-3 n} d^2 e^{\frac{8 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{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 4.21847, size = 989, normalized size = 1.09 \[ \frac{2^{-3 n-11} 3^{-n-1} d^3 e^{-\frac{8 i a}{b}} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^n \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (i 3^{n+1} 4^{n+2} b e^{\frac{10 i a}{b}} (n+1) \text{Gamma}\left (n+1,\frac{2 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-i 2^{n+3} 3^{n+1} b e^{\frac{12 i a}{b}} \text{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-i 2^{n+3} 3^{n+1} b e^{\frac{12 i a}{b}} n \text{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-i 4^{n+2} b e^{\frac{14 i a}{b}} \text{Gamma}\left (n+1,\frac{6 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-i 4^{n+2} b e^{\frac{14 i a}{b}} n \text{Gamma}\left (n+1,\frac{6 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-i 3^{n+1} b e^{\frac{16 i a}{b}} \text{Gamma}\left (n+1,\frac{8 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-i 3^{n+1} b e^{\frac{16 i a}{b}} n \text{Gamma}\left (n+1,\frac{8 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+5\ 2^{3 n+4} 3^{n+1} a e^{\frac{8 i a}{b}} \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n+5\ 2^{3 n+4} 3^{n+1} b e^{\frac{8 i a}{b}} \sin ^{-1}(c x) \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^n-i 3^{n+1} 4^{n+2} b e^{\frac{6 i a}{b}} (n+1) \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 )+i 2^{n+3} 3^{n+1} b e^{\frac{4 i a}{b}} \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 )+i 2^{n+3} 3^{n+1} b e^{\frac{4 i a}{b}} 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 )+i 4^{n+2} b e^{\frac{2 i a}{b}} \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 )+i 4^{n+2} b e^{\frac{2 i a}{b}} 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 )+i 3^{n+1} b \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+i 3^{n+1} b n \left (\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )^n \text{Gamma}\left (n+1,-\frac{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.259, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]