3.221 \(\int \sqrt{c e+d e x} (a+b \cosh ^{-1}(c+d x))^4 \, dx\)

Optimal. Leaf size=88 \[ \frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e}-\frac{8 b \text{Unintegrable}\left (\frac{(e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt{c+d x-1} \sqrt{c+d x+1}},x\right )}{3 e} \]

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^4)/(3*d*e) - (8*b*Unintegrable[((e*(c + d*x))^(3/2)*(a + b*Arc
Cosh[c + d*x])^3)/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]), x])/(3*e)

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Rubi [A]  time = 0.308667, 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 = {} \[ \int \sqrt{c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^4)/(3*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(3/2)*(a + b*
ArcCosh[x])^3)/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, c + d*x])/(3*d*e)

Rubi steps

\begin{align*} \int \sqrt{c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{e x} \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{(e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e}\\ \end{align*}

Mathematica [F]  time = 180.001, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

$Aborted

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Maple [A]  time = 0.424, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{4}\sqrt{dex+ce}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^4*(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arccosh(d*x+c))^4*(d*e*x+c*e)^(1/2),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((a+b*arccosh(d*x+c))^4*(d*e*x+c*e)^(1/2),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^{4} \operatorname{arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arcosh}\left (d x + c\right ) + a^{4}\right )} \sqrt{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

integral((b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a^3*b*arccosh
(d*x + c) + a^4)*sqrt(d*e*x + c*e), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \left (c + d x\right )} \left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**4*(d*e*x+c*e)**(1/2),x)

[Out]

Integral(sqrt(e*(c + d*x))*(a + b*acosh(c + d*x))**4, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d e x + c e}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^4*(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a)^4, x)