3.217 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{6 b \text{Unintegrable}\left (\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}},x\right )}{e}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt{e (c+d x)}} \]

[Out]

(-2*(a + b*ArcCosh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)]) + (6*b*Unintegrable[(a + b*ArcCosh[c + d*x])^2/(Sqrt[-
1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x]), x])/e

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Rubi [A]  time = 0.302954, 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 = {} \[ \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(3/2),x]

[Out]

(-2*(a + b*ArcCosh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)]) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^2/(S
qrt[-1 + x]*Sqrt[e*x]*Sqrt[1 + x]), x], x, c + d*x])/(d*e)

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt{e (c+d x)}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}

Mathematica [A]  time = 30.6674, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(3/2),x]

[Out]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(3/2), x]

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Maple [A]  time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{3} \left ( dex+ce \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(3/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**3/(d*e*x+c*e)**(3/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**3/(e*(c + d*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^3/(d*e*x + c*e)^(3/2), x)