3.2482 \(\int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx\)

Optimal. Leaf size=30 \[ -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {718, 424} \[ -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 424

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

Rule 718

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*} \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx &=-\left (\frac {1}{3} \sqrt {19} \operatorname {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*}

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Mathematica [B]  time = 0.08, size = 86, normalized size = 2.87 \[ \frac {\sqrt {19} \sqrt {-4 x-1} \sqrt {2-3 x} \left (F\left (\sin ^{-1}\left (\frac {2 \sqrt {5 x+3}}{\sqrt {7}}\right )|\frac {21}{76}\right )-E\left (\sin ^{-1}\left (\frac {2 \sqrt {5 x+3}}{\sqrt {7}}\right )|\frac {21}{76}\right )\right )}{3 \sqrt {-12 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[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])

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-12 \, x^{2} + 5 \, x + 2} \sqrt {5 \, x + 3}}{12 \, x^{2} - 5 \, x - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.10, size = 77, normalized size = 2.57 \[ -\frac {\left (-\EllipticE \left (\frac {\sqrt {285 x +171}}{19}, \frac {2 \sqrt {399}}{21}\right )+\EllipticF \left (\frac {\sqrt {285 x +171}}{19}, \frac {2 \sqrt {399}}{21}\right )\right ) \sqrt {-285 x +190}\, \sqrt {-140 x -35}\, \sqrt {57}\, \sqrt {-12 x^{2}+5 x +2}}{570 \left (12 x^{2}-5 x -2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/570*(EllipticF(1/19*(285*x+171)^(1/2),2/21*399^(1/2))-EllipticE(1/19*(285*x+171)^(1/2),2/21*399^(1/2)))*(-2
85*x+190)^(1/2)*(-140*x-35)^(1/2)*57^(1/2)*(-12*x^2+5*x+2)^(1/2)/(12*x^2-5*x-2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {5\,x+3}}{\sqrt {-12\,x^2+5\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 x + 3}}{\sqrt {- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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