Optimal. Leaf size=216 \[ -\frac{15 \sqrt{\pi } b^{5/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 c^2}-\frac{15 \sqrt{\pi } b^{5/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 c^2}+\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.742519, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {4629, 4707, 4641, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac{15 \sqrt{\pi } b^{5/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 c^2}-\frac{15 \sqrt{\pi } b^{5/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 c^2}+\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4629
Rule 4707
Rule 4641
Rule 4723
Rule 3312
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int x \left (a+b \sin ^{-1}(c x)\right )^{5/2} \, dx &=\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{1}{4} (5 b c) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{1}{16} \left (15 b^2\right ) \int x \sqrt{a+b \sin ^{-1}(c x)} \, dx-\frac{(5 b) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^{3/2}}{\sqrt{1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{1}{64} \left (15 b^3 c\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}} \, dx\\ &=-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}\\ &=-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}+\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cos (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{128 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^3 \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 )}{128 c^2}-\frac{\left (15 b^3 \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 )}{128 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{\left (15 b^2 \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 )}{64 c^2}-\frac{\left (15 b^2 \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 )}{64 c^2}\\ &=\frac{15 b^2 \sqrt{a+b \sin ^{-1}(c x)}}{64 c^2}-\frac{15}{32} b^2 x^2 \sqrt{a+b \sin ^{-1}(c x)}+\frac{5 b x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}}{8 c}-\frac{\left (a+b \sin ^{-1}(c x)\right )^{5/2}}{4 c^2}+\frac{1}{2} x^2 \left (a+b \sin ^{-1}(c x)\right )^{5/2}-\frac{15 b^{5/2} \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 )}{128 c^2}-\frac{15 b^{5/2} \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 )}{128 c^2}\\ \end{align*}
Mathematica [C] time = 0.0945766, size = 141, normalized size = 0.65 \[ \frac{e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c x)\right )^{5/2} \left (\sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{32 \sqrt{2} c^2 \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.068, size = 394, normalized size = 1.8 \begin{align*} -{\frac{1}{128\,{c}^{2}} \left ( 15\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{3}+15\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{3}+32\, \left ( \arcsin \left ( cx \right ) \right ) ^{3}\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}+96\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}-40\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}\sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}+96\,\arcsin \left ( cx \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}b-30\,\arcsin \left ( cx \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}-80\,\arcsin \left ( cx \right ) \sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}+32\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{3}-30\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}-40\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}b \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \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 (b \arcsin \left (c x\right ) + a\right )}^{\frac{5}{2}} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [C] time = 2.79751, size = 1570, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]