Optimal. Leaf size=440 \[ \frac{\sqrt{2 \pi } c^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{2 \pi } c^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{\pi } c \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } c \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{\sqrt{b} d^3} \]
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Rubi [A] time = 1.01881, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4805, 4747, 6741, 6742, 3354, 3352, 3351, 3353, 4574} \[ \frac{\sqrt{2 \pi } c^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{2 \pi } c^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{\pi } c \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{\sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } c \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{\sqrt{b} d^3} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4747
Rule 6741
Rule 6742
Rule 3354
Rule 3352
Rule 3351
Rule 3353
Rule 4574
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{c}{d}+\frac{x}{d}\right )^2}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \left (-\frac{c}{d}+\frac{\sin (x)}{d}\right )^2}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \cos \left (\frac{a-x^2}{b}\right ) \left (c+\sin \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \left (c+\sin \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (c^2 \cos \left (\frac{a}{b}-\frac{x^2}{b}\right )+c \sin \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )+\cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sin ^2\left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sin ^2\left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac{(2 c) \operatorname{Subst}\left (\int \sin \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{1}{4} \cos \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right )+\frac{1}{4} \cos \left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac{\left (2 c^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}-\frac{\left (2 c \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 \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac{\left (2 c^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{b d^3}+\frac{\left (2 c \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 \sin ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{c^2 \sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}-\frac{c \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{\sqrt{b} d^3}+\frac{c^2 \sqrt{2 \pi } S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} d^3}+\frac{c \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{\sqrt{b} d^3}-\frac{\operatorname{Subst}\left (\int \cos \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}+\frac{\operatorname{Subst}\left (\int \cos \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}\\ &=\frac{c^2 \sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}-\frac{c \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{\sqrt{b} d^3}+\frac{c^2 \sqrt{2 \pi } S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} d^3}+\frac{c \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{\sqrt{b} d^3}+\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}-\frac{\cos \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}-\frac{\sin \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d^3}\\ &=\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c^2 \sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{c \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{\sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 \sqrt{b} d^3}+\frac{c^2 \sqrt{2 \pi } S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} d^3}+\frac{c \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{\sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{2 \sqrt{b} d^3}\\ \end{align*}
Mathematica [A] time = 1.11203, size = 335, normalized size = 0.76 \[ \frac{\sqrt{\pi } \sqrt{\frac{1}{b}} \left (3 \sqrt{2} \left (4 c^2+1\right ) \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+12 \sqrt{2} c^2 \sin \left (\frac{a}{b}\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+12 c \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )-\sqrt{6} \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}\right )+3 \sqrt{2} \sin \left (\frac{a}{b}\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-\sqrt{6} \sin \left (\frac{3 a}{b}\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right )-12 c \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right )\right )}{12 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 339, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{\pi }}{12\,{d}^{3}}\sqrt{{b}^{-1}} \left ( -12\,\cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}{c}^{2}-12\,\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}{c}^{2}+\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{3}\sqrt{2}+\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) \sqrt{3}\sqrt{2}+12\,\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) c-12\,\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) c-3\,\cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}-3\,\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2} \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{x^{2}}{\sqrt{b \arcsin \left (d x + c\right ) + a}}\,{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^{2}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.62234, size = 898, normalized size = 2.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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