Optimal. Leaf size=485 \[ \frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \cos \left (\frac{8 a}{b}\right ) \text{FresnelC}\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 b^{3/2} c^4}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \sin \left (\frac{8 a}{b}\right ) S\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 b^{3/2} c^4}-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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Rubi [A] time = 1.66205, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {4721, 4723, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \cos \left (\frac{6 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \cos \left (\frac{8 a}{b}\right ) \text{FresnelC}\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 b^{3/2} c^4}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \sin \left (\frac{6 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \sin \left (\frac{8 a}{b}\right ) S\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 b^{3/2} c^4}-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}} \]
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 )^2}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (6 d^2\right ) \int \frac{x^2 \left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac{\left (16 c d^2\right ) \int \frac{x^4 \left (1-c^2 x^2\right )^{3/2}}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac{\left (16 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin ^4(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (6 d^2\right ) \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{\left (16 d^2\right ) \operatorname{Subst}\left (\int \left (\frac{3}{128 \sqrt{a+b x}}-\frac{\cos (4 x)}{32 \sqrt{a+b x}}+\frac{\cos (8 x)}{128 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\cos (8 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos (6 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac{\left (3 d^2\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{d^2 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^4}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (3 d^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 )}{16 b c^4}-\frac{\left (3 d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^4}-\frac{\left (3 d^2 \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 )}{16 b c^4}-\frac{\left (d^2 \cos \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{8 a}{b}+8 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (3 d^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 )}{16 b c^4}-\frac{\left (3 d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}+\frac{\left (d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^4}-\frac{\left (3 d^2 \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 )}{16 b c^4}-\frac{\left (d^2 \sin \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{8 a}{b}+8 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{\left (3 d^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 )}{8 b^2 c^4}-\frac{\left (3 d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}+\frac{\left (d^2 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2 c^4}-\frac{\left (3 d^2 \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 )}{8 b^2 c^4}-\frac{\left (d^2 \cos \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{8 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}+\frac{\left (3 d^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 )}{8 b^2 c^4}-\frac{\left (3 d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}+\frac{\left (d^2 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b^2 c^4}-\frac{\left (3 d^2 \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 )}{8 b^2 c^4}-\frac{\left (d^2 \sin \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{8 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{4 b^2 c^4}\\ &=-\frac{2 d^2 x^3 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{d^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^4}-\frac{d^2 \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 )}{16 b^{3/2} c^4}+\frac{3 d^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 )}{16 b^{3/2} c^4}-\frac{d^2 \sqrt{\pi } \cos \left (\frac{8 a}{b}\right ) C\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{16 b^{3/2} c^4}+\frac{3 d^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 )}{16 b^{3/2} c^4}+\frac{d^2 \sqrt{\frac{\pi }{2}} S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{8 b^{3/2} c^4}-\frac{d^2 \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 )}{16 b^{3/2} c^4}-\frac{d^2 \sqrt{\pi } S\left (\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{8 a}{b}\right )}{16 b^{3/2} c^4}\\ \end{align*}
Mathematica [C] time = 2.74498, size = 540, normalized size = 1.11 \[ -\frac{i d^2 e^{-\frac{8 i a}{b}} \left (3 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-3 \sqrt{2} e^{\frac{10 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+2 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-2 e^{\frac{12 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{4 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{6} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{6} e^{\frac{14 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{6 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{2} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt{2} e^{\frac{16 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{8 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-6 i e^{\frac{8 i a}{b}} \sin \left (2 \sin ^{-1}(c x)\right )-2 i e^{\frac{8 i a}{b}} \sin \left (4 \sin ^{-1}(c x)\right )+2 i e^{\frac{8 i a}{b}} \sin \left (6 \sin ^{-1}(c x)\right )+i e^{\frac{8 i a}{b}} \sin \left (8 \sin ^{-1}(c x)\right )\right )}{64 b c^4 \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.105, size = 551, normalized size = 1.1 \begin{align*} \text{result too large to display} \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 )}^{2} 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^{2} \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{2 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 + \int \frac{c^{4} x^{7}}{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 )}^{2} 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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