Optimal. Leaf size=187 \[ \frac {x \sqrt {\text {ArcSin}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\text {ArcSin}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\text {ArcSin}(a x)}},x\right )}{6 c^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right ) \sqrt {\text {ArcSin}(a x)}},x\right )}{3 c^2 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {\text {ArcSin}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {x \sqrt {\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\sqrt {\sin ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\sin ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \sqrt {\sin ^{-1}(a x)}}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \sqrt {\sin ^{-1}(a x)}}{3 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \sqrt {\sin ^{-1}(a x)}} \, dx}{6 c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right ) \sqrt {\sin ^{-1}(a x)}} \, dx}{3 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\text {ArcSin}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\arcsin \left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx\]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {asin}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
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]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\mathrm {asin}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________