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

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

[Out]

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

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Rubi [A]  time = 0.312231, 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} \[ -\frac{16 b^2 \sqrt{e (c+d x)} \, _3F_2\left (\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};(c+d x)^2\right )}{3 d e^3}-\frac{8 b \sqrt{1-(c+d x)^2} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} \sqrt{e (c+d x)}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e (e (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*ArcCosh[c + d*x])^2)/(3*d*e*(e*(c + d*x))^(3/2)) - (8*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcCosh[c + d*
x])*Hypergeometric2F1[-1/4, 1/2, 3/4, (c + d*x)^2])/(3*d*e^2*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c +
d*x]) - (16*b^2*Sqrt[e*(c + d*x)]*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, (c + d*x)^2])/(3*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 \frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{(c e+d e x)^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \cosh ^{-1}(x)\right )^2}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e (e (c+d x))^{3/2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{\sqrt{-1+x} (e x)^{3/2} \sqrt{1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d e (e (c+d x))^{3/2}}-\frac{8 b \sqrt{1-(c+d x)^2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right )}{3 d e^2 \sqrt{-1+c+d x} \sqrt{e (c+d x)} \sqrt{1+c+d x}}-\frac{16 b^2 \sqrt{e (c+d x)} \, _3F_2\left (\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};(c+d x)^2\right )}{3 d e^3}\\ \end{align*}

Mathematica [A]  time = 0.357624, size = 140, normalized size = 0.92 \[ \frac{2 \left (4 b (c+d x) \left (-2 b (c+d x) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{4},1\right \},\left \{\frac{3}{4},\frac{5}{4}\right \},(c+d x)^2\right )-\frac{\sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},(c+d x)^2\right ) \left (a+b \cosh ^{-1}(c+d x)\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}\right )-\left (a+b \cosh ^{-1}(c+d x)\right )^2\right )}{3 d e (e (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(a + b*ArcCosh[c + d*x])^2 + 4*b*(c + d*x)*(-((Sqrt[1 - (c + d*x)^2]*(a + b*ArcCosh[c + d*x])*Hypergeomet
ric2F1[-1/4, 1/2, 3/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])) - 2*b*(c + d*x)*HypergeometricPFQ
[{1/4, 1/4, 1}, {3/4, 5/4}, (c + d*x)^2])))/(3*d*e*(e*(c + d*x))^(3/2))

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Maple [F]  time = 0.354, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2} \left ( dex+ce \right ) ^{-{\frac{5}{2}}}}\, 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)^(5/2),x)

[Out]

int((a+b*arccosh(d*x+c))^2/(d*e*x+c*e)^(5/2),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)^(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{{\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}}{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))^2/(d*e*x+c*e)^(5/2),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)/(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{\left (a + b \operatorname{acosh}{\left (c + d x \right )}\right )^{2}}{\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))**2/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*acosh(c + d*x))**2/(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))^2/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError