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

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

[Out]

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

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Rubi [A]  time = 0.31, 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 = {} \[ \int \sqrt {c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

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*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

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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fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]

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)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4} \sqrt {d e x +c e}\, dx \]

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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b^{4} d \sqrt {e} x + b^{4} c \sqrt {e}\right )} \sqrt {d x + c} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4}}{3 \, d} + \frac {2 \, {\left (d e x + c e\right )}^{\frac {3}{2}} a^{4}}{3 \, d e} + \int -\frac {2 \, {\left (2 \, {\left ({\left (2 \, b^{4} c^{2} \sqrt {e} - 3 \, {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a b^{3} - {\left (3 \, a b^{3} d^{2} \sqrt {e} - 2 \, b^{4} d^{2} \sqrt {e}\right )} x^{2} - 2 \, {\left (3 \, a b^{3} c d \sqrt {e} - 2 \, b^{4} c d \sqrt {e}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} - {\left (3 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a b^{3} - 2 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} b^{4} + {\left (3 \, a b^{3} d^{3} \sqrt {e} - 2 \, b^{4} d^{3} \sqrt {e}\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c d^{2} \sqrt {e} - 2 \, b^{4} c d^{2} \sqrt {e}\right )} x^{2} + {\left (3 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a b^{3} - 2 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} b^{4}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} - 9 \, {\left ({\left (a^{2} b^{2} d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b^{2} c d \sqrt {e} x + {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (a^{2} b^{2} d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b^{2} c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a^{2} b^{2} x + {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} - 6 \, {\left ({\left (a^{3} b d^{2} \sqrt {e} x^{2} + 2 \, a^{3} b c d \sqrt {e} x + {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a^{3} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (a^{3} b d^{3} \sqrt {e} x^{3} + 3 \, a^{3} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a^{3} b x + {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a^{3} b\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c\right )}}\,{d x} \]

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]

2/3*(b^4*d*sqrt(e)*x + b^4*c*sqrt(e))*sqrt(d*x + c)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/d + 2
/3*(d*e*x + c*e)^(3/2)*a^4/(d*e) + integrate(-2/3*(2*((2*b^4*c^2*sqrt(e) - 3*(c^2*sqrt(e) - sqrt(e))*a*b^3 - (
3*a*b^3*d^2*sqrt(e) - 2*b^4*d^2*sqrt(e))*x^2 - 2*(3*a*b^3*c*d*sqrt(e) - 2*b^4*c*d*sqrt(e))*x)*sqrt(d*x + c + 1
)*sqrt(d*x + c)*sqrt(d*x + c - 1) - (3*(c^3*sqrt(e) - c*sqrt(e))*a*b^3 - 2*(c^3*sqrt(e) - c*sqrt(e))*b^4 + (3*
a*b^3*d^3*sqrt(e) - 2*b^4*d^3*sqrt(e))*x^3 + 3*(3*a*b^3*c*d^2*sqrt(e) - 2*b^4*c*d^2*sqrt(e))*x^2 + (3*(3*c^2*d
*sqrt(e) - d*sqrt(e))*a*b^3 - 2*(3*c^2*d*sqrt(e) - d*sqrt(e))*b^4)*x)*sqrt(d*x + c))*log(d*x + sqrt(d*x + c +
1)*sqrt(d*x + c - 1) + c)^3 - 9*((a^2*b^2*d^2*sqrt(e)*x^2 + 2*a^2*b^2*c*d*sqrt(e)*x + (c^2*sqrt(e) - sqrt(e))*
a^2*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^2*b^2*d^3*sqrt(e)*x^3 + 3*a^2*b^2*c*d^2*sqrt(e
)*x^2 + (3*c^2*d*sqrt(e) - d*sqrt(e))*a^2*b^2*x + (c^3*sqrt(e) - c*sqrt(e))*a^2*b^2)*sqrt(d*x + c))*log(d*x +
sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 - 6*((a^3*b*d^2*sqrt(e)*x^2 + 2*a^3*b*c*d*sqrt(e)*x + (c^2*sqrt(e)
- sqrt(e))*a^3*b)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^3*b*d^3*sqrt(e)*x^3 + 3*a^3*b*c*d^2*s
qrt(e)*x^2 + (3*c^2*d*sqrt(e) - d*sqrt(e))*a^3*b*x + (c^3*sqrt(e) - c*sqrt(e))*a^3*b)*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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{4}\, dx \]

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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