3.2586 \(\int (5-x) \sqrt{3+2 x} (2+5 x+3 x^2)^{3/2} \, dx\)

Optimal. Leaf size=202 \[ \frac{4153 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-(Sqrt[3 + 2*x]*(1246 + 3987*x)*Sqrt[2 + 5*x + 3*x^2])/8910 + (Sqrt[3 + 2*x]*(119 + 127*x)*(2 + 5*x + 3*x^2)^(
3/2))/99 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/33 - (15283*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[
3]*Sqrt[1 + x]], -2/3])/(17820*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (4153*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[
Sqrt[3]*Sqrt[1 + x]], -2/3])/(3564*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.129675, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{33} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}+\frac{1}{99} \sqrt{2 x+3} (127 x+119) \left (3 x^2+5 x+2\right )^{3/2}-\frac{\sqrt{2 x+3} (3987 x+1246) \sqrt{3 x^2+5 x+2}}{8910}+\frac{4153 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{15283 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3 + 2*x]*(1246 + 3987*x)*Sqrt[2 + 5*x + 3*x^2])/8910 + (Sqrt[3 + 2*x]*(119 + 127*x)*(2 + 5*x + 3*x^2)^(
3/2))/99 - (2*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/33 - (15283*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[
3]*Sqrt[1 + x]], -2/3])/(17820*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (4153*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[
Sqrt[3]*Sqrt[1 + x]], -2/3])/(3564*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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (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 (5-x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac{2}{33} \int \frac{\left (287+\frac{381 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\int \frac{\left (\frac{9807}{2}+\frac{9303 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{2079}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac{\int \frac{-131691-\frac{320943 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{187110}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{15283 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{35640}+\frac{4153 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{7128}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\left (15283 \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 )}{17820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (4153 \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 )}{3564 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{\sqrt{3+2 x} (1246+3987 x) \sqrt{2+5 x+3 x^2}}{8910}+\frac{1}{99} \sqrt{3+2 x} (119+127 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{33} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{15283 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{17820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{4153 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{3564 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.352769, size = 208, normalized size = 1.03 \[ -\frac{-2824 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (87480 x^7-48600 x^6-2001510 x^5-6002964 x^4-8112483 x^3-5740860 x^2-2059597 x-293686\right ) \sqrt{2 x+3}+15283 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{53460 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[3 + 2*x]*(-293686 - 2059597*x - 5740860*x^2 - 8112483*x^3 - 6002964*x^4 - 2001510*x^5 - 48600*x^6 + 8
7480*x^7) + 15283*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] - 2824*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Ellipti
cF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(53460*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.012, size = 156, normalized size = 0.8 \begin{align*}{\frac{1}{3207600\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3207600}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -1749600\,{x}^{7}+972000\,{x}^{6}[phone]\,{x}^{5}+5482\,\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 ) +15283\,\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 ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+6485040 \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]

1/534600*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-1749600*x^7+972000*x^6+40030200*x^5+5482*(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))+15283*(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))[phone]*x^4+162249660*x^3+115734180*x^
2+42720240*x+6485040)/(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{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{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((3*x^2 + 5*x + 2)^(3/2)*sqrt(2*x + 3)*(x - 5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, 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(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 10 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 23 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 10 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 3 x^{3} \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(-10*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-23*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2),
x) - Integral(-10*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(3*x**3*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 -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} \sqrt{2 \, x + 3}{\left (x - 5\right )}\,{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(-(3*x^2 + 5*x + 2)^(3/2)*sqrt(2*x + 3)*(x - 5), x)