Optimal. Leaf size=227 \[ -\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac {\sqrt {\frac {\pi }{2}} c \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\sqrt {\pi } c \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4635
Rule 4641
Rule 4647
Rule 4649
Rule 4723
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)} \, dx &=\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)} \, dx-\frac {\left (a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x \left (1-a^2 x^2\right )}{\sqrt {\sin ^{-1}(a x)}} \, dx}{8 \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{32 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{16 a \sqrt {1-a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \sqrt {\sin ^{-1}(a x)}+\frac {c \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{8 a \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.26, size = 166, normalized size = 0.73 \[ \frac {c \sqrt {c-a^2 c x^2} \left (32 \sin ^{-1}(a x)^2+8 \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}(a x)\right )+8 \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {3}{2},4 i \sin ^{-1}(a x)\right )\right )}{128 a \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
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 = 0.47, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arcsin \left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {\mathrm {asin}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________