\(\int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx\) [2482]

Optimal result

Integrand size = 24, antiderivative size = 30 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {1}{3} \sqrt {19} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]

[Out]

-1/3*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/38*1045^(1/2))*19^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {732, 435} \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {1}{3} \sqrt {19} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]

[In]

Int[Sqrt[3 + 5*x]/Sqrt[2 + 5*x - 12*x^2],x]

[Out]

-1/3*(Sqrt[19]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], 55/76])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{3} \sqrt {19} \text {Subst}\left (\int \frac {\sqrt {1-\frac {55 x^2}{76}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {16-24 x}}{\sqrt {22}}\right )\right ) \\ & = -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(30)=60\).

Time = 30.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.87 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\frac {\sqrt {19} \sqrt {-1-4 x} \sqrt {2-3 x} \left (-E\left (\arcsin \left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right )|\frac {21}{76}\right )+\operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right ),\frac {21}{76}\right )\right )}{3 \sqrt {2+5 x-12 x^2}} \]

[In]

Integrate[Sqrt[3 + 5*x]/Sqrt[2 + 5*x - 12*x^2],x]

[Out]

(Sqrt[19]*Sqrt[-1 - 4*x]*Sqrt[2 - 3*x]*(-EllipticE[ArcSin[(2*Sqrt[3 + 5*x])/Sqrt[7]], 21/76] + EllipticF[ArcSi
n[(2*Sqrt[3 + 5*x])/Sqrt[7]], 21/76]))/(3*Sqrt[2 + 5*x - 12*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(23)=46\).

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57

method	result	size
default	\(-\frac {\left (F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )-E\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )\right ) \sqrt {190-285 x}\, \sqrt {-35-140 x}\, \sqrt {57}\, \sqrt {-12 x^{2}+5 x +2}}{570 \left (12 x^{2}-5 x -2\right )}\)	\(77\)
elliptic	\(\frac {\sqrt {-\left (12 x^{2}-5 x -2\right ) \left (3+5 x \right )}\, \left (\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{665 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}+\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, \left (-\frac {7 E\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{20}-\frac {F\left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{4}\right )}{399 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}\right )}{\sqrt {3+5 x}\, \sqrt {-12 x^{2}+5 x +2}}\)	\(171\)

[In]

int((3+5*x)^(1/2)/(-12*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/570*(EllipticF(1/19*(171+285*x)^(1/2),2/21*399^(1/2))-EllipticE(1/19*(171+285*x)^(1/2),2/21*399^(1/2)))*(19
0-285*x)^(1/2)*(-35-140*x)^(1/2)*57^(1/2)*(-12*x^2+5*x+2)^(1/2)/(12*x^2-5*x-2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=-\frac {97}{540} \, \sqrt {-15} {\rm weierstrassPInverse}\left (\frac {4621}{2700}, \frac {216019}{729000}, x + \frac {11}{180}\right ) + \frac {1}{3} \, \sqrt {-15} {\rm weierstrassZeta}\left (\frac {4621}{2700}, \frac {216019}{729000}, {\rm weierstrassPInverse}\left (\frac {4621}{2700}, \frac {216019}{729000}, x + \frac {11}{180}\right )\right ) \]

[In]

integrate((3+5*x)^(1/2)/(-12*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

-97/540*sqrt(-15)*weierstrassPInverse(4621/2700, 216019/729000, x + 11/180) + 1/3*sqrt(-15)*weierstrassZeta(46
21/2700, 216019/729000, weierstrassPInverse(4621/2700, 216019/729000, x + 11/180))

Sympy [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \]

[In]

integrate((3+5*x)**(1/2)/(-12*x**2+5*x+2)**(1/2),x)

[Out]

Integral(sqrt(5*x + 3)/sqrt(-(3*x - 2)*(4*x + 1)), x)

Maxima [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}} \,d x } \]

[In]

integrate((3+5*x)^(1/2)/(-12*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2), x)

Giac [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}} \,d x } \]

[In]

integrate((3+5*x)^(1/2)/(-12*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)/sqrt(-12*x^2 + 5*x + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx=\int \frac {\sqrt {5\,x+3}}{\sqrt {-12\,x^2+5\,x+2}} \,d x \]

[In]

int((5*x + 3)^(1/2)/(5*x - 12*x^2 + 2)^(1/2),x)

[Out]

int((5*x + 3)^(1/2)/(5*x - 12*x^2 + 2)^(1/2), x)