Optimal. Leaf size=380 \[ \frac{3 \sqrt{\pi } b^{3/2} e^3 \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \sin \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}-\frac{3 \sqrt{\pi } b^{3/2} e^3 \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{64 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d} \]
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Rubi [A] time = 1.12443, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {4805, 12, 4629, 4707, 4641, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{3 \sqrt{\pi } b^{3/2} e^3 \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{64 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \sin \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}-\frac{3 \sqrt{\pi } b^{3/2} e^3 \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{64 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4629
Rule 4707
Rule 4641
Rule 4635
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (3 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (9 b e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sin ^{-1}(x)}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{64 d}-\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}-\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{128 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{a+b x}}-\frac{\sin (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}-\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}-\frac{\left (3 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}-\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (9 b^2 e^3\right ) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}-\frac{\left (3 b^2 e^3 \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,\sin ^{-1}(c+d x)\right )}{256 d}+\frac{\left (3 b^2 e^3 \cos \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+d x)\right )}{512 d}+\frac{\left (3 b^2 e^3 \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,\sin ^{-1}(c+d x)\right )}{256 d}-\frac{\left (3 b^2 e^3 \sin \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+d x)\right )}{512 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac{\left (3 b e^3 \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 )}{128 d}-\frac{\left (9 b^2 e^3 \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,\sin ^{-1}(c+d x)\right )}{256 d}+\frac{\left (3 b e^3 \cos \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+d x)}\right )}{256 d}+\frac{\left (3 b e^3 \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 )}{128 d}+\frac{\left (9 b^2 e^3 \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,\sin ^{-1}(c+d x)\right )}{256 d}-\frac{\left (3 b e^3 \sin \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+d x)}\right )}{256 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{3 b^{3/2} e^3 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}-\frac{3 b^{3/2} e^3 \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 )}{256 d}+\frac{3 b^{3/2} e^3 \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 )}{256 d}-\frac{3 b^{3/2} e^3 \sqrt{\frac{\pi }{2}} C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{512 d}-\frac{\left (9 b e^3 \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 )}{128 d}+\frac{\left (9 b e^3 \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 )}{128 d}\\ &=\frac{9 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{3 b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}-\frac{3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{3 b^{3/2} e^3 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{512 d}-\frac{3 b^{3/2} e^3 \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 )}{64 d}+\frac{3 b^{3/2} e^3 \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 )}{64 d}-\frac{3 b^{3/2} e^3 \sqrt{\frac{\pi }{2}} C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{512 d}\\ \end{align*}
Mathematica [C] time = 0.186122, size = 273, normalized size = 0.72 \[ -\frac{i b e^3 e^{-\frac{4 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left (8 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-8 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},-\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{8 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{5}{2},\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{512 d \sqrt{\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.12, size = 582, normalized size = 1.5 \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{\left (d e x + c e\right )}^{3}{\left (b \arcsin \left (d x + c\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(-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.
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Giac [B] time = 2.27437, size = 1890, normalized size = 4.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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