3.2596 \(\int (5-x) \sqrt{3+2 x} (2+5 x+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=234 \[ \frac{5021353 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{5837832 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{45} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac{\sqrt{2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac{\sqrt{2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac{(287729-2667537 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{14594580}-\frac{2742319 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{4169880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((287729 - 2667537*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/14594580 - (Sqrt[3 + 2*x]*(15076 + 34643*x)*(2 + 5*
x + 3*x^2)^(3/2))/162162 + (Sqrt[3 + 2*x]*(15467 + 17193*x)*(2 + 5*x + 3*x^2)^(5/2))/19305 - (2*Sqrt[3 + 2*x]*
(2 + 5*x + 3*x^2)^(7/2))/45 - (2742319*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(4
169880*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5021353*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(5837832*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.155343, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{2}{45} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}+\frac{\sqrt{2 x+3} (17193 x+15467) \left (3 x^2+5 x+2\right )^{5/2}}{19305}-\frac{\sqrt{2 x+3} (34643 x+15076) \left (3 x^2+5 x+2\right )^{3/2}}{162162}+\frac{(287729-2667537 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{14594580}+\frac{5021353 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5837832 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2742319 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{4169880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((287729 - 2667537*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/14594580 - (Sqrt[3 + 2*x]*(15076 + 34643*x)*(2 + 5*
x + 3*x^2)^(3/2))/162162 + (Sqrt[3 + 2*x]*(15467 + 17193*x)*(2 + 5*x + 3*x^2)^(5/2))/19305 - (2*Sqrt[3 + 2*x]*
(2 + 5*x + 3*x^2)^(7/2))/45 - (2742319*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(4
169880*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5021353*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]],
-2/3])/(5837832*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 )^{5/2} \, dx &=-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}+\frac{2}{45} \int \frac{\left (392+\frac{521 x}{2}\right ) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac{\int \frac{\left (\frac{16573}{2}+\frac{14847 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx}{3861}\\ &=-\frac{\sqrt{3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}+\frac{\int \frac{\left (-356538-\frac{889179 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{486486}\\ &=\frac{(287729-2667537 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{14594580}-\frac{\sqrt{3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac{\int \frac{\frac{48722901}{2}+\frac{57588699 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{43783740}\\ &=\frac{(287729-2667537 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{14594580}-\frac{\sqrt{3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac{2742319 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{8339760}+\frac{5021353 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{11675664}\\ &=\frac{(287729-2667537 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{14594580}-\frac{\sqrt{3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac{\left (2742319 \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 )}{4169880 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (5021353 \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 )}{5837832 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{(287729-2667537 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{14594580}-\frac{\sqrt{3+2 x} (15076+34643 x) \left (2+5 x+3 x^2\right )^{3/2}}{162162}+\frac{\sqrt{3+2 x} (15467+17193 x) \left (2+5 x+3 x^2\right )^{5/2}}{19305}-\frac{2}{45} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}-\frac{2742319 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{4169880 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{5021353 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5837832 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.384633, size = 218, normalized size = 0.93 \[ -\frac{-4132174 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (315242928 x^9+468822816 x^8-6333945660 x^7-30512259036 x^6-63978029658 x^5-76896556902 x^4-56607962679 x^3-25296672765 x^2-6298405666 x-666434848\right ) \sqrt{2 x+3}[phone] \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{87567480 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[3 + 2*x]*(-666434848 - 6298405666*x - 25296672765*x^2 - 56607962679*x^3 - 76896556902*x^4 - 639780296
58*x^5 - 30512259036*x^6 - 6333945660*x^7 + 468822816*x^8 + 315242928*x^9) + 19196233*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] - 4132174*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
])/(87567480*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.013, size = 166, normalized size = 0.7 \begin{align*}{\frac{1}{5254048800\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+5254048800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -6304858560\,{x}^{9}-9376456320\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+5910532\,\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 ) [phone]\,\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+14096546280 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/875674800*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-6304858560*x^9-9376456320*x^8+126678913200*x^7+610245180720*x^
6+1279560593160*x^5+5910532*(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))[phone]*(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+1132159253580*x^3+507085229280*x^2+127887736620*x+14096546280)/(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{5}{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)^(5/2)*(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*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 (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\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)^(5/2)*(3+2*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*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 - 20 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 96 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 165 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 113 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 15 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 9 x^{5} \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)**(5/2)*(3+2*x)**(1/2),x)

[Out]

-Integral(-20*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-96*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2),
x) - Integral(-165*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-113*x**3*sqrt(2*x + 3)*sqrt(3*x**
2 + 5*x + 2), x) - Integral(-15*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(9*x**5*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{5}{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)^(5/2)*(3+2*x)^(1/2),x, algorithm="giac")

[Out]

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