3.323 \(\int x^m \sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=198 \[ \frac{d \text{Unintegrable}\left (\frac{x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}},x\right )}{m+2}+\frac{2 b^2 c^2 x^{m+3} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right )}{(m+2)^2 (m+3) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{m+2}-\frac{2 b c x^{m+2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+2)^2 \sqrt{c^2 x^2+1}} \]

[Out]

(-2*b*c*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((2 + m)^2*Sqrt[1 + c^2*x^2]) + (x^(1 + m)*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2 + m) + (2*b^2*c^2*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometric2F1[1/2,
(3 + m)/2, (5 + m)/2, -(c^2*x^2)])/((2 + m)^2*(3 + m)*Sqrt[1 + c^2*x^2]) + (d*Unintegrable[(x^m*(a + b*ArcSin
h[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(2 + m)

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Rubi [A]  time = 0.136242, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

Defer[Int][x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x]

Rubi steps

\begin{align*} \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 0.29456, size = 0, normalized size = 0. \[ \int x^m \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

Integrate[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x]

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Maple [A]  time = 0.852, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{c}^{2}d{x}^{2}+d} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x)

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Maxima [A]  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 )}^{2} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^2*x^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}\right )} x^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*x^m, x)

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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.

[In]

integrate(x**m*(c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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Giac [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.

[In]

integrate(x^m*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Timed out