3.3.1 \(\int \sqrt {c e+d e x} (a+b \cosh ^{-1}(c+d x)) \, dx\) [201]

Optimal. Leaf size=127 \[ -\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {-1+c+d x}} \]

[Out]

2/3*(e*(d*x+c))^(3/2)*(a+b*arccosh(d*x+c))/d/e-4/9*b*EllipticF((e*(d*x+c))^(1/2)/e^(1/2),I)*e^(1/2)*(-d*x-c+1)
^(1/2)/d/(d*x+c-1)^(1/2)-4/9*b*(d*x+c-1)^(1/2)*(e*(d*x+c))^(1/2)*(d*x+c+1)^(1/2)/d

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Rubi [A]
time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 = {5996, 5883, 104, 12, 118, 117} \begin {gather*} \frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} \sqrt {-c-d x+1} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {c+d x-1}}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*Sqrt[-1 + c + d*x]*Sqrt[e*(c + d*x)]*Sqrt[1 + c + d*x])/(9*d) + (2*(e*(c + d*x))^(3/2)*(a + b*ArcCosh[c
+ d*x]))/(3*d*e) - (4*b*Sqrt[e]*Sqrt[1 - c - d*x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(9*d*Sqrt[
-1 + c + d*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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)/(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 117

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

Rule 118

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5996

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 \sqrt {c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \sqrt {e x} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {\left (2 b e \sqrt {1-c-d x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d \sqrt {-1+c+d x}}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {-1+c+d x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.33, size = 131, normalized size = 1.03 \begin {gather*} \frac {\sqrt {e (c+d x)} \left (\frac {2}{3} (c+d x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {4 b \left (-1+c^2+2 c d x+d^2 x^2+\sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )\right )}{9 \sqrt {\frac {-1+c+d x}{c+d x}} \sqrt {1+c+d x}}\right )}{d \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[e*(c + d*x)]*((2*(c + d*x)^(3/2)*(a + b*ArcCosh[c + d*x]))/3 - (4*b*(-1 + c^2 + 2*c*d*x + d^2*x^2 + Sqrt
[1 - (c + d*x)^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(9*Sqrt[(-1 + c + d*x)/(c + d*x)]*Sqrt[1 + c
+ d*x])))/(d*Sqrt[c + d*x])

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Maple [A]
time = 0.05, size = 194, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(194\)
default	\(\frac {\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\)	\(194\)

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,method=_RETURNVERBOSE)

[Out]

2/d/e*(1/3*a*(d*e*x+c*e)^(3/2)+b*(1/3*(d*e*x+c*e)^(3/2)*arccosh((d*e*x+c*e)/e)-2/9/e*((-1/e)^(1/2)*(d*e*x+c*e)
^(5/2)+((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)*e^2-(-1/e)
^(1/2)*e^2*(d*e*x+c*e)^(1/2))/(-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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

2/3*(d*x*e + c*e)^(3/2)*a*e^(-1)/d + 1/9*(6*(d*x*e^(1/2) + c*e^(1/2))*sqrt(d*x + c)*log(d*x + sqrt(d*x + c + 1
)*sqrt(d*x + c - 1) + c)/d + (3*I*(log(I*sqrt(d*x + c) + 1) - log(-I*sqrt(d*x + c) + 1))*e^(1/2) + 3*e^(1/2)*l
og(sqrt(d*x + c) + 1) - 3*e^(1/2)*log(sqrt(d*x + c) - 1) - 4*e^(3/2*log(d*x + c) + 1/2))/d + 9*integrate(2/3*(
d*x*e^(1/2) + c*e^(1/2))*sqrt(d*x + c)/(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (d^2*x^2 + 2*c*d*x + c^2 - 1)*sqrt(d*x +
c + 1)*sqrt(d*x + c - 1) + (3*c^2*d - d)*x - c), x))*b

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 159, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} b {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (3 \, a d^{3} x + 3 \, a c d^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b d^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{9 \, d^{3}} \end {gather*}

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]

2/9*(3*(b*d^3*x + b*c*d^2)*sqrt((d*x + c)*cosh(1) + (d*x + c)*sinh(1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x +
c^2 - 1)) - 2*sqrt(d^3*cosh(1) + d^3*sinh(1))*b*weierstrassPInverse(4/d^2, 0, (d*x + c)/d) + (3*a*d^3*x + 3*a*
c*d^2 - 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*b*d^2)*sqrt((d*x + c)*cosh(1) + (d*x + c)*sinh(1)))/d^3

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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