3.526 \(\int \frac{(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^n}{x^2} \, dx\)

Optimal. Leaf size=454 \[ d^3 \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c x)\right )^n}{x^2 \sqrt{c^2 d x^2+d}},x\right )+\frac{c d^3 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}+\frac{c d^3 2^{-n-2} e^{-\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-n-2} e^{\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text{Gamma}\left (n+1,\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt{c^2 d x^2+d}}+\frac{15 c d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{8 b (n+1) \sqrt{c^2 d x^2+d}} \]

[Out]

(15*c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d + c^2*d*x^2]) + (c*d^3*Sqrt[1 +
c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*E^((4*a)/b)*Sqrt[d +
c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (2^(-2 - n)*c*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[
1 + n, (-2*(a + b*ArcSinh[c*x]))/b])/(E^((2*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (2^(-2
- n)*c*d^3*E^((2*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(Sqr
t[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) - (c*d^3*E^((4*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gam
ma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(2^(2*(3 + n))*Sqrt[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) + d^3*Un
integrable[(a + b*ArcSinh[c*x])^n/(x^2*Sqrt[d + c^2*d*x^2]), x]

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Rubi [A]  time = 0.155804, 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 = {} \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2,x]

[Out]

Defer[Int][((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2, x]

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx &=\int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.638545, size = 0, normalized size = 0. \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2,x]

[Out]

Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2, x]

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Maple [A]  time = 0.162, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}}{{x}^{2}} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n/x^2,x)

[Out]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n/x^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n/x^2,x, algorithm="maxima")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n/x^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n/x^2,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^4 + 2*c^2*d^2*x^2 + d^2)*sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)^n/x^2, 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((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**n/x**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^n/x^2,x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(b*arcsinh(c*x) + a)^n/x^2, x)