3.433 \(\int \frac{(1+c^2 x^2)^{5/2}}{x^3 (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{\left (c^2 x^2+1\right )^{5/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]

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Rubi [A]  time = 0.143702, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (1+c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2),x]

[Out]

Defer[Int][(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]

Rubi steps

\begin{align*} \int \frac{\left (1+c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \frac{\left (1+c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 14.0174, size = 0, normalized size = 0. \[ \int \frac{\left (1+c^2 x^2\right )^{5/2}}{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]

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Maple [A]  time = 0.807, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}} \left ({c}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x))^2,x)

[Out]

int((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{6} x^{6} + 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{7} x^{7} + 3 \, c^{5} x^{5} + 3 \, c^{3} x^{3} + c x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{3} x^{5} + \sqrt{c^{2} x^{2} + 1} a b c^{2} x^{4} + a b c x^{3} +{\left (b^{2} c^{3} x^{5} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x^{4} + b^{2} c x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (3 \, c^{7} x^{7} + 2 \, c^{5} x^{5} - 5 \, c^{3} x^{3} - 4 \, c x\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 3 \,{\left (2 \, c^{8} x^{8} + 3 \, c^{6} x^{6} - c^{4} x^{4} - 3 \, c^{2} x^{2} - 1\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (3 \, c^{9} x^{9} + 7 \, c^{7} x^{7} + 3 \, c^{5} x^{5} - 3 \, c^{3} x^{3} - 2 \, c x\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} x^{8} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{6} + 2 \, a b c^{3} x^{6} + a b c x^{4} +{\left (b^{2} c^{5} x^{8} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{6} + 2 \, b^{2} c^{3} x^{6} + b^{2} c x^{4} + 2 \,{\left (b^{2} c^{4} x^{7} + b^{2} c^{2} x^{5}\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^{7} + a b c^{2} x^{5}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^6 + 3*c^4*x^4 + 3*c^2*x^2 + 1)*(c^2*x^2 + 1) + (c^7*x^7 + 3*c^5*x^5 + 3*c^3*x^3 + c*x)*sqrt(c^2*x^2 +
1))/(a*b*c^3*x^5 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^4 + a*b*c*x^3 + (b^2*c^3*x^5 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^4 +
b^2*c*x^3)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((3*c^7*x^7 + 2*c^5*x^5 - 5*c^3*x^3 - 4*c*x)*(c^2*x^2 +
1)^(3/2) + 3*(2*c^8*x^8 + 3*c^6*x^6 - c^4*x^4 - 3*c^2*x^2 - 1)*(c^2*x^2 + 1) + (3*c^9*x^9 + 7*c^7*x^7 + 3*c^5*
x^5 - 3*c^3*x^3 - 2*c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^8 + (c^2*x^2 + 1)*a*b*c^3*x^6 + 2*a*b*c^3*x^6 + a*b*c*x
^4 + (b^2*c^5*x^8 + (c^2*x^2 + 1)*b^2*c^3*x^6 + 2*b^2*c^3*x^6 + b^2*c*x^4 + 2*(b^2*c^4*x^7 + b^2*c^2*x^5)*sqrt
(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^7 + a*b*c^2*x^5)*sqrt(c^2*x^2 + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )} \sqrt{c^{2} x^{2} + 1}}{b^{2} x^{3} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^4 + 2*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1)/(b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^
3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*x**2+1)**(5/2)/x**3/(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*x^2+1)^(5/2)/x^3/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate((c^2*x^2 + 1)^(5/2)/((b*arcsinh(c*x) + a)^2*x^3), x)