Optimal. Leaf size=256 \[ -\frac{15 \sqrt{\pi } b^{5/2} e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 d}-\frac{15 \sqrt{\pi } b^{5/2} e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
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Rubi [A] time = 0.719408, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4805, 12, 4629, 4707, 4641, 4723, 3312, 3306, 3305, 3351, 3304, 3352} \[ -\frac{15 \sqrt{\pi } b^{5/2} e \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{128 d}-\frac{15 \sqrt{\pi } b^{5/2} e \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{128 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4629
Rule 4707
Rule 4641
Rule 4723
Rule 3312
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int (c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{(5 b e) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{(5 b e) \operatorname{Subst}\left (\int \frac{\left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac{\left (15 b^2 e\right ) \operatorname{Subst}\left (\int x \sqrt{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2} \sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{\sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}-\frac{\cos (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^3 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 )}{128 d}-\frac{\left (15 b^3 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 )}{128 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^2 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 )}{64 d}-\frac{\left (15 b^2 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 )}{64 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{8 d}-\frac{e \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{15 b^{5/2} 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 )}{128 d}-\frac{15 b^{5/2} 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 )}{128 d}\\ \end{align*}
Mathematica [C] time = 0.0939008, size = 154, normalized size = 0.6 \[ \frac{e e^{-\frac{2 i a}{b}} \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \left (\sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{7}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{32 \sqrt{2} d \left (\frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.095, size = 435, normalized size = 1.7 \begin{align*} -{\frac{e}{128\,d} \left ( 15\,\sqrt{{b}^{-1}}\sqrt{\pi }\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{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{3}+15\,\sqrt{{b}^{-1}}\sqrt{\pi }\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{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{3}+32\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{3}\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}+96\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}-40\, \left ( \arcsin \left ( dx+c \right ) \right ) ^{2}\sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}+96\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}b-30\,\arcsin \left ( dx+c \right ) \cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){b}^{3}-80\,\arcsin \left ( dx+c \right ) \sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}+32\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{3}-30\,\cos \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ) a{b}^{2}-40\,\sin \left ( 2\,{\frac{a+b\arcsin \left ( dx+c \right ) }{b}}-2\,{\frac{a}{b}} \right ){a}^{2}b \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{\left (d e x + c e\right )}{\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(-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 = 2.17891, size = 1848, normalized size = 7.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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