Optimal. Leaf size=251 \[ -\frac{\sqrt{3 \pi } d \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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Rubi [A] time = 1.44441, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {4721, 4723, 4406, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{3 \pi } d \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 4721
Rule 4723
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x^3 \left (d-c^2 d x^2\right )}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{(6 d) \int \frac{x^2 \sqrt{1-c^2 x^2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{(12 c d) \int \frac{x^4 \sqrt{1-c^2 x^2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{(6 d) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{(12 d) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin ^4(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{(6 d) \operatorname{Subst}\left (\int \left (\frac{1}{8 \sqrt{a+b x}}-\frac{\cos (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{(12 d) \operatorname{Subst}\left (\int \left (\frac{1}{16 \sqrt{a+b x}}-\frac{\cos (2 x)}{32 \sqrt{a+b x}}-\frac{\cos (4 x)}{16 \sqrt{a+b x}}+\frac{\cos (6 x)}{32 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{\cos (6 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (3 d \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 )}{8 b c^4}-\frac{\left (3 d \cos \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{6 a}{b}+6 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (3 d \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 )}{8 b c^4}-\frac{\left (3 d \sin \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{6 a}{b}+6 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (3 d \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 )}{4 b^2 c^4}-\frac{\left (3 d \cos \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{6 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}+\frac{\left (3 d \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 )}{4 b^2 c^4}-\frac{\left (3 d \sin \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{6 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}\\ &=-\frac{2 d x^3 \left (1-c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{d \sqrt{3 \pi } \cos \left (\frac{6 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}+\frac{3 d \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 )}{8 b^{3/2} c^4}+\frac{3 d \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 )}{8 b^{3/2} c^4}-\frac{d \sqrt{3 \pi } S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{6 a}{b}\right )}{8 b^{3/2} c^4}\\ \end{align*}
Mathematica [C] time = 1.4099, size = 287, normalized size = 1.14 \[ -\frac{i d e^{-\frac{6 i a}{b}} \left (3 \sqrt{2} e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 \sqrt{2} e^{\frac{8 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{6} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{6} e^{\frac{12 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-6 i e^{\frac{6 i a}{b}} \sin \left (2 \sin ^{-1}(c x)\right )+2 i e^{\frac{6 i a}{b}} \sin \left (6 \sin ^{-1}(c x)\right )\right )}{32 b c^4 \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.097, size = 287, normalized size = 1.1 \begin{align*} -{\frac{d}{16\,b{c}^{4}} \left ( 2\,\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ( 6\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}+2\,\sqrt{3}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ( 6\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{\pi }\sqrt{{b}^{-1}}-6\,\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{\pi }\sqrt{{b}^{-1}}b}} \right ) -6\,\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{\pi }\sqrt{{b}^{-1}}b}} \right ) +3\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) -\sin \left ( 6\,{\frac{a+b\arcsin \left ( cx \right ) }{b}}-6\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \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{{\left (c^{2} d x^{2} - d\right )} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{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*} - d \left (\int - \frac{x^{3}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx + \int \frac{c^{2} x^{5}}{a \sqrt{a + b \operatorname{asin}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c x \right )}} \operatorname{asin}{\left (c x \right )}}\, dx\right ) \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{{\left (c^{2} d x^{2} - d\right )} x^{3}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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