Optimal. Leaf size=545 \[ \frac{c^2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac{3^{-n-1} e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac{c 2^{-n-2} e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}+\frac{e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}-\frac{c 2^{-n-2} e^{\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \]
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Rubi [A] time = 1.15162, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5865, 5805, 6741, 12, 6742, 3307, 2181, 5448, 3308} \[ \frac{c^2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac{3^{-n-1} e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac{c 2^{-n-2} e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}+\frac{e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}-\frac{c 2^{-n-2} e^{\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 5805
Rule 6741
Rule 12
Rule 6742
Rule 3307
Rule 2181
Rule 5448
Rule 3308
Rubi steps
\begin{align*} \int x^2 \left (a+b \sinh ^{-1}(c+d x)\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{c}{d}+\frac{x}{d}\right )^2 \left (a+b \sinh ^{-1}(x)\right )^n \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \left (-\frac{c}{d}+\frac{\sinh (x)}{d}\right )^2 \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n \cosh (x) (c-\sinh (x))^2}{d^2} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cosh (x) (c-\sinh (x))^2 \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (c^2 (a+b x)^n \cosh (x)-2 c (a+b x)^n \cosh (x) \sinh (x)+(a+b x)^n \cosh (x) \sinh ^2(x)\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}-\frac{(2 c) \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \sinh (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}+\frac{c^2 \operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{4} (a+b x)^n \cosh (x)+\frac{1}{4} (a+b x)^n \cosh (3 x)\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{2} (a+b x)^n \sinh (2 x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d^3}\\ &=\frac{c^2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{\operatorname{Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{4 d^3}+\frac{\operatorname{Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{4 d^3}-\frac{c \operatorname{Subst}\left (\int (a+b x)^n \sinh (2 x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d^3}\\ &=\frac{c^2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac{\operatorname{Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d^3}-\frac{\operatorname{Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d^3}-\frac{\operatorname{Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d^3}+\frac{\operatorname{Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d^3}+\frac{c \operatorname{Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d^3}-\frac{c \operatorname{Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 d^3}\\ &=\frac{3^{-1-n} e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}-\frac{2^{-2-n} c e^{-\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}+\frac{c^2 e^{-\frac{a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}+\frac{e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 d^3}-\frac{c^2 e^{a/b} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{2 d^3}-\frac{2^{-2-n} c e^{\frac{2 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{d^3}-\frac{3^{-1-n} e^{\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{-n} \Gamma \left (1+n,\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 d^3}\\ \end{align*}
Mathematica [A] time = 0.966835, size = 345, normalized size = 0.63 \[ \frac{2^{-n-3} 3^{-n-1} e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^n \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{-n} \left (\left (4 c^2-1\right ) 2^n 3^{n+1} e^{\frac{2 a}{b}} \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^n \text{Gamma}\left (n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-\left (4 c^2-1\right ) 2^n 3^{n+1} e^{\frac{4 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^n \text{Gamma}\left (n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+2^n \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^n \text{Gamma}\left (n+1,-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-2 c 3^{n+1} e^{a/b} \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )^n \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-e^{\frac{5 a}{b}} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^n \left (2 c 3^{n+1} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+2^n e^{a/b} \text{Gamma}\left (n+1,\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.216, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{n}\, 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 (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2}\,{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 ({\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2}, 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^{2} \left (a + b \operatorname{asinh}{\left (c + d 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{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{n} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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