\(\int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx\) [215]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))^3}{3 d e}-\frac {2 b \text {Int}\left (\frac {(e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))^2}{\sqrt {-1+c+d x} \sqrt {1+c+d x}},x\right )}{e} \]

[Out]

2/3*(e*(d*x+c))^(3/2)*(a+b*arccosh(d*x+c))^3/d/e-2*b*Unintegrable((e*(d*x+c))^(3/2)*(a+b*arccosh(d*x+c))^2/(d*
x+c-1)^(1/2)/(d*x+c+1)^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 94.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{3} \sqrt {d e x +c e}d x\]

[In]

int((a+b*arccosh(d*x+c))^3*(d*e*x+c*e)^(1/2),x)

[Out]

int((a+b*arccosh(d*x+c))^3*(d*e*x+c*e)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((a+b*arccosh(d*x+c))^3*(d*e*x+c*e)^(1/2),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), x)

Sympy [N/A]

Not integrable

Time = 6.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}\, dx \]

[In]

integrate((a+b*acosh(d*x+c))**3*(d*e*x+c*e)**(1/2),x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccosh(d*x+c))^3*(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((a+b*arccosh(d*x+c))^3*(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x))^3 \, dx=\int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((c*e + d*e*x)^(1/2)*(a + b*acosh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^(1/2)*(a + b*acosh(c + d*x))^3, x)