3.6.26 \(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^n}{x^2} \, dx\) [526]

Optimal. Leaf size=455 \[ \frac {15 c d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt {d+c^2 d x^2}}+\frac {2^{-2 (3+n)} c d^3 e^{-\frac {4 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d+c^2 d x^2}}+\frac {2^{-2-n} c d^3 e^{-\frac {2 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-2-n} c d^3 e^{\frac {2 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d+c^2 d x^2}}-\frac {2^{-2 (3+n)} c d^3 e^{\frac {4 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d+c^2 d x^2}}+d^3 \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^n}{x^2 \sqrt {d+c^2 d x^2}},x\right ) \]

[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)

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Rubi [A]
time = 0.84, 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^2} \, 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^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*} \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*}

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Mathematica [A]
time = 0.45, 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^2} \, 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^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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}}{x^{2}}\, dx\]

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^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 = 0.00, 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^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 = 0.00, 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^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(-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**2,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^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

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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^2} \,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^2,x)

[Out]

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

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