3.2608 \(\int \frac{(5-x) \sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{5 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{9} \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{101 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/9 + (101*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(9*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(9*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.0781303, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {832, 843, 718, 424, 419} \[ -\frac{2}{9} \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{5 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{101 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/9 + (101*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(9*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(9*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 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]

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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{(5-x) \sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx &=-\frac{2}{9} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{2}{9} \int \frac{77+\frac{101 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2}{9} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{5}{18} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{101}{18} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2}{9} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{\left (5 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{9 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (101 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{9 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{2}{9} \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}+\frac{101 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{5 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{9 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.337048, size = 188, normalized size = 1.36 \[ -\frac{104 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+4 \left (9 x^3-123 x^2-224 x-92\right ) \sqrt{2 x+3}-101 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{27 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4*Sqrt[3 + 2*x]*(-92 - 224*x - 123*x^2 + 9*x^3) - 101*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 +
3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 104*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x
)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(27*(3 + 2*x)*Sqrt[2 + 5*x + 3*
x^2])

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Maple [A]  time = 0.013, size = 136, normalized size = 1. \begin{align*}{\frac{1}{1620\,{x}^{3}+5130\,{x}^{2}+5130\,x+1620}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 106\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -101\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -360\,{x}^{3}-1140\,{x}^{2}-1140\,x-360 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/270*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(106*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticF(
1/5*(30*x+45)^(1/2),1/3*15^(1/2))-101*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticE(1/5*(30
*x+45)^(1/2),1/3*15^(1/2))-360*x^3-1140*x^2-1140*x-360)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(sqrt(2*x + 3)*(x - 5)/sqrt(3*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{2 \, x + 3}{\left (x - 5\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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