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

Optimal. Leaf size=84 \[ d \text {Int}\left (\frac {\left (a+b \sinh ^{-1}(c x)\right )^n}{x^2 \sqrt {c^2 d x^2+d}},x\right )+\frac {c d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{b (n+1) \sqrt {c^2 d x^2+d}} \]

[Out]

c*d*(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)+d*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 = 0.15, 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 = {} \[ \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arcsinh \left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,x)

[Out]

int((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{n}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^n*(c^2*d*x^2+d)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2))/x^2,x)

[Out]

int(((a + b*asinh(c*x))^n*(d + c^2*d*x^2)^(1/2))/x^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**n*(c**2*d*x**2+d)**(1/2)/x**2,x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**n/x**2, x)

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