3.458 \(\int (f x)^m \sqrt{d-c^2 d x^2} (a+b \cosh ^{-1}(c x))^n \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.401181, 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 = {} \[ \int (f x)^m \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.104005, size = 0, normalized size = 0. \[ \int (f x)^m \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.401, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m}\sqrt{-{c}^{2}d{x}^{2}+d} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n,x)

[Out]

int((f*x)^m*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(f*x)^m*(b*arccosh(c*x) + a)^n, 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*(-c**2*d*x**2+d)**(1/2)*(a+b*acosh(c*x))**n,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x