Optimal. Leaf size=355 \[ \frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{a/b} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}} \]
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Rubi [A] time = 0.473072, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5782, 5779, 5448, 3308, 2181} \[ \frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{a/b} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5782
Rule 5779
Rule 5448
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac{\sqrt{d+c^2 d x^2} \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \cosh ^2(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int \left (\frac{1}{4} (a+b x)^n \sinh (x)+\frac{1}{4} (a+b x)^n \sinh (3 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2 \sqrt{1+c^2 x^2}}+\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2 \sqrt{1+c^2 x^2}}\\ &=-\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt{1+c^2 x^2}}-\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt{1+c^2 x^2}}+\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt{1+c^2 x^2}}+\frac{\sqrt{d+c^2 d x^2} \operatorname{Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt{1+c^2 x^2}}\\ &=\frac{3^{-1-n} e^{-\frac{3 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1+c^2 x^2}}+\frac{e^{-\frac{a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{1+c^2 x^2}}+\frac{e^{a/b} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt{1+c^2 x^2}}+\frac{3^{-1-n} e^{\frac{3 a}{b}} \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.923611, size = 229, normalized size = 0.65 \[ \frac{d e^{-\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \left (3^{-n} e^{\frac{6 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3 e^{\frac{2 a}{b}} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right )\right )+3 e^{\frac{4 a}{b}} \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )^{-n} \text{Gamma}\left (n+1,\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )}{24 c^2 \sqrt{d \left (c^2 x^2+1\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}\sqrt{{c}^{2}d{x}^{2}+d}\, 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 \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n} x, x\right ) \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*} \int x \sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{n}\, dx \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 \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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