Optimal. Leaf size=262 \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}-\frac{5 b e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
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Rubi [A] time = 0.731318, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5865, 12, 5663, 5758, 5675, 5779, 3312, 3307, 2180, 2204, 2205} \[ -\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}-\frac{5 b e (c+d x) \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5663
Rule 5758
Rule 5675
Rule 5779
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{(5 b e) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \sinh ^{-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 \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac{(5 b e) \operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-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 \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{16 d}\\ &=\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-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 \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}\\ &=\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-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{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{128 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{256 d}-\frac{\left (15 b^3 e\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{256 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{\left (15 b^2 e\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{128 d}-\frac{\left (15 b^2 e\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{128 d}\\ &=\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}-\frac{5 b e (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{8 d}+\frac{e \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}-\frac{15 b^{5/2} e e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{15 b^{5/2} e e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}\\ \end{align*}
Mathematica [A] time = 0.0757203, size = 126, normalized size = 0.48 \[ \frac{e e^{-\frac{2 a}{b}} \left (b^3 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{7}{2},\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-b^3 \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{7}{2},-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{32 \sqrt{2} d \sqrt{a+b \sinh ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\, dx \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 \operatorname{arsinh}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\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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