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

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

[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

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Rubi [A]  time = 0.32, 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 = {} \[ \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

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 {\operatorname {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) \operatorname {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*}

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Mathematica [A]  time = 75.45, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

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]

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

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fricas [A]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} d e x + a^{4} c e + {\left (b^{4} d e x + b^{4} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, {\left (a^{2} b^{2} d e x + a^{2} b^{2} c e\right )} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, {\left (a^{3} b d e x + a^{3} b c e\right )} \operatorname {arcosh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [F(-2)]  time = 180.00, 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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

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*e*x + c*e)^(5/2)*a^4/(d*e) + 2/5*(b^4*d^2*e^(3/2)*x^2 + 2*b^4*c*d*e^(3/2)*x + 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*((2*b^4*c^3*e^(3/2) - 5*(c^3*
e^(3/2) - c*e^(3/2))*a*b^3 - (5*a*b^3*d^3*e^(3/2) - 2*b^4*d^3*e^(3/2))*x^3 - 3*(5*a*b^3*c*d^2*e^(3/2) - 2*b^4*
c*d^2*e^(3/2))*x^2 + (6*b^4*c^2*d*e^(3/2) - 5*(3*c^2*d*e^(3/2) - d*e^(3/2))*a*b^3)*x)*sqrt(d*x + c + 1)*sqrt(d
*x + c)*sqrt(d*x + c - 1) - (5*(c^4*e^(3/2) - c^2*e^(3/2))*a*b^3 - 2*(c^4*e^(3/2) - c^2*e^(3/2))*b^4 + (5*a*b^
3*d^4*e^(3/2) - 2*b^4*d^4*e^(3/2))*x^4 + 4*(5*a*b^3*c*d^3*e^(3/2) - 2*b^4*c*d^3*e^(3/2))*x^3 + (5*(6*c^2*d^2*e
^(3/2) - d^2*e^(3/2))*a*b^3 - 2*(6*c^2*d^2*e^(3/2) - d^2*e^(3/2))*b^4)*x^2 + 2*(5*(2*c^3*d*e^(3/2) - c*d*e^(3/
2))*a*b^3 - 2*(2*c^3*d*e^(3/2) - c*d*e^(3/2))*b^4)*x)*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*e^(3/2)*x^3 + 3*a^2*b^2*c*d^2*e^(3/2)*x^2 + (3*c^2*d*e^(3/2) - d*e^(3/2))*a^2*b
^2*x + (c^3*e^(3/2) - c*e^(3/2))*a^2*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^2*b^2*d^4*e^(
3/2)*x^4 + 4*a^2*b^2*c*d^3*e^(3/2)*x^3 + (6*c^2*d^2*e^(3/2) - d^2*e^(3/2))*a^2*b^2*x^2 + 2*(2*c^3*d*e^(3/2) -
c*d*e^(3/2))*a^2*b^2*x + (c^4*e^(3/2) - c^2*e^(3/2))*a^2*b^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*e^(3/2)*x^3 + 3*a^3*b*c*d^2*e^(3/2)*x^2 + (3*c^2*d*e^(3/2) - d*e^(3/2))*a
^3*b*x + (c^3*e^(3/2) - c*e^(3/2))*a^3*b)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^3*b*d^4*e^(3/
2)*x^4 + 4*a^3*b*c*d^3*e^(3/2)*x^3 + (6*c^2*d^2*e^(3/2) - d^2*e^(3/2))*a^3*b*x^2 + 2*(2*c^3*d*e^(3/2) - c*d*e^
(3/2))*a^3*b*x + (c^4*e^(3/2) - c^2*e^(3/2))*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 \[ \int {\left (c\,e+d\,e\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^(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)

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

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)

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