Optimal. Leaf size=130 \[ \frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4631, 3306, 3305, 3351, 3304, 3352} \[ \frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4631
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}+\frac {\left (2 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (4 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}+\frac {\left (4 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.15, size = 155, normalized size = 1.19 \[ \frac {i e^{-\frac {2 i a}{b}} \left (2 i e^{\frac {2 i a}{b}} \sin \left (2 \sin ^{-1}(c x)\right )-\sqrt {2} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 b c^2 \sqrt {a+b \sin ^{-1}(c x)}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 143, normalized size = 1.10 \[ -\frac {-2 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-2 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )+\sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right )}{c^{2} b \sqrt {a +b \arcsin \left (c x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________