3.202 \(\int \frac{a+b \cosh ^{-1}(c+d x)}{\sqrt{c e+d e x}} \, dx\)

Optimal. Leaf size=104 \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac{4 b \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{d e \sqrt{-c-d x} \sqrt{c+d x-1}} \]

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x]))/(d*e) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcS
in[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(d*e*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])

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Rubi [A]  time = 0.0878043, antiderivative size = 104, normalized size of antiderivative = 1., 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, 114, 113} \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac{4 b \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{d e \sqrt{-c-d x} \sqrt{c+d x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x]))/(d*e) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcS
in[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(d*e*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])

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
]

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 114

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

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

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

Mathematica [C]  time = 0.177272, size = 94, normalized size = 0.9 \[ \frac{2 \sqrt{e (c+d x)} \left (3 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{2 b (c+d x) \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},(c+d x)^2\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )}{3 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2,
3/4, 7/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(3*d*e)

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Maple [C]  time = 0.019, size = 138, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{de} \left ( a\sqrt{dex+ce}+b \left ( \sqrt{dex+ce}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )-2\,{ \left ({\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) -{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) \right ) \sqrt{-{\frac{dex+ce-e}{e}}}{\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce-e}{e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (d x + c\right ) + a}{\sqrt{d e x + c e}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acosh}{\left (c + d x \right )}}{\sqrt{e \left (c + d x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (d x + c\right ) + a}{\sqrt{d e x + c e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)/sqrt(d*e*x + c*e), x)