3.36 \(\int (d+e x)^m (a+b \cosh ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=81 \[ \frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (m+1)}-\frac{3 b c \text{Unintegrable}\left (\frac{(d+e x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}},x\right )}{e (m+1)} \]

[Out]

((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^3)/(e*(1 + m)) - (3*b*c*Unintegrable[((d + e*x)^(1 + m)*(a + b*ArcCosh
[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))

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Rubi [A]  time = 0.383444, 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 = {} \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d + e*x)^m*(a + b*ArcCosh[c*x])^3,x]

[Out]

((d + e*x)^(1 + m)*(a + b*ArcCosh[c*x])^3)/(e*(1 + m)) - (3*b*c*Defer[Int][((d + e*x)^(1 + m)*(a + b*ArcCosh[c
*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(e*(1 + m))

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx &=\frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^3}{e (1+m)}-\frac{(3 b c) \int \frac{(d+e x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 6.27858, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+b \cosh ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d + e*x)^m*(a + b*ArcCosh[c*x])^3,x]

[Out]

Integrate[(d + e*x)^m*(a + b*ArcCosh[c*x])^3, x]

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Maple [A]  time = 3.002, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(a+b*arccosh(c*x))^3,x)

[Out]

int((e*x+d)^m*(a+b*arccosh(c*x))^3,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="fricas")

[Out]

integral((b^3*arccosh(c*x)^3 + 3*a*b^2*arccosh(c*x)^2 + 3*a^2*b*arccosh(c*x) + a^3)*(e*x + d)^m, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(a+b*acosh(c*x))**3,x)

[Out]

Integral((a + b*acosh(c*x))**3*(d + e*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(a+b*arccosh(c*x))^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^3*(e*x + d)^m, x)