\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx\) [526]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\frac {15 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b (1+n) \sqrt {d+c^2 d x^2}}+\frac {2^{-2 (3+n)} c d^3 e^{-\frac {4 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+\frac {2^{-2-n} c d^3 e^{-\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-2-n} c d^3 e^{\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-2 (3+n)} c d^3 e^{\frac {4 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+d^3 \text {Int}\left (\frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}},x\right ) \]

[Out]

15/8*c*d^3*(a+b*arcsinh(c*x))^(1+n)*(c^2*x^2+1)^(1/2)/b/(1+n)/(c^2*d*x^2+d)^(1/2)+c*d^3*(a+b*arcsinh(c*x))^n*G
AMMA(1+n,-4*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/(2^(6+2*n))/exp(4*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*d*
x^2+d)^(1/2)+2^(-2-n)*c*d^3*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/exp(2*a/
b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)-2^(-2-n)*c*d^3*exp(2*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,2*
(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/(((a+b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)-c*d^3*exp(4*a/b)*(a+b*a
rcsinh(c*x))^n*GAMMA(1+n,4*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/(2^(6+2*n))/(((a+b*arcsinh(c*x))/b)^n)/(c^2
*d*x^2+d)^(1/2)+d^3*Unintegrable((a+b*arcsinh(c*x))^n/x^2/(c^2*d*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]

[In]

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

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 c^2 d^3 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}+\frac {d^3 (a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}}+\frac {3 c^4 d^3 x^2 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}+\frac {c^6 d^3 x^4 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}}\right ) \, dx \\ & = d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\left (3 c^2 d^3\right ) \int \frac {(a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx+\left (3 c^4 d^3\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx+\left (c^6 d^3\right ) \int \frac {x^4 (a+b \text {arcsinh}(c x))^n}{\sqrt {d+c^2 d x^2}} \, dx \\ & = \frac {3 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{b (1+n) \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}} \\ & = \frac {3 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{b (1+n) \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 x^n}{8}+\frac {1}{8} x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )-\frac {1}{2} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}}-\frac {\left (3 c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {x^n}{2}-\frac {1}{2} x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b \sqrt {d+c^2 d x^2}} \\ & = \frac {15 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b (1+n) \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {4 a}{b}-\frac {4 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b \sqrt {d+c^2 d x^2}}-\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int x^n \cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b \sqrt {d+c^2 d x^2}} \\ & = \frac {15 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b (1+n) \sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx+\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}+\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {4 i a}{b}-\frac {4 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b \sqrt {d+c^2 d x^2}}-\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}}-\frac {\left (c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}}+\frac {\left (3 c d^3 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )} x^n \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b \sqrt {d+c^2 d x^2}} \\ & = \frac {15 c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{1+n}}{8 b (1+n) \sqrt {d+c^2 d x^2}}+\frac {4^{-3-n} c d^3 e^{-\frac {4 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+\frac {2^{-2-n} c d^3 e^{-\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-2-n} c d^3 e^{\frac {2 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {4^{-3-n} c d^3 e^{\frac {4 a}{b}} \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^n \left (\frac {a+b \text {arcsinh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {d+c^2 d x^2}}+d^3 \int \frac {(a+b \text {arcsinh}(c x))^n}{x^2 \sqrt {d+c^2 d x^2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{n}}{x^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 2.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^n}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^2} \,d x \]

[In]

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

[Out]

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