3.3.17 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^{3/2}} \, dx\) [217]

Optimal. Leaf size=85 \[ -\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt {e (c+d x)}}+\frac {6 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}},x\right )}{e} \]

[Out]

-2*(a+b*arccosh(d*x+c))^3/d/e/(e*(d*x+c))^(1/2)+6*b*Unintegrable((a+b*arccosh(d*x+c))^2/(d*x+c-1)^(1/2)/(e*(d*
x+c))^(1/2)/(d*x+c+1)^(1/2),x)/e

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Rubi [A]
time = 0.19, 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)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(a + b*ArcCosh[c + d*x])^3)/(d*e*Sqrt[e*(c + d*x)]) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^2/(S
qrt[-1 + x]*Sqrt[e*x]*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)^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt {e (c+d x)}}+\frac {(6 b) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}

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Mathematica [A]
time = 25.20, 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)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-2*b^3*e^(-3/2)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/(sqrt(d*x + c)*d) - 2*a^3*e^(-1)/(sqrt(d*
x*e + c*e)*d) + integrate(3*(((c^3 - c)*a*b^2 + 2*(c^3 - c)*b^3 + (a*b^2*d^3 + 2*b^3*d^3)*x^3 + 3*(a*b^2*c*d^2
+ 2*b^3*c*d^2)*x^2 + (2*b^3*c^2 + (c^2 - 1)*a*b^2 + (a*b^2*d^2 + 2*b^3*d^2)*x^2 + 2*(a*b^2*c*d + 2*b^3*c*d)*x
)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + ((3*c^2*d - d)*a*b^2 + 2*(3*c^2*d - d)*b^3)*x)*log(d*x + sqrt(d*x + c
+ 1)*sqrt(d*x + c - 1) + c)^2 + (a^2*b*d^3*x^3 + 3*a^2*b*c*d^2*x^2 + (3*c^2*d - d)*a^2*b*x + (c^3 - c)*a^2*b +
(a^2*b*d^2*x^2 + 2*a^2*b*c*d*x + (c^2 - 1)*a^2*b)*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^3*x^3*e^(3/2) + 3*c*d^2*x^2*e^(3/2) + (3*c^2*d - d)*x*e^(3/2) + (c^3 - c)*e^
(3/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (d^4*x^4*e^(3/2) + 4*c*d^3*x^3*e^(3/2) + (6*c^2*d^2
- d^2)*x^2*e^(3/2) + 2*(2*c^3*d - c*d)*x*e^(3/2) + (c^4 - c^2)*e^(3/2))*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)^(3/2),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^(-3/2)/(d^2*x^2 + 2*c*d*x + c^2), 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 {3}{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)**(3/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**3/(e*(c + d*x))**(3/2), 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))^3/(d*e*x+c*e)^(3/2),x, algorithm="giac")

[Out]

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

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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 )}^{3/2}} \,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)^(3/2),x)

[Out]

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

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