3.3.18 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^{5/2}} \, dx\) [218]

Optimal. Leaf size=87 \[ -\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e (e (c+d x))^{3/2}}+\frac {2 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {-1+c+d x} (e (c+d x))^{3/2} \sqrt {1+c+d x}},x\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.21, 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 \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

(-2*(a + b*ArcCosh[c + d*x])^3)/(3*d*e*(e*(c + d*x))^(3/2)) + (2*b*Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^
2/(Sqrt[-1 + x]*(e*x)^(3/2)*Sqrt[1 + x]), x], x, c + d*x])/(d*e)

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} (e x)^{3/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 29.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(5/2),x]

[Out]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(5/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(5/2),x)

[Out]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(5/2),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))^3/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

-2/3*b^3*e^(1/2)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/((d^2*x*e^3 + c*d*e^3)*sqrt(d*x + c)) -
2/3*a^3*e^(-1)/((d*x*e + c*e)^(3/2)*d) + integrate((((3*a*b^2*d^3 + 2*b^3*d^3)*x^3*e^(1/2) + 3*(3*a*b^2*c*d^2
+ 2*b^3*c*d^2)*x^2*e^(1/2) + (3*(3*c^2*d - d)*a*b^2 + 2*(3*c^2*d - d)*b^3)*x*e^(1/2) + ((3*a*b^2*d^2 + 2*b^3*d
^2)*x^2*e^(1/2) + 2*(3*a*b^2*c*d + 2*b^3*c*d)*x*e^(1/2) + (2*b^3*c^2 + 3*(c^2 - 1)*a*b^2)*e^(1/2))*sqrt(d*x +
c + 1)*sqrt(d*x + c - 1) + (3*(c^3 - c)*a*b^2 + 2*(c^3 - c)*b^3)*e^(1/2))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x
+ c - 1) + c)^2 + 3*(a^2*b*d^3*x^3*e^(1/2) + 3*a^2*b*c*d^2*x^2*e^(1/2) + (3*c^2*d - d)*a^2*b*x*e^(1/2) + (c^3
- c)*a^2*b*e^(1/2) + (a^2*b*d^2*x^2*e^(1/2) + 2*a^2*b*c*d*x*e^(1/2) + (c^2 - 1)*a^2*b*e^(1/2))*sqrt(d*x + c +
1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/((d^4*x^4*e^3 + 4*c*d^3*x^3*e^3 + (
6*c^2*d^2 - d^2)*x^2*e^3 + 2*(2*c^3*d - c*d)*x*e^3 + (c^4 - c^2)*e^3)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x
+ c - 1) + (d^5*x^5*e^3 + 5*c*d^4*x^4*e^3 + (10*c^2*d^3 - d^3)*x^3*e^3 + (10*c^3*d^2 - 3*c*d^2)*x^2*e^3 + (5*
c^4*d - 3*c^2*d)*x*e^3 + (c^5 - c^3)*e^3)*sqrt(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))^3/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

[Out]

integral((b^3*arccosh(d*x + c)^3 + 3*a*b^2*arccosh(d*x + c)^2 + 3*a^2*b*arccosh(d*x + c) + a^3)*sqrt(d*x + c)*
e^(-5/2)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**3/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**3/(e*(c + d*x))**(5/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^(5/2), x)

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