3.205 \(\int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac{4 b \sqrt{-c-d x+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{15 d e^{7/2} \sqrt{c+d x-1}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{15 d e^2 (e (c+d x))^{3/2}} \]

[Out]

(4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*d*e^2*(e*(c + d*x))^(3/2)) - (2*(a + b*ArcCosh[c + d*x]))/(5*d*
e*(e*(c + d*x))^(5/2)) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(15*d*e^(7/2
)*Sqrt[-1 + c + d*x])

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Rubi [A]  time = 0.111154, antiderivative size = 130, 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, 117, 116} \[ -\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{4 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{15 d e^2 (e (c+d x))^{3/2}}+\frac{4 b \sqrt{-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{15 d e^{7/2} \sqrt{c+d x-1}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(7/2),x]

[Out]

(4*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*d*e^2*(e*(c + d*x))^(3/2)) - (2*(a + b*ArcCosh[c + d*x]))/(5*d*
e*(e*(c + d*x))^(5/2)) + (4*b*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(15*d*e^(7/2
)*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 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{(e x)^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} (e x)^{5/2} \sqrt{1+x}} \, dx,x,c+d x\right )}{5 d e}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e x}{2 \sqrt{-1+x} (e x)^{3/2} \sqrt{1+x}} \, dx,x,c+d x\right )}{15 d e^3}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} (e x)^{3/2} \sqrt{1+x}} \, dx,x,c+d x\right )}{15 d e^2}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{15 d e^3}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{\left (2 b \sqrt{1-c-d x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{e x} \sqrt{1+x}} \, dx,x,c+d x\right )}{15 d e^3 \sqrt{-1+c+d x}}\\ &=\frac{4 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d e (e (c+d x))^{5/2}}+\frac{4 b \sqrt{1-c-d x} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{15 d e^{7/2} \sqrt{-1+c+d x}}\\ \end{align*}

Mathematica [C]  time = 0.151375, size = 94, normalized size = 0.72 \[ \frac{2 \left (-\frac{2 b (c+d x) \sqrt{1-(c+d x)^2} \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{1}{2},\frac{1}{4},(c+d x)^2\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}-3 \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{15 d e (e (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(7/2),x]

[Out]

(2*(-3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[-3/4, 1/2, 1/4, (c +
d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(15*d*e*(e*(c + d*x))^(5/2))

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Maple [A]  time = 0.027, size = 201, normalized size = 1.6 \begin{align*} 2\,{\frac{1}{de} \left ( -1/5\,{\frac{a}{ \left ( dex+ce \right ) ^{5/2}}}+b \left ( -1/5\,{\frac{1}{ \left ( dex+ce \right ) ^{5/2}}{\rm arccosh} \left ({\frac{dex+ce}{e}}\right )}+2/15\,{\frac{1}{{e}^{3} \left ( dex+ce \right ) ^{3/2}} \left ( \sqrt{{\frac{dex+ce+e}{e}}}\sqrt{-{\frac{dex+ce-e}{e}}}{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{-{e}^{-1}},i \right ) \left ( dex+ce \right ) ^{3/2}+\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)^(7/2),x)

[Out]

2/d/e*(-1/5*a/(d*e*x+c*e)^(5/2)+b*(-1/5/(d*e*x+c*e)^(5/2)*arccosh(1/e*(d*e*x+c*e))+2/15/e^3*(((d*e*x+c*e+e)/e)
^(1/2)*(-(d*e*x+c*e-e)/e)^(1/2)*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*(d*e*x+c*e)^(3/2)+(-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)^(3/2)/((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)^(7/2),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^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, 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)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*e*x + c*e)*(b*arccosh(d*x + c) + a)/(d^4*e^4*x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3
*d*e^4*x + c^4*e^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(7/2),x)

[Out]

Timed out

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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}{{\left (d e x + c e\right )}^{\frac{7}{2}}}\,{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)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^(7/2), x)