3.3.21 \(\int \sqrt {c e+d e x} (a+b \cosh ^{-1}(c+d x))^4 \, dx\) [221]

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

[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 = 0.20, 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 \sqrt {c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}

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 {\text {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) \text {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 [A]
time = 136.84, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*(d*x*e + c*e)^(3/2)*a^4*e^(-1)/d + 2/3*(b^4*d*x*e^(1/2) + b^4*c*e^(1/2))*sqrt(d*x + c)*log(d*x + sqrt(d*x
+ c + 1)*sqrt(d*x + c - 1) + c)^4/d + integrate(2/3*(2*(((3*a*b^3*d^2 - 2*b^4*d^2)*x^2*e^(1/2) + 2*(3*a*b^3*c*
d - 2*b^4*c*d)*x*e^(1/2) - (2*b^4*c^2 - 3*(c^2 - 1)*a*b^3)*e^(1/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x +
c - 1) + ((3*a*b^3*d^3 - 2*b^4*d^3)*x^3*e^(1/2) + 3*(3*a*b^3*c*d^2 - 2*b^4*c*d^2)*x^2*e^(1/2) + (3*(3*c^2*d -
d)*a*b^3 - 2*(3*c^2*d - d)*b^4)*x*e^(1/2) + (3*(c^3 - c)*a*b^3 - 2*(c^3 - c)*b^4)*e^(1/2))*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*x^2*e^(1/2) + 2*a^2*b^2*c*d*x*e^(1/2) + (c
^2 - 1)*a^2*b^2*e^(1/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^2*b^2*d^3*x^3*e^(1/2) + 3*a^2*
b^2*c*d^2*x^2*e^(1/2) + (3*c^2*d - d)*a^2*b^2*x*e^(1/2) + (c^3 - c)*a^2*b^2*e^(1/2))*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*x^2*e^(1/2) + 2*a^3*b*c*d*x*e^(1/2) + (c^2 - 1)*a^3
*b*e^(1/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^3*b*d^3*x^3*e^(1/2) + 3*a^3*b*c*d^2*x^2*e^(
1/2) + (3*c^2*d - d)*a^3*b*x*e^(1/2) + (c^3 - c)*a^3*b*e^(1/2))*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqr
t(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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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((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*x + c)*e^(1/2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {e \left (c + d x\right )} \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((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 = 0.00, 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((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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {c\,e+d\,e\,x}\,{\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)^(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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