3.2617 \(\int \frac{5-x}{\sqrt{3+2 x} (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{70 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{6 \sqrt{2 x+3} (47 x+37)}{5 \sqrt{3 x^2+5 x+2}}+\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-6*Sqrt[3 + 2*x]*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (94*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 + 5*x + 3*x^2]) - (70*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.0850018, antiderivative size = 141, 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 = {822, 843, 718, 424, 419} \[ -\frac{6 \sqrt{2 x+3} (47 x+37)}{5 \sqrt{3 x^2+5 x+2}}-\frac{70 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[3 + 2*x]*(37 + 47*x))/(5*Sqrt[2 + 5*x + 3*x^2]) + (94*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 + 5*x + 3*x^2]) - (70*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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} \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{6 \sqrt{3+2 x} (37+47 x)}{5 \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{-124-141 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 \sqrt{3+2 x} (37+47 x)}{5 \sqrt{2+5 x+3 x^2}}+\frac{141}{5} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx-35 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 \sqrt{3+2 x} (37+47 x)}{5 \sqrt{2+5 x+3 x^2}}-\frac{\left (70 \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 )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (94 \sqrt{3} \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 )}{5 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{6 \sqrt{3+2 x} (37+47 x)}{5 \sqrt{2+5 x+3 x^2}}+\frac{94 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{2+5 x+3 x^2}}-\frac{70 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.328643, size = 178, normalized size = 1.26 \[ \frac{-24 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )-10 (35 x+29) \sqrt{2 x+3}+94 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{5 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 2*x]*(29 + 35*x) + 94*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Elli
pticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 24*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])/(5*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.02, size = 131, normalized size = 0.9 \begin{align*} -{\frac{1}{450\,{x}^{3}+1425\,{x}^{2}+1425\,x+450}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 34\,\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 ) +141\,\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 ) +8460\,{x}^{2}+19350\,x+9990 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/75*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(34*(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))+141*(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))+8460*x^2+19350*x+9990)/(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{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*(x - 5)/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12), x)

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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