3.2618 \(\int \frac{5-x}{(3+2 x)^{3/2} (2+5 x+3 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=170 \[ -\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}-\frac{6 (47 x+37)}{5 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}-\frac{908 \sqrt{3 x^2+5 x+2}}{25 \sqrt{2 x+3}}+\frac{454 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-6*(37 + 47*x))/(5*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) - (908*Sqrt[2 + 5*x + 3*x^2])/(25*Sqrt[3 + 2*x]) + (4
54*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) - (
94*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.110856, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {822, 834, 843, 718, 424, 419} \[ -\frac{6 (47 x+37)}{5 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}-\frac{908 \sqrt{3 x^2+5 x+2}}{25 \sqrt{2 x+3}}-\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}+\frac{454 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(37 + 47*x))/(5*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) - (908*Sqrt[2 + 5*x + 3*x^2])/(25*Sqrt[3 + 2*x]) + (4
54*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) - (
94*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{98+141 x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}-\frac{908 \sqrt{2+5 x+3 x^2}}{25 \sqrt{3+2 x}}+\frac{4}{25} \int \frac{\frac{669}{2}+\frac{681 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}-\frac{908 \sqrt{2+5 x+3 x^2}}{25 \sqrt{3+2 x}}+\frac{681}{25} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{141}{5} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}-\frac{908 \sqrt{2+5 x+3 x^2}}{25 \sqrt{3+2 x}}+\frac{\left (454 \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 )}{25 \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{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{5 \sqrt{2+5 x+3 x^2}}\\ &=-\frac{6 (37+47 x)}{5 \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}-\frac{908 \sqrt{2+5 x+3 x^2}}{25 \sqrt{3+2 x}}+\frac{454 \sqrt{3} \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{25 \sqrt{2+5 x+3 x^2}}-\frac{94 \sqrt{3} \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.333955, size = 172, normalized size = 1.01 \[ -\frac{2 \left (86 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+705 x-227 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+555\right )}{25 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(555 + 705*x - 227*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[Arc
Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 86*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*
x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(25*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 131, normalized size = 0.8 \begin{align*} -{\frac{1}{750\,{x}^{3}+2375\,{x}^{2}+2375\,x+750} \left ( 8\,\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 ) +227\,\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 ) +13620\,{x}^{2}+29750\,x+14630 \right ) \sqrt{3\,{x}^{2}+5\,x+2}\sqrt{3+2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/125*(8*(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))+2
27*(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))+13620*x^
2+29750*x+14630)*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)/(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}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x - 5)/((3*x^2 + 5*x + 2)^(3/2)*(2*x + 3)^(3/2)), 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 )}}{36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*(x - 5)/(36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x
+ 36), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{6 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 6 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{6 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 6 \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)**(3/2)/(3*x**2+5*x+2)**(3/2),x)

[Out]

-Integral(x/(6*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x
*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(6*x**3*sqrt
(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x*sqrt(2*x + 3)*sqrt(3*x*
*2 + 5*x + 2) + 6*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}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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