Optimal. Leaf size=144 \[ \frac{2 \sqrt{\pi } e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2} d}+\frac{2 \sqrt{\pi } e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} d}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]
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Rubi [A] time = 0.230992, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {4805, 12, 4631, 3306, 3305, 3351, 3304, 3352} \[ \frac{2 \sqrt{\pi } e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2} d}+\frac{2 \sqrt{\pi } e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} d}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4631
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left (2 e \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+d x)\right )}{b d}+\frac{\left (2 e \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+d x)\right )}{b d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left (4 e \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+d x)}\right )}{b^2 d}+\frac{\left (4 e \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+d x)}\right )}{b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{2 e \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} d}+\frac{2 e \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2} d}\\ \end{align*}
Mathematica [C] time = 0.147742, size = 168, normalized size = 1.17 \[ \frac{i e e^{-\frac{2 i a}{b}} \left (-\sqrt{2} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{2} e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+2 i e^{\frac{2 i a}{b}} \sin \left (2 \sin ^{-1}(c+d x)\right )\right )}{2 b d \sqrt{a+b \sin ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.073, size = 156, normalized size = 1.1 \begin{align*} -{\frac{e}{bd} \left ( -2\,\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( dx+c \right ) }\cos \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 ) \sqrt{\pi }-2\,\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( dx+c \right ) }\sin \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 ) \sqrt{\pi }+\sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arcsin \left ( dx+c \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{d e x + c e}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} e \left (\int \frac{c}{a \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d x \right )}}\, dx + \int \frac{d x}{a \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} + b \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d 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{d e x + c e}{{\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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