Optimal. Leaf size=180 \[ -\frac{8 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} c^2}+\frac{8 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} c^2}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]
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Rubi [A] time = 0.48727, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {4634, 4720, 4636, 4406, 12, 3306, 3305, 3351, 3304, 3352, 4642} \[ -\frac{8 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} c^2}+\frac{8 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} c^2}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4634
Rule 4720
Rule 4636
Rule 4406
Rule 12
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rule 4642
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \cos ^{-1}(c x)\right )^{5/2}} \, dx &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{2 \int \frac{1}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}+\frac{(4 c) \int \frac{x^2}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx}{3 b}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}-\frac{16 \int \frac{x}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx}{3 b^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{\left (8 \cos \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,\cos ^{-1}(c x)\right )}{3 b^2 c^2}-\frac{\left (8 \sin \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,\cos ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{\left (16 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{3 b^3 c^2}-\frac{\left (16 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{3 b^3 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{3 b c \left (a+b \cos ^{-1}(c x)\right )^{3/2}}-\frac{4}{3 b^2 c^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 x^2}{3 b^2 \sqrt{a+b \cos ^{-1}(c x)}}+\frac{8 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} c^2}-\frac{8 \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{3 b^{5/2} c^2}\\ \end{align*}
Mathematica [A] time = 0.667768, size = 176, normalized size = 0.98 \[ \frac{-8 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left (\frac{2 a}{b}\right ) \left (a+b \cos ^{-1}(c x)\right )^{3/2} \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )+8 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left (\frac{2 a}{b}\right ) \left (a+b \cos ^{-1}(c x)\right )^{3/2} S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )+4 a \cos \left (2 \cos ^{-1}(c x)\right )+4 b \cos ^{-1}(c x) \cos \left (2 \cos ^{-1}(c x)\right )+b \sin \left (2 \cos ^{-1}(c x)\right )}{3 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.114, size = 311, normalized size = 1.7 \begin{align*}{\frac{1}{3\,{b}^{2}{c}^{2}} \left ( 8\,\arccos \left ( cx \right ) \sqrt{\pi }\sqrt{{b}^{-1}}\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{a+b\arccos \left ( cx \right ) }b-8\,\arccos \left ( cx \right ) \sqrt{\pi }\sqrt{{b}^{-1}}\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{a+b\arccos \left ( cx \right ) }b+8\,\sqrt{\pi }\sqrt{{b}^{-1}}\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{a+b\arccos \left ( cx \right ) }a-8\,\sqrt{\pi }\sqrt{{b}^{-1}}\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{a+b\arccos \left ( cx \right ) }a+4\,\arccos \left ( cx \right ) \cos \left ( 2\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+\sin \left ( 2\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+4\,\cos \left ( 2\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) a \right ) \left ( a+b\arccos \left ( cx \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b \operatorname{acos}{\left (c x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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