Optimal. Leaf size=217 \[ -\frac{\sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d} \]
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Rubi [A] time = 0.455371, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {5865, 12, 5669, 5448, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5669
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^3}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^3 x^3}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{a+b x}}+\frac{\sinh (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}-\frac{e^3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac{e^3 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}+\frac{e^3 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}+\frac{e^3 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}-\frac{e^3 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac{e^3 \operatorname{Subst}\left (\int e^{\frac{4 a}{b}-\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 b d}+\frac{e^3 \operatorname{Subst}\left (\int e^{-\frac{4 a}{b}+\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 b d}+\frac{e^3 \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 )}{4 b d}-\frac{e^3 \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 )}{4 b d}\\ &=-\frac{e^3 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}+\frac{e^3 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 )}{8 \sqrt{b} d}+\frac{e^3 e^{-\frac{4 a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32 \sqrt{b} d}-\frac{e^3 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 )}{8 \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.224919, size = 205, normalized size = 0.94 \[ \frac{e^3 e^{-\frac{4 a}{b}} \left (\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-2 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{1}{2},\frac{4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-2 \sqrt{2} \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{32 d \sqrt{a+b \sinh ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.219, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dex+ce \right ) ^{3}{\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( dx+c \right ) }}}}\, 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 \frac{{\left (d e x + c e\right )}^{3}}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{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^{3} \left (\int \frac{c^{3}}{\sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}\, dx + \int \frac{d^{3} x^{3}}{\sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}\, dx + \int \frac{3 c d^{2} x^{2}}{\sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}}\, dx + \int \frac{3 c^{2} d x}{\sqrt{a + b \operatorname{asinh}{\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{{\left (d e x + c e\right )}^{3}}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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