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

Optimal. Leaf size=104 \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {4 b \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-c-d x} \sqrt {c+d x-1}} \]

[Out]

2*(a+b*arccosh(d*x+c))*(e*(d*x+c))^(1/2)/d/e-4*b*EllipticE(1/2*(d*x+c+1)^(1/2)*2^(1/2),2^(1/2))*(-d*x-c+1)^(1/
2)*(e*(d*x+c))^(1/2)/d/e/(-d*x-c)^(1/2)/(d*x+c-1)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5866, 5662, 114, 113} \[ \frac {2 \sqrt {e (c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e}-\frac {4 b \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-c-d x} \sqrt {c+d x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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 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
]

Rubi steps

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

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Mathematica [C]  time = 0.19, size = 94, normalized size = 0.90 \[ \frac {2 \sqrt {e (c+d x)} \left (3 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};(c+d x)^2\right )}{\sqrt {c+d x-1} \sqrt {c+d x+1}}\right )}{3 d e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*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])))/(3*d*e)

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{\sqrt {d e x + c e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{\sqrt {d e x + c e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 138, normalized size = 1.33 \[ \frac {2 a \sqrt {d e x +c e}+2 b \left (\sqrt {d e x +c e}\, \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\EllipticE \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {-\frac {d e x +c e -e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/e*(a*(d*e*x+c*e)^(1/2)+b*((d*e*x+c*e)^(1/2)*arccosh((d*e*x+c*e)/e)-2*(EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(
1/2),I)-EllipticE((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I))*(-(d*e*x+c*e-e)/e)^(1/2)/(-1/e)^(1/2)/((d*e*x+c*e-e)/e)^(
1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -b {\left (\frac {\frac {i \, {\left (\log \left (i \, \sqrt {d x + c} + 1\right ) - \log \left (-i \, \sqrt {d x + c} + 1\right )\right )}}{\sqrt {e}} - \frac {\log \left (\sqrt {d x + c} + 1\right )}{\sqrt {e}} + \frac {\log \left (\sqrt {d x + c} - 1\right )}{\sqrt {e}} + \frac {4 \, \sqrt {d x + c}}{\sqrt {e}}}{d} - \frac {2 \, {\left (d \sqrt {e} x + c \sqrt {e}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{\sqrt {d x + c} d e} - \int \frac {2 \, {\left (d \sqrt {e} x + c \sqrt {e}\right )}}{{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + c^{3} e + {\left (d^{2} e x^{2} + 2 \, c d e x + c^{2} e - e\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} - c e + {\left (3 \, c^{2} d e - d e\right )} x\right )} \sqrt {d x + c}}\,{d x}\right )} + \frac {2 \, \sqrt {d e x + c e} a}{d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*((I*(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c) + 1))/sqrt(e) - log(sqrt(d*x + c) + 1)/sqrt(e) + log(s
qrt(d*x + c) - 1)/sqrt(e) + 4*sqrt(d*x + c)/sqrt(e))/d - 2*(d*sqrt(e)*x + c*sqrt(e))*log(d*x + sqrt(d*x + c +
1)*sqrt(d*x + c - 1) + c)/(sqrt(d*x + c)*d*e) - integrate(2*(d*sqrt(e)*x + c*sqrt(e))/((d^3*e*x^3 + 3*c*d^2*e*
x^2 + c^3*e + (d^2*e*x^2 + 2*c*d*e*x + c^2*e - e)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) - c*e + (3*c^2*d*e - d*e
)*x)*sqrt(d*x + c)), x)) + 2*sqrt(d*e*x + c*e)*a/(d*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{\sqrt {c\,e+d\,e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acosh}{\left (c + d x \right )}}{\sqrt {e \left (c + d x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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