3.3.20 \(\int (c e+d e x)^{3/2} (a+b \cosh ^{-1}(c+d x))^4 \, dx\) [220]

Optimal. Leaf size=89 \[ \frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt {-1+c+d x} \sqrt {1+c+d x}},x\right )}{5 e} \]

[Out]

2/5*(e*(d*x+c))^(5/2)*(a+b*arccosh(d*x+c))^4/d/e-8/5*b*Unintegrable((e*(d*x+c))^(5/2)*(a+b*arccosh(d*x+c))^3/(
d*x+c-1)^(1/2)/(d*x+c+1)^(1/2),x)/e

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh[c + d*x])^4)/(5*d*e) - (8*b*Defer[Subst][Defer[Int][((e*x)^(5/2)*(a + b*
ArcCosh[x])^3)/(Sqrt[-1 + x]*Sqrt[1 + x]), x], x, c + d*x])/(5*d*e)

Rubi steps

\begin {align*} \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int (e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e}-\frac {(8 b) \text {Subst}\left (\int \frac {(e x)^{5/2} \left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]
time = 180.01, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcCosh[c + d*x])^4,x]

[Out]

$Aborted

________________________________________________________________________________________

Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c))^4,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

2/5*(d*x*e + c*e)^(5/2)*a^4*e^(-1)/d + 2/5*(b^4*d^2*x^2*e^(3/2) + 2*b^4*c*d*x*e^(3/2) + b^4*c^2*e^(3/2))*sqrt(
d*x + c)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/d + integrate(2/5*(2*(((5*a*b^3*d^3 - 2*b^4*d^3)
*x^3*e^(3/2) + 3*(5*a*b^3*c*d^2 - 2*b^4*c*d^2)*x^2*e^(3/2) - (6*b^4*c^2*d - 5*(3*c^2*d - d)*a*b^3)*x*e^(3/2) -
(2*b^4*c^3 - 5*(c^3 - c)*a*b^3)*e^(3/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + ((5*a*b^3*d^4 -
2*b^4*d^4)*x^4*e^(3/2) + 4*(5*a*b^3*c*d^3 - 2*b^4*c*d^3)*x^3*e^(3/2) + (5*(6*c^2*d^2 - d^2)*a*b^3 - 2*(6*c^2*d
^2 - d^2)*b^4)*x^2*e^(3/2) + 2*(5*(2*c^3*d - c*d)*a*b^3 - 2*(2*c^3*d - c*d)*b^4)*x*e^(3/2) + (5*(c^4 - c^2)*a*
b^3 - 2*(c^4 - c^2)*b^4)*e^(3/2))*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 15*((a
^2*b^2*d^3*x^3*e^(3/2) + 3*a^2*b^2*c*d^2*x^2*e^(3/2) + (3*c^2*d - d)*a^2*b^2*x*e^(3/2) + (c^3 - c)*a^2*b^2*e^(
3/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^2*b^2*d^4*x^4*e^(3/2) + 4*a^2*b^2*c*d^3*x^3*e^(3/
2) + (6*c^2*d^2 - d^2)*a^2*b^2*x^2*e^(3/2) + 2*(2*c^3*d - c*d)*a^2*b^2*x*e^(3/2) + (c^4 - c^2)*a^2*b^2*e^(3/2)
)*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 + 10*((a^3*b*d^3*x^3*e^(3/2) + 3*a^3*b*c
*d^2*x^2*e^(3/2) + (3*c^2*d - d)*a^3*b*x*e^(3/2) + (c^3 - c)*a^3*b*e^(3/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sq
rt(d*x + c - 1) + (a^3*b*d^4*x^4*e^(3/2) + 4*a^3*b*c*d^3*x^3*e^(3/2) + (6*c^2*d^2 - d^2)*a^3*b*x^2*e^(3/2) + 2
*(2*c^3*d - c*d)*a^3*b*x*e^(3/2) + (c^4 - c^2)*a^3*b*e^(3/2))*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(
d*x + c - 1) + c))/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x + c + 1)*sqrt(d*x + c
- 1) + (3*c^2*d - d)*x - c), x)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral(((b^4*d*x + b^4*c)*arccosh(d*x + c)^4*e + 4*(a*b^3*d*x + a*b^3*c)*arccosh(d*x + c)^3*e + 6*(a^2*b^2*d
*x + a^2*b^2*c)*arccosh(d*x + c)^2*e + 4*(a^3*b*d*x + a^3*b*c)*arccosh(d*x + c)*e + (a^4*d*x + a^4*c)*e)*sqrt(
d*x + c)*e^(1/2), x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e \left (c + d x\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(3/2)*(a+b*acosh(d*x+c))**4,x)

[Out]

Integral((e*(c + d*x))**(3/2)*(a + b*acosh(c + d*x))**4, x)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^(3/2)*(b*arccosh(d*x + c) + a)^4, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(3/2)*(a + b*acosh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^(3/2)*(a + b*acosh(c + d*x))^4, x)

________________________________________________________________________________________