3.204 \(\int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{5/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}+\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{4 b \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{3 d e^3 \sqrt{-c-d x} \sqrt{c+d x-1}} \]

[Out]

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

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Rubi [A]  time = 0.120569, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {5866, 5662, 104, 12, 16, 114, 113} \[ -\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}+\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{4 b \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{3 d e^3 \sqrt{-c-d x} \sqrt{c+d x-1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(5/2),x]

[Out]

(4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(3*d*e^2*Sqrt[e*(c + d*x)]) - (2*(a + b*ArcCosh[c + d*x]))/(3*d*e*(
e*(c + d*x))^(3/2)) - (4*b*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2]
)/(3*d*e^3*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 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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)}{(c e+d e x)^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} (e x)^{3/2} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}+\frac{(4 b) \operatorname{Subst}\left (\int -\frac{e x}{2 \sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e^3}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e^2}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e^3}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}-\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 )}{3 d e^3 \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}}-\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 )}{3 d e^3 \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.139273, size = 94, normalized size = 0.63 \[ \frac{2 \left (-\frac{2 b (c+d x) \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},(c+d x)^2\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}-a-b \cosh ^{-1}(c+d x)\right )}{3 d e (e (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(5/2),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, 3/4, (c + d*x)
^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(3*d*e*(e*(c + d*x))^(3/2))

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Maple [C]  time = 0.026, size = 269, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{de} \left ( -1/3\,{\frac{a}{ \left ( dex+ce \right ) ^{3/2}}}+b \left ( -1/3\,{\frac{1}{ \left ( dex+ce \right ) ^{3/2}}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )}+2/3\,{\frac{1}{{e}^{3}\sqrt{dex+ce}} \left ( -\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}\sqrt{dex+ce}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) e+\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}\sqrt{dex+ce}{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) e+\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{2}-\sqrt{-{e}^{-1}}{e}^{2} \right ){\frac{1}{\sqrt{-{e}^{-1}}}}{\frac{1}{\sqrt{{\frac{dex+ce+e}{e}}}}}{\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)^(5/2),x)

[Out]

2/d/e*(-1/3*a/(d*e*x+c*e)^(3/2)+b*(-1/3/(d*e*x+c*e)^(3/2)*arccosh(1/e*(d*e*x+c*e))+2/3/e^3*(-((d*e*x+c*e+e)/e)
^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*(d*e*x+c*e)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*e+((d*e*x+c*e+e)
/e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*(d*e*x+c*e)^(1/2)*EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*e+(-1/e)^(1/2
)*(d*e*x+c*e)^2-(-1/e)^(1/2)*e^2)/(-1/e)^(1/2)/(d*e*x+c*e)^(1/2)/((d*e*x+c*e+e)/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)^(5/2),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{\sqrt{d e x + c e}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, 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)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3),
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 )}}{\left (e \left (c + d x\right )\right )^{\frac{5}{2}}}\, 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)**(5/2),x)

[Out]

Integral((a + b*acosh(c + d*x))/(e*(c + d*x))**(5/2), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError