3.2598 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{3/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{4145485 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt{2 x+3}}+\frac{5 \sqrt{2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac{(21871-471213 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{24948}+\frac{451331 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((21871 - 471213*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/24948 + (5*Sqrt[3 + 2*x]*(218 + 3031*x)*(2 + 5*x + 3
*x^2)^(3/2))/1386 - ((73 + x)*(2 + 5*x + 3*x^2)^(5/2))/(11*Sqrt[3 + 2*x]) + (451331*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(7128*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (4145485*Sqrt[-2 - 5*x - 3*x
^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(49896*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.129384, antiderivative size = 205, 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 = {812, 814, 843, 718, 424, 419} \[ -\frac{(x+73) \left (3 x^2+5 x+2\right )^{5/2}}{11 \sqrt{2 x+3}}+\frac{5 \sqrt{2 x+3} (3031 x+218) \left (3 x^2+5 x+2\right )^{3/2}}{1386}-\frac{(21871-471213 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{24948}-\frac{4145485 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{451331 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((21871 - 471213*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/24948 + (5*Sqrt[3 + 2*x]*(218 + 3031*x)*(2 + 5*x + 3
*x^2)^(3/2))/1386 - ((73 + x)*(2 + 5*x + 3*x^2)^(5/2))/(11*Sqrt[3 + 2*x]) + (451331*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(7128*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (4145485*Sqrt[-2 - 5*x - 3*x
^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(49896*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 812

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

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}}{(3+2 x)^{3/2}} \, dx &=-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}-\frac{5}{22} \int \frac{(-361-433 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{5 \int \frac{(43918+52357 x) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{2772}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}-\frac{\int \frac{-2666233-3159317 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{49896}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{451331 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{14256}-\frac{4145485 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{99792}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{\left (451331 \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 )}{7128 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (4145485 \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 )}{49896 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(21871-471213 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{24948}+\frac{5 \sqrt{3+2 x} (218+3031 x) \left (2+5 x+3 x^2\right )^{3/2}}{1386}-\frac{(73+x) \left (2+5 x+3 x^2\right )^{5/2}}{11 \sqrt{3+2 x}}+\frac{451331 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{7128 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{4145485 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{49896 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.442033, size = 205, normalized size = 1. \[ \frac{2 \left (-336013 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{3/2} \sqrt{\frac{3 x+2}{2 x+3}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )-183708 x^7+401436 x^6+3305934 x^5+7163046 x^4+6935769 x^3+6834513 x^2+6998740 x+2657740\right )+3159317 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{149688 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3159317*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/
Sqrt[3 + 2*x]], 3/5] + 2*(2657740 + 6998740*x + 6834513*x^2 + 6935769*x^3 + 7163046*x^4 + 3305934*x^5 + 401436
*x^6 - 183708*x^7 - 336013*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]))/(149688*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.019, size = 156, normalized size = 0.8 \begin{align*} -{\frac{1}{8981280\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+8981280}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 3674160\,{x}^{7}-8028720\,{x}^{6}+986168\,\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 ) +3159317\,\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 ) -66118680\,{x}^{5}-143260920\,{x}^{4}-138715380\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+73217880 \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)^(3/2),x)

[Out]

-1/1496880*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(3674160*x^7-8028720*x^6+986168*(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))+3159317*(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))-66118680*x^5-143260920*x^4-138715380*x^3+528687
60*x^2+175956900*x+73217880)/(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 )}}{{\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*x^2+5*x+2)^(5/2)/(3+2*x)^(3/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(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{{\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}}{4 \, x^{2} + 12 \, x + 9}, 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)^(3/2),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)/(4*x^2 + 12*x +
9), 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}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int - \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \sqrt{2 x + 3}}\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{2 x \sqrt{2 x + 3} + 3 \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)**(3/2),x)

[Out]

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

[Out]

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