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

Optimal. Leaf size=755 \[ d^3 \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c x)\right )^n}{x \sqrt{c^2 d x^2+d}},x\right )+\frac{d^3 5^{-n-1} e^{-\frac{5 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{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt{c^2 d x^2+d}}-\frac{5 d^3 3^{-n-1} e^{-\frac{3 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{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt{c^2 d x^2+d}}+\frac{d^3 3^{-n} e^{-\frac{3 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{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt{c^2 d x^2+d}}+\frac{11 d^3 e^{-\frac{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{a+b \sinh ^{-1}(c x)}{b}\right )}{16 \sqrt{c^2 d x^2+d}}+\frac{11 d^3 e^{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{a+b \sinh ^{-1}(c x)}{b}\right )}{16 \sqrt{c^2 d x^2+d}}-\frac{5 d^3 3^{-n-1} e^{\frac{3 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{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt{c^2 d x^2+d}}+\frac{d^3 3^{-n} e^{\frac{3 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{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt{c^2 d x^2+d}}+\frac{d^3 5^{-n-1} e^{\frac{5 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{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt{c^2 d x^2+d}} \]

[Out]

(5^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b])/(32*E^((5*
a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) - (5*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[
c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(32*E^((3*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])
/b))^n) + (d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b])/(8*3^n*E^((
3*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (11*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*
Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)])/(16*E^(a/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n) + (11*
d^3*E^(a/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (a + b*ArcSinh[c*x])/b])/(16*Sqrt[d + c^2*d*
x^2]*((a + b*ArcSinh[c*x])/b)^n) - (5*3^(-1 - n)*d^3*E^((3*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamm
a[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(32*Sqrt[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) + (d^3*E^((3*a)/b)*S
qrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(8*3^n*Sqrt[d + c^2*d*x^2]*(
(a + b*ArcSinh[c*x])/b)^n) + (5^(-1 - n)*d^3*E^((5*a)/b)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*Gamma[1 + n,
(5*(a + b*ArcSinh[c*x]))/b])/(32*Sqrt[d + c^2*d*x^2]*((a + b*ArcSinh[c*x])/b)^n) + d^3*Unintegrable[(a + b*Ar
cSinh[c*x])^n/(x*Sqrt[d + c^2*d*x^2]), x]

________________________________________________________________________________________

Rubi [A]  time = 0.155439, 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 = {} \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Mathematica [A]  time = 0.270226, size = 0, normalized size = 0. \[ \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.152, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{n}}{x} \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,x)

[Out]

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

________________________________________________________________________________________

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}\,{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,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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}, 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,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, x)

________________________________________________________________________________________

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,x)

[Out]

Timed out

________________________________________________________________________________________

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}\,{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,x, algorithm="giac")

[Out]

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