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

Optimal. Leaf size=153 \[ -\frac{16 b^2 (e (c+d x))^{7/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},(c+d x)^2\right )}{105 d e^3}-\frac{8 b \sqrt{-c-d x+1} (e (c+d x))^{5/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{15 d e^2 \sqrt{c+d x-1}}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e} \]

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^2)/(3*d*e) - (8*b*Sqrt[1 - c - d*x]*(e*(c + d*x))^(5/2)*(a + b
*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2])/(15*d*e^2*Sqrt[-1 + c + d*x]) - (16*b^2*(e*(
c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2])/(105*d*e^3)

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Rubi [A]  time = 0.315728, antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5866, 5662, 5763} \[ -\frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};(c+d x)^2\right )}{105 d e^3}-\frac{8 b \sqrt{1-(c+d x)^2} (e (c+d x))^{5/2} \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{15 d e^2 \sqrt{c+d x-1} \sqrt{c+d x+1}}+\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c + d*x])^2)/(3*d*e) - (8*b*(e*(c + d*x))^(5/2)*Sqrt[1 - (c + d*x)^2]*(a
+ b*ArcCosh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2])/(15*d*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c
+ d*x]) - (16*b^2*(e*(c + d*x))^(7/2)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2])/(105*d*e^3)

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 5763

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_
)]), x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2,
(3 + m)/2, c^2*x^2])/(f*(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Simp[(b*c*(f*x)^(m + 2)*Hypergeometric
PFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(Sqrt[-(d1*d2)]*f^2*(m + 1)*(m + 2)), x] /; FreeQ[{
a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1, 0] && LtQ[d2, 0] &&  !
IntegerQ[m]

Rubi steps

\begin{align*} \int \sqrt{c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{e x} \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{(e x)^{3/2} \left (a+b \cosh ^{-1}(x)\right )}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=\frac{2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e}-\frac{8 b (e (c+d x))^{5/2} \sqrt{1-(c+d x)^2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, _2F_1\left (\frac{1}{2},\frac{5}{4};\frac{9}{4};(c+d x)^2\right )}{15 d e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}-\frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left (1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};(c+d x)^2\right )}{105 d e^3}\\ \end{align*}

Mathematica [A]  time = 0.435795, size = 140, normalized size = 0.92 \[ \frac{2 (e (c+d x))^{3/2} \left (35 \left (a+b \cosh ^{-1}(c+d x)\right )^2-4 b (c+d x) \left (2 b (c+d x) \text{HypergeometricPFQ}\left (\left \{1,\frac{7}{4},\frac{7}{4}\right \},\left \{\frac{9}{4},\frac{11}{4}\right \},(c+d x)^2\right )+\frac{7 \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{5}{4},\frac{9}{4},(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )\right )}{105 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*(c + d*x))^(3/2)*(35*(a + b*ArcCosh[c + d*x])^2 - 4*b*(c + d*x)*((7*Sqrt[1 - (c + d*x)^2]*(a + b*ArcCosh
[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]) + 2*b*(c + d*
x)*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2])))/(105*d*e)

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Maple [F]  time = 0.37, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}\sqrt{dex+ce}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))^2*(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 ({\left (b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcosh}\left (d x + c\right ) + a^{2}\right )} \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))^2*(d*e*x+c*e)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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