Optimal. Leaf size=245 \[ -\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}+\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d} \]
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Rubi [A] time = 0.650181, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {5865, 12, 5663, 5779, 3312, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}+\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5663
Rule 5779
Rule 3312
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c e+d e x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}-\frac{\left (i b e^2\right ) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 \sqrt{a+b x}}-\frac{i \sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{24 d}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{48 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{48 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}+\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}+\frac{e^2 \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d}-\frac{e^2 \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d}-\frac{e^2 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d}+\frac{e^2 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d}\\ &=\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}-\frac{\sqrt{b} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{\sqrt{b} e^2 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}+\frac{\sqrt{b} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{\sqrt{b} e^2 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{48 d}\\ \end{align*}
Mathematica [A] time = 0.389586, size = 238, normalized size = 0.97 \[ \frac{e^2 e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{72 d \sqrt{-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.209, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{2}\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{\left (d e x + c e\right )}^{2} \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^{2} \left (\int c^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}\, dx + \int d^{2} x^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}}\, dx + \int 2 c 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{\left (d e x + c e\right )}^{2} \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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