Optimal. Leaf size=207 \[ \frac{8 \sqrt{\pi } e \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} d}-\frac{8 \sqrt{\pi } e \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} d}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]
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Rubi [A] time = 0.535887, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4805, 12, 4633, 4719, 4635, 4406, 3306, 3305, 3351, 3304, 3352, 4641} \[ \frac{8 \sqrt{\pi } e \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{3 b^{5/2} d}-\frac{8 \sqrt{\pi } e \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{3 b^{5/2} d}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4633
Rule 4719
Rule 4635
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rule 4641
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sin ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}-\frac{(4 e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{(16 e) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{(16 e) \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{(16 e) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{(8 e) \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (8 e \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 )}{3 b^2 d}+\frac{\left (8 e \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 )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{\left (16 e \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 )}{3 b^3 d}+\frac{\left (16 e \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 )}{3 b^3 d}\\ &=-\frac{2 e (c+d x) \sqrt{1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{8 e \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 )}{3 b^{5/2} d}+\frac{8 e \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 )}{3 b^{5/2} d}\\ \end{align*}
Mathematica [C] time = 1.17985, size = 192, normalized size = 0.93 \[ -\frac{e \left (b \sin \left (2 \sin ^{-1}(c+d x)\right )+2 \left (a+b \sin ^{-1}(c+d x)\right ) \left (-\sqrt{2} e^{-\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{-2 i \sin ^{-1}(c+d x)}+e^{2 i \sin ^{-1}(c+d x)}\right )\right )}{3 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.086, size = 342, normalized size = 1.7 \begin{align*} -{\frac{e}{3\,{b}^{2}d} \left ( 8\,\arcsin \left ( dx+c \right ) \sqrt{\pi }\sqrt{{b}^{-1}}\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 ) \sqrt{a+b\arcsin \left ( dx+c \right ) }b-8\,\arcsin \left ( dx+c \right ) \sqrt{\pi }\sqrt{{b}^{-1}}\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 ) \sqrt{a+b\arcsin \left ( dx+c \right ) }b+8\,\sqrt{\pi }\sqrt{{b}^{-1}}\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 ) \sqrt{a+b\arcsin \left ( dx+c \right ) }a-8\,\sqrt{\pi }\sqrt{{b}^{-1}}\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 ) \sqrt{a+b\arcsin \left ( dx+c \right ) }a+4\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+\sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) b+4\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) a \right ) \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \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{5}{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^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} + 2 a b \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d x \right )} + b^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac{d x}{a^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} + 2 a b \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}{\left (c + d x \right )} + b^{2} \sqrt{a + b \operatorname{asin}{\left (c + d x \right )}} \operatorname{asin}^{2}{\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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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