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

Optimal. Leaf size=84 \[ \frac {4 b \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \sqrt {c+d x-1}}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5866, 5662, 117, 116} \[ \frac {4 b \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \sqrt {c+d x-1}}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*ArcCosh[c + d*x]))/(d*e*Sqrt[e*(c + d*x)]) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*
x)]/Sqrt[e]], -1])/(d*e^(3/2)*Sqrt[-1 + c + d*x])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {\left (2 b \sqrt {1-c-d x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e \sqrt {-1+c+d x}}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2} \sqrt {-1+c+d x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.17, size = 92, normalized size = 1.10 \[ \frac {2 \left (-a+\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )}{\sqrt {c+d x-1} \sqrt {c+d x+1}}-b \cosh ^{-1}(c+d x)\right )}{d e \sqrt {e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-a - b*ArcCosh[c + d*x] + (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^
2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(d*e*Sqrt[e*(c + d*x)])

________________________________________________________________________________________

fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.02, size = 119, normalized size = 1.42 \[ \frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {-\frac {d e x +c e -e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*((-I*(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c) + 1))/e^(3/2) - log(sqrt(d*x + c) + 1)/e^(3/2) + log(s
qrt(d*x + c) - 1)/e^(3/2))/d - 2*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)/(sqrt(d*x + c)*d*e^(3/2))
- 2*integrate(1/((d^2*e^(3/2)*x^2 + 2*c*d*e^(3/2)*x + c^2*e^(3/2) - e^(3/2))*sqrt(d*x + c + 1)*sqrt(d*x + c)*s
qrt(d*x + c - 1) + (d^3*e^(3/2)*x^3 + 3*c*d^2*e^(3/2)*x^2 + c^3*e^(3/2) - c*e^(3/2) + (3*c^2*d*e^(3/2) - d*e^(
3/2))*x)*sqrt(d*x + c)), x)) - 2*a/(sqrt(d*e*x + c*e)*d*e)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________