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

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {\sqrt {c^2 x^2+1} x^m}{\left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(x^m*(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x))^2,x)

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {x^m \sqrt {1+c^2 x^2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]

[Out]

Defer[Int][(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {x^m \sqrt {1+c^2 x^2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]

[Out]

Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x]

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{m}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c^2*x^2 + 1)*x^m/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^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(x^m*(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*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 [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \sqrt {c^{2} x^{2}+1}}{\left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{2} x^{2} + 1\right )}^{2} x^{m} + {\left (c^{3} x^{3} + c x\right )} \sqrt {c^{2} x^{2} + 1} x^{m}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (c^{3} {\left (m + 2\right )} x^{3} + c {\left (m - 1\right )} x\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{m} + {\left (2 \, c^{4} {\left (m + 2\right )} x^{4} + c^{2} {\left (3 \, m + 2\right )} x^{2} + m\right )} {\left (c^{2} x^{2} + 1\right )} x^{m} + {\left (c^{5} {\left (m + 2\right )} x^{5} + c^{3} {\left (2 \, m + 3\right )} x^{3} + c {\left (m + 1\right )} x\right )} \sqrt {c^{2} x^{2} + 1} x^{m}}{a b c^{5} x^{5} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{3} + 2 \, a b c^{3} x^{3} + a b c x + {\left (b^{2} c^{5} x^{5} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{3} + 2 \, b^{2} c^{3} x^{3} + b^{2} c x + 2 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{4} + a b c^{2} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*x^2 + 1)^2*x^m + (c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1)*x^m)/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*
b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((c^3*(m +
2)*x^3 + c*(m - 1)*x)*(c^2*x^2 + 1)^(3/2)*x^m + (2*c^4*(m + 2)*x^4 + c^2*(3*m + 2)*x^2 + m)*(c^2*x^2 + 1)*x^m
+ (c^5*(m + 2)*x^5 + c^3*(2*m + 3)*x^3 + c*(m + 1)*x)*sqrt(c^2*x^2 + 1)*x^m)/(a*b*c^5*x^5 + (c^2*x^2 + 1)*a*b
*c^3*x^3 + 2*a*b*c^3*x^3 + a*b*c*x + (b^2*c^5*x^5 + (c^2*x^2 + 1)*b^2*c^3*x^3 + 2*b^2*c^3*x^3 + b^2*c*x + 2*(b
^2*c^4*x^4 + b^2*c^2*x^2)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^4 + a*b*c^2*x^2)*sqrt
(c^2*x^2 + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m\,\sqrt {c^2\,x^2+1}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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