3.238 \(\int \frac{(f x)^m (a+b \cosh ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}},x\right ) \]

[Out]

Unintegrable[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]

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Rubi [A]  time = 0.556631, 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 = {} \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Defer[Int][((f*x)^m*(a + b*ArcCosh[c*x])^2)/((-1 + c*x)^(5/2)*(1 + c*x)^(5/2)),
x])/(d^2*Sqrt[d - c^2*d*x^2])

Rubi steps

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

Mathematica [A]  time = 4.62682, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]

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Maple [A]  time = 0.55, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)

[Out]

int((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} \left (f x\right )^{m}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)^2*(f*x)^m/(-c^2*d*x^2 + d)^(5/2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )} \left (f x\right )^{m}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*(f*x)^m/(c^6*d^3*x^6 - 3*c^4*d^
3*x^4 + 3*c^2*d^3*x^2 - d^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((f*x)**m*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2*(f*x)^m/(-c^2*d*x^2 + d)^(5/2), x)