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

Optimal. Leaf size=94 \[ \frac {(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d e (m+1)}-\frac {4 b \text {Int}\left (\frac {(e (c+d x))^{m+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}},x\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.29, 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 = {} \[ \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 4.12, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^m \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^4,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)*(d*e*x + c*e)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(d*x + c) + a)^4*(d*e*x + c*e)^m, x)

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maple [A]  time = 2.65, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{m} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^m*(a+b*arccosh(d*x+c))^4,x)

[Out]

int((d*e*x+c*e)^m*(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)^m*(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^m*(a + b*acosh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)^m*(a + b*acosh(c + d*x))^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**m*(a+b*acosh(d*x+c))**4,x)

[Out]

Integral((e*(c + d*x))**m*(a + b*acosh(c + d*x))**4, x)

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