3.2597 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{\sqrt{3+2 x}} \, dx\)

Optimal. Leaf size=207 \[ \frac{5983645 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{648648 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{1}{429} (224-33 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac{5 \sqrt{2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac{(34372-676791 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{324324}-\frac{651617 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{92664 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((34372 - 676791*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/324324 - (5*Sqrt[3 + 2*x]*(563 + 4669*x)*(2 + 5*x + 3
*x^2)^(3/2))/18018 + ((224 - 33*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/429 - (651617*Sqrt[-2 - 5*x - 3*x^2]
*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(92664*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5983645*Sqrt[-2 - 5*x
- 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(648648*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.127158, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ \frac{1}{429} (224-33 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac{5 \sqrt{2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac{(34372-676791 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{324324}+\frac{5983645 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{648648 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{651617 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{92664 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((34372 - 676791*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/324324 - (5*Sqrt[3 + 2*x]*(563 + 4669*x)*(2 + 5*x + 3
*x^2)^(3/2))/18018 + ((224 - 33*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/429 - (651617*Sqrt[-2 - 5*x - 3*x^2]
*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(92664*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5983645*Sqrt[-2 - 5*x
- 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(648648*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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 \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt{3+2 x}} \, dx &=\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{5}{858} \int \frac{(1744+2001 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\\ &=-\frac{5 \sqrt{3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac{5 \int \frac{(-188643-225597 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{108108}\\ &=\frac{(34372-676791 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{324324}-\frac{5 \sqrt{3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\int \frac{11550468+13683957 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1945944}\\ &=\frac{(34372-676791 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{324324}-\frac{5 \sqrt{3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{651617 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{185328}+\frac{5983645 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1297296}\\ &=\frac{(34372-676791 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{324324}-\frac{5 \sqrt{3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\left (651617 \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 )}{92664 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (5983645 \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 )}{648648 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{(34372-676791 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{324324}-\frac{5 \sqrt{3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac{1}{429} (224-33 x) \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac{651617 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{92664 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{5983645 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{648648 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.361123, size = 213, normalized size = 1.03 \[ -\frac{-971132 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (4041576 x^8-1163484 x^7-83553120 x^6-268524558 x^5-406647648 x^4-349849791 x^3-170798082 x^2-39284147 x-1864706\right ) \sqrt{2 x+3}+4561319 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{1945944 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[3 + 2*x]*(-1864706 - 39284147*x - 170798082*x^2 - 349849791*x^3 - 406647648*x^4 - 268524558*x^5 - 835
53120*x^6 - 1163484*x^7 + 4041576*x^8) + 4561319*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] - 971132*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])/(1945944*(3 + 2*x)*Sqrt[2 + 5*x + 3
*x^2])

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Maple [A]  time = 0.012, size = 161, normalized size = 0.8 \begin{align*}{\frac{1}{116756640\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+116756640}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -80831520\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+1422326\,\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 ) +4561319\,\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+219746880 \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]

1/19459440*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-80831520*x^8+23269680*x^7+1671062400*x^6+5370491160*x^5+1422326
*(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))+4561319*(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+6996995820*x^3+3689640780*x^2+1241814840*x+219746880)/(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{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{\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)^(5/2)/(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\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 - \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{2 x + 3}}\, 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(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3),
x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/s
qrt(2*x + 3), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(9*x**5*sqrt(3*x**2 +
5*x + 2)/sqrt(2*x + 3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{\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)^(5/2)/(3+2*x)^(1/2),x, algorithm="giac")

[Out]

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