3.219 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^3}{(c e+d e x)^{7/2}} \, dx\)

Optimal. Leaf size=89 \[ \frac {6 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}},x\right )}{5 e}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}} \]

[Out]

-2/5*(a+b*arccosh(d*x+c))^3/d/e/(e*(d*x+c))^(5/2)+6/5*b*Unintegrable((a+b*arccosh(d*x+c))^2/(e*(d*x+c))^(5/2)/
(d*x+c-1)^(1/2)/(d*x+c+1)^(1/2),x)/e

________________________________________________________________________________________

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 = {} \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(7/2),x]

[Out]

(-2*(a + b*ArcCosh[c + d*x])^3)/(5*d*e*(e*(c + d*x))^(5/2)) + (6*b*Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^
2/(Sqrt[-1 + x]*(e*x)^(5/2)*Sqrt[1 + x]), x], x, c + d*x])/(5*d*e)

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{5 d e (e (c+d x))^{5/2}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} (e x)^{5/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 123.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{7/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(7/2),x]

[Out]

Integrate[(a + b*ArcCosh[c + d*x])^3/(c*e + d*e*x)^(7/2), x]

________________________________________________________________________________________

fricas [A]  time = 1.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(7/2),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*e*x + c
*e)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^3/(d*e*x + c*e)^(7/2), x)

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(7/2),x)

[Out]

int((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(7/2),x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} \sqrt {e} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3}}{5 \, {\left (d^{3} e^{4} x^{2} + 2 \, c d^{2} e^{4} x + c^{2} d e^{4}\right )} \sqrt {d x + c}} - \frac {2 \, a^{3}}{5 \, {\left (d e x + c e\right )}^{\frac {5}{2}} d e} + \int \frac {3 \, {\left ({\left (5 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a b^{2} + 2 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} b^{3} + {\left (5 \, a b^{2} d^{3} \sqrt {e} + 2 \, b^{3} d^{3} \sqrt {e}\right )} x^{3} + 3 \, {\left (5 \, a b^{2} c d^{2} \sqrt {e} + 2 \, b^{3} c d^{2} \sqrt {e}\right )} x^{2} + {\left (2 \, b^{3} c^{2} \sqrt {e} + 5 \, {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a b^{2} + {\left (5 \, a b^{2} d^{2} \sqrt {e} + 2 \, b^{3} d^{2} \sqrt {e}\right )} x^{2} + 2 \, {\left (5 \, a b^{2} c d \sqrt {e} + 2 \, b^{3} c d \sqrt {e}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (5 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a b^{2} + 2 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} b^{3}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} + 5 \, {\left (a^{2} b d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a^{2} b x + {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a^{2} b + {\left (a^{2} b d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b c d \sqrt {e} x + {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a^{2} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{5 \, {\left ({\left (d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + c^{5} e^{4} - c^{3} e^{4} + {\left (10 \, c^{2} d^{3} e^{4} - d^{3} e^{4}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{4} - 3 \, c d^{2} e^{4}\right )} x^{2} + {\left (5 \, c^{4} d e^{4} - 3 \, c^{2} d e^{4}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} - c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} - d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} - c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} - 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} - 2 \, c^{3} d e^{4}\right )} x\right )} \sqrt {d x + c}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))^3/(d*e*x+c*e)^(7/2),x, algorithm="maxima")

[Out]

-2/5*b^3*sqrt(e)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3/((d^3*e^4*x^2 + 2*c*d^2*e^4*x + c^2*d*e^
4)*sqrt(d*x + c)) - 2/5*a^3/((d*e*x + c*e)^(5/2)*d*e) + integrate(3/5*((5*(c^3*sqrt(e) - c*sqrt(e))*a*b^2 + 2*
(c^3*sqrt(e) - c*sqrt(e))*b^3 + (5*a*b^2*d^3*sqrt(e) + 2*b^3*d^3*sqrt(e))*x^3 + 3*(5*a*b^2*c*d^2*sqrt(e) + 2*b
^3*c*d^2*sqrt(e))*x^2 + (2*b^3*c^2*sqrt(e) + 5*(c^2*sqrt(e) - sqrt(e))*a*b^2 + (5*a*b^2*d^2*sqrt(e) + 2*b^3*d^
2*sqrt(e))*x^2 + 2*(5*a*b^2*c*d*sqrt(e) + 2*b^3*c*d*sqrt(e))*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (5*(3*c^
2*d*sqrt(e) - d*sqrt(e))*a*b^2 + 2*(3*c^2*d*sqrt(e) - d*sqrt(e))*b^3)*x)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x
+ c - 1) + c)^2 + 5*(a^2*b*d^3*sqrt(e)*x^3 + 3*a^2*b*c*d^2*sqrt(e)*x^2 + (3*c^2*d*sqrt(e) - d*sqrt(e))*a^2*b*x
+ (c^3*sqrt(e) - c*sqrt(e))*a^2*b + (a^2*b*d^2*sqrt(e)*x^2 + 2*a^2*b*c*d*sqrt(e)*x + (c^2*sqrt(e) - sqrt(e))*
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^5*e^4*x^5
+ 5*c*d^4*e^4*x^4 + c^5*e^4 - c^3*e^4 + (10*c^2*d^3*e^4 - d^3*e^4)*x^3 + (10*c^3*d^2*e^4 - 3*c*d^2*e^4)*x^2 +
(5*c^4*d*e^4 - 3*c^2*d*e^4)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (d^6*e^4*x^6 + 6*c*d^5*e^4*
x^5 + c^6*e^4 - c^4*e^4 + (15*c^2*d^4*e^4 - d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 - c*d^3*e^4)*x^3 + 3*(5*c^4*d^2*e^
4 - 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 - 2*c^3*d*e^4)*x)*sqrt(d*x + c)), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^(7/2),x)

[Out]

int((a + b*acosh(c + d*x))^3/(c*e + d*e*x)^(7/2), x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**3/(d*e*x+c*e)**(7/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**3/(e*(c + d*x))**(7/2), x)

________________________________________________________________________________________