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

Optimal. Leaf size=145 \[ \frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{3/2}}{25 d}-\frac{12 b e \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{25 d \sqrt{-c-d x} \sqrt{c+d x-1}} \]

[Out]

(-4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(25*d) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh
[c + d*x]))/(5*d*e) - (12*b*e*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]],
2])/(25*d*Sqrt[-c - d*x]*Sqrt[-1 + c + d*x])

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Rubi [A]  time = 0.111114, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {5866, 5662, 102, 12, 114, 113} \[ \frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1} (e (c+d x))^{3/2}}{25 d}-\frac{12 b e \sqrt{-c-d x+1} \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c+d x+1}}{\sqrt{2}}\right )\right |2\right )}{25 d \sqrt{-c-d x} \sqrt{c+d x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*Sqrt[-1 + c + d*x]*(e*(c + d*x))^(3/2)*Sqrt[1 + c + d*x])/(25*d) + (2*(e*(c + d*x))^(5/2)*(a + b*ArcCosh
[c + d*x]))/(5*d*e) - (12*b*e*Sqrt[1 - c - d*x]*Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]],
2])/(25*d*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 102

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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 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 (c e+d e x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(e x)^{5/2}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{5 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{3 e^2 \sqrt{e x}}{2 \sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{25 d e}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{(6 b e) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{\left (3 \sqrt{2} b e \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 )}{25 d \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ &=-\frac{4 b \sqrt{-1+c+d x} (e (c+d x))^{3/2} \sqrt{1+c+d x}}{25 d}+\frac{2 (e (c+d x))^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e}-\frac{12 b e \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 )}{25 d \sqrt{-c-d x} \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.451433, size = 109, normalized size = 0.75 \[ \frac{2 (e (c+d x))^{3/2} \left (5 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{2 b \left (\sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},(c+d x)^2\right )+c^2+2 c d x+d^2 x^2-1\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )}{25 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(5*(c + d*x)*(a + b*ArcCosh[c + d*x]) - (2*b*(-1 + c^2 + 2*c*d*x + d^2*x^2 + 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])))/(25*d)

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Maple [C]  time = 0.02, size = 254, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{de} \left ( 1/5\, \left ( dex+ce \right ) ^{5/2}a+b \left ( 1/5\, \left ( dex+ce \right ) ^{5/2}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )-{\frac{2}{25}}\,{\frac{1}{e} \left ( \sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{7/2}+3\,\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{e}^{3}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) -3\,{e}^{3}\sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) -\sqrt{-{e}^{-1}} \left ( dex+ce \right ) ^{3/2}{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((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),x)

[Out]

2/d/e*(1/5*(d*e*x+c*e)^(5/2)*a+b*(1/5*(d*e*x+c*e)^(5/2)*arccosh(1/e*(d*e*x+c*e))-2/25/e*((-1/e)^(1/2)*(d*e*x+c
*e)^(7/2)+3*((d*e*x+c*e+e)/e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*e^3*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-3
*e^3*((d*e*x+c*e+e)/e)^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-(-1/e)^(1/2)
*(d*e*x+c*e)^(3/2)*e^2)/(-1/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((d*e*x+c*e)^(3/2)*(a+b*arccosh(d*x+c)),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 ({\left (a d e x + a c e +{\left (b d e x + b c e\right )} \operatorname{arcosh}\left (d x + c\right )\right )} \sqrt{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \left (c + d x\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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