3.3.25 \(\int \frac {(a+b \cosh ^{-1}(c+d x))^4}{(c e+d e x)^{7/2}} \, dx\) [225]

Optimal. Leaf size=89 \[ -\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e (e (c+d x))^{5/2}}+\frac {8 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}},x\right )}{5 e} \]

[Out]

-2/5*(a+b*arccosh(d*x+c))^4/d/e/(e*(d*x+c))^(5/2)+8/5*b*Unintegrable((a+b*arccosh(d*x+c))^3/(e*(d*x+c))^(5/2)/
(d*x+c-1)^(1/2)/(d*x+c+1)^(1/2),x)/e

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Rubi [A]
time = 0.20, 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 )^4}{(c e+d e x)^{7/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(7/2),x]

[Out]

(-2*(a + b*ArcCosh[c + d*x])^4)/(5*d*e*(e*(c + d*x))^(5/2)) + (8*b*Defer[Subst][Defer[Int][(a + b*ArcCosh[x])^
3/(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 )^4}{(c e+d e x)^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{5 d e (e (c+d x))^{5/2}}+\frac {(8 b) \text {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} (e x)^{5/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e}\\ \end {align*}

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Mathematica [A]
time = 121.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{7/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(7/2),x]

[Out]

Integrate[(a + b*ArcCosh[c + d*x])^4/(c*e + d*e*x)^(7/2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arccosh(d*x+c))^4/(d*e*x+c*e)^(7/2),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))^4/(d*e*x+c*e)^(7/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(d*x + c)*b^4*e^(1/2)*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^4/(d^4*x^3*e^4 + 3*c*d^3*x^2
*e^4 + 3*c^2*d^2*x*e^4 + c^3*d*e^4) - 2/5*a^4*e^(-1)/((d*x*e + c*e)^(5/2)*d) + integrate(2/5*(2*(((5*a*b^3*d^2
+ 2*b^4*d^2)*x^2*e^(1/2) + 2*(5*a*b^3*c*d + 2*b^4*c*d)*x*e^(1/2) + (2*b^4*c^2 + 5*(c^2 - 1)*a*b^3)*e^(1/2))*s
qrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + ((5*a*b^3*d^3 + 2*b^4*d^3)*x^3*e^(1/2) + 3*(5*a*b^3*c*d^2 +
2*b^4*c*d^2)*x^2*e^(1/2) + (5*(3*c^2*d - d)*a*b^3 + 2*(3*c^2*d - d)*b^4)*x*e^(1/2) + (5*(c^3 - c)*a*b^3 + 2*(
c^3 - c)*b^4)*e^(1/2))*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^3 + 15*((a^2*b^2*d^2*
x^2*e^(1/2) + 2*a^2*b^2*c*d*x*e^(1/2) + (c^2 - 1)*a^2*b^2*e^(1/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x +
c - 1) + (a^2*b^2*d^3*x^3*e^(1/2) + 3*a^2*b^2*c*d^2*x^2*e^(1/2) + (3*c^2*d - d)*a^2*b^2*x*e^(1/2) + (c^3 - c)*
a^2*b^2*e^(1/2))*sqrt(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 + 10*((a^3*b*d^2*x^2*e^(1
/2) + 2*a^3*b*c*d*x*e^(1/2) + (c^2 - 1)*a^3*b*e^(1/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*sqrt(d*x + c - 1) + (a^
3*b*d^3*x^3*e^(1/2) + 3*a^3*b*c*d^2*x^2*e^(1/2) + (3*c^2*d - d)*a^3*b*x*e^(1/2) + (c^3 - c)*a^3*b*e^(1/2))*sqr
t(d*x + c))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c))/(d^7*x^7*e^4 + 7*c*d^6*x^6*e^4 + (21*c^2*d^5 -
d^5)*x^5*e^4 + 5*(7*c^3*d^4 - c*d^4)*x^4*e^4 + 5*(7*c^4*d^3 - 2*c^2*d^3)*x^3*e^4 + (21*c^5*d^2 - 10*c^3*d^2)*
x^2*e^4 + (7*c^6*d - 5*c^4*d)*x*e^4 + (d^6*x^6*e^4 + 6*c*d^5*x^5*e^4 + (15*c^2*d^4 - d^4)*x^4*e^4 + 4*(5*c^3*d
^3 - c*d^3)*x^3*e^4 + 3*(5*c^4*d^2 - 2*c^2*d^2)*x^2*e^4 + 2*(3*c^5*d - 2*c^3*d)*x*e^4 + (c^6 - c^4)*e^4)*sqrt(
d*x + c + 1)*sqrt(d*x + c - 1) + (c^7 - c^5)*e^4), 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))^4/(d*e*x+c*e)^(7/2),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)*sqrt(d*x + c)*e^(-7/2)/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4), 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 )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))**4/(d*e*x+c*e)**(7/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**4/(e*(c + d*x))**(7/2), 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))^4/(d*e*x+c*e)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)^4/(d*e*x + c*e)^(7/2), 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 )}^4}{{\left (c\,e+d\,e\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^(7/2),x)

[Out]

int((a + b*acosh(c + d*x))^4/(c*e + d*e*x)^(7/2), x)

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