3.6.25 \(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^n}{x} \, dx\) [525]

Optimal. Leaf size=756 \[ \frac {5^{-1-n} d^3 e^{-\frac {5 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{-\frac {3 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{-\frac {3 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{-\frac {a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}+\frac {11 d^3 e^{a/b} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 \sqrt {d+c^2 d x^2}}-\frac {5\ 3^{-1-n} d^3 e^{\frac {3 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+\frac {3^{-n} d^3 e^{\frac {3 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d+c^2 d x^2}}+\frac {5^{-1-n} d^3 e^{\frac {5 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 \sqrt {d+c^2 d x^2}}+d^3 \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^n}{x \sqrt {d+c^2 d x^2}},x\right ) \]

[Out]

1/32*5^(-1-n)*d^3*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-5*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/exp(5*a/b)/(((-a-b
*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)-5/32*3^(-1-n)*d^3*(a+b*arcsinh(c*x))^n*GAMMA(1+n,-3*(a+b*arcsinh(c*x)
)/b)*(c^2*x^2+1)^(1/2)/exp(3*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)+1/8*d^3*(a+b*arcsinh(c*x))^n
*GAMMA(1+n,-3*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/(3^n)/exp(3*a/b)/(((-a-b*arcsinh(c*x))/b)^n)/(c^2*d*x^2+
d)^(1/2)+11/16*d^3*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(-a-b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/exp(a/b)/(((-a-b*ar
csinh(c*x))/b)^n)/(c^2*d*x^2+d)^(1/2)+11/16*d^3*exp(a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,(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)-5/32*3^(-1-n)*d^3*exp(3*a/b)*(a+b*arcsinh(c*x
))^n*GAMMA(1+n,3*(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)+1/8*d^
3*exp(3*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,3*(a+b*arcsinh(c*x))/b)*(c^2*x^2+1)^(1/2)/(3^n)/(((a+b*arcsinh(c*x
))/b)^n)/(c^2*d*x^2+d)^(1/2)+1/32*5^(-1-n)*d^3*exp(5*a/b)*(a+b*arcsinh(c*x))^n*GAMMA(1+n,5*(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)+d^3*Unintegrable((a+b*arcsinh(c*x))^n/x/(c
^2*d*x^2+d)^(1/2),x)

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Rubi [A]
time = 1.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

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]

(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*Defer[Int][(a + b*ArcS
inh[c*x])^n/(x*Sqrt[d + c^2*d*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} \, 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*}

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Mathematica [A]
time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

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]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}}{x}\, dx\]

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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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