3.322 \(\int x^m (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=487 \[ \frac{3 d^2 \text{Unintegrable}\left (\frac{x^m \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{c^2 d x^2+d}},x\right )}{m^2+6 m+8}+\frac{2 b^2 c^2 d (3 m+10) x^{m+3} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right )}{(m+2) (m+3) (m+4)^3 \sqrt{c^2 x^2+1}}+\frac{6 b^2 c^2 d x^{m+3} \sqrt{c^2 d x^2+d} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},-c^2 x^2\right )}{(m+2)^2 (m+3) (m+4) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{m^2+6 m+8}-\frac{2 b c d x^{m+2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{\left (m^2+6 m+8\right ) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{m+4}-\frac{6 b c d x^{m+2} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+2)^2 (m+4) \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^{m+4} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{(m+4)^2 \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d x^{m+3} \sqrt{c^2 d x^2+d}}{(m+4)^3} \]

[Out]

(2*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2])/(4 + m)^3 - (6*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[
c*x]))/((2 + m)^2*(4 + m)*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^(2 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((
8 + 6*m + m^2)*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*x^(4 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/((4 + m)^2*
Sqrt[1 + c^2*x^2]) + (3*d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8 + 6*m + m^2) + (x^(1 + m)*(
d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(4 + m) + (6*b^2*c^2*d*x^(3 + m)*Sqrt[d + c^2*d*x^2]*Hypergeometr
ic2F1[1/2, (3 + m)/2, (5 + m)/2, -(c^2*x^2)])/((2 + m)^2*(3 + m)*(4 + m)*Sqrt[1 + c^2*x^2]) + (2*b^2*c^2*d*(10
+ 3*m)*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)*(3 +
m)*(4 + m)^3*Sqrt[1 + c^2*x^2]) + (3*d^2*Unintegrable[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x])/(8
+ 6*m + m^2)

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Rubi [A]  time = 0.151321, 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 = {} \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

Defer[Int][x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x]

Rubi steps

\begin{align*} \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 0.420097, size = 0, normalized size = 0. \[ \int x^m \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

Integrate[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x]

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Maple [A]  time = 0.997, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}} \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)^(3/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int(x^m*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\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)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^(3/2)*(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 ({\left (a^{2} c^{2} d x^{2} + a^{2} d +{\left (b^{2} c^{2} d x^{2} + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d x^{2} + a b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d} 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)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((a^2*c^2*d*x^2 + a^2*d + (b^2*c^2*d*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*c^2*d*x^2 + a*b*d)*arcsinh(c
*x))*sqrt(c^2*d*x^2 + d)*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)**(3/2)*(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)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Timed out