### 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]

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

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Rubi [A]  time = 0.02086, antiderivative size = 30, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [B]  time = 0.0766456, size = 86, normalized size = 2.87 $\frac{\sqrt{19} \sqrt{-4 x-1} \sqrt{2-3 x} \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right ),\frac{21}{76}\right )-E\left (\sin ^{-1}\left (\frac{2 \sqrt{5 x+3}}{\sqrt{7}}\right )|\frac{21}{76}\right )\right )}{3 \sqrt{-12 x^2+5 x+2}}$

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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Maple [B]  time = 0.119, size = 77, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{57}}{6840\,{x}^{2}-2850\,x-1140} \left ({\it EllipticF} \left ({\frac{1}{19}\sqrt{171+285\,x}},{\frac{2\,\sqrt{399}}{21}} \right ) -{\it EllipticE} \left ({\frac{1}{19}\sqrt{171+285\,x}},{\frac{2\,\sqrt{399}}{21}} \right ) \right ) \sqrt{190-285\,x}\sqrt{-140\,x-35}\sqrt{-12\,{x}^{2}+5\,x+2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*(-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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-12 \, x^{2} + 5 \, x + 2} \sqrt{5 \, x + 3}}{12 \, x^{2} - 5 \, x - 2}, x\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 x + 3}}{\sqrt{- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 \, x + 3}}{\sqrt{-12 \, x^{2} + 5 \, x + 2}}\,{d x} \end{align*}

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)