3.169 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^{5/2}}{c e+d e x} \, dx\)

Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{c+d x},x\right )}{e} \]

[Out]

Unintegrable((a+b*arccosh(d*x+c))^(5/2)/(d*x+c),x)/e

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCosh[c + d*x])^(5/2)/(c*e + d*e*x),x]

[Out]

Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^(5/2)/x, x], x, c + d*x]/(d*e)

Rubi steps

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

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Mathematica [A]  time = 1.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{c e+d e x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^(5/2)/(c*e + d*e*x),x]

[Out]

Integrate[(a + b*ArcCosh[c + d*x])^(5/2)/(c*e + d*e*x), x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^(5/2)/(d*e*x+c*e),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{d e x + c e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^(5/2)/(d*e*x+c*e),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^(5/2)/(d*e*x + c*e), x)

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maple [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}{d e x +c e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^(5/2)/(d*e*x+c*e),x)

[Out]

int((a+b*arccosh(d*x+c))^(5/2)/(d*e*x+c*e),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{d e x + c e}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^(5/2)/(d*e*x+c*e),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x + c) + a)^(5/2)/(d*e*x + c*e), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}}{c\,e+d\,e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^(5/2)/(c*e + d*e*x),x)

[Out]

int((a + b*acosh(c + d*x))^(5/2)/(c*e + d*e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**(5/2)/(d*e*x+c*e),x)

[Out]

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

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