Optimal. Leaf size=247 \[ -\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \text {ArcSin}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4741, 4737,
4725, 4795, 4731, 4491, 12, 3386, 3432} \begin {gather*} \frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\text {ArcSin}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}+\frac {\text {ArcSin}(a x)^{7/2} \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {1}{2} x \text {ArcSin}(a x)^{5/2} \sqrt {c-a^2 c x^2}-\frac {5 a x^2 \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{8 \sqrt {1-a^2 x^2}}+\frac {5 \text {ArcSin}(a x)^{3/2} \sqrt {c-a^2 c x^2}}{16 a \sqrt {1-a^2 x^2}}-\frac {15}{32} x \sqrt {\text {ArcSin}(a x)} \sqrt {c-a^2 c x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4725
Rule 4731
Rule 4737
Rule 4741
Rule 4795
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \int \frac {\sin ^{-1}(a x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {1-a^2 x^2}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \sin ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {1-a^2 x^2}}\\ &=-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{64 \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{128 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{64 a \sqrt {1-a^2 x^2}}\\ &=-\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\sin ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt {1-a^2 x^2}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt {1-a^2 x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac {\sqrt {c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {\pi } \sqrt {c-a^2 c x^2} S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{128 a \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.08, size = 158, normalized size = 0.64 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \sqrt {\text {ArcSin}(a x)} \left (64 \left (\text {ArcSin}(a x)^2\right )^{3/2} \left (7 a x \sqrt {1-a^2 x^2}+2 \text {ArcSin}(a x)\right )+35 i \sqrt {2} \sqrt {i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},-2 i \text {ArcSin}(a x)\right )-35 i \sqrt {2} \sqrt {-i \text {ArcSin}(a x)} \text {Gamma}\left (\frac {5}{2},2 i \text {ArcSin}(a x)\right )\right )}{896 a \sqrt {1-a^2 x^2} \sqrt {\text {ArcSin}(a x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} c \,x^{2}+c}\, \arcsin \left (a x \right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\mathrm {asin}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________