3.2603 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{13/2}} \, dx\)

Optimal. Leaf size=207 \[ \frac{198109 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{46200 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac{(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac{(948443 x+1301762) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}-\frac{107857 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{33000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((1301762 + 948443*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 + 2*x)^(3/2)) + ((24161 + 18699*x)*(2 + 5*x + 3*x^2)^(
3/2))/(34650*(3 + 2*x)^(7/2)) + ((114 + 115*x)*(2 + 5*x + 3*x^2)^(5/2))/(99*(3 + 2*x)^(11/2)) - (107857*Sqrt[-
2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(33000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (198109
*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(46200*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.133028, 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 = {810, 843, 718, 424, 419} \[ \frac{(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac{(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac{(948443 x+1301762) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}+\frac{198109 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{46200 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{107857 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{33000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]

[Out]

((1301762 + 948443*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 + 2*x)^(3/2)) + ((24161 + 18699*x)*(2 + 5*x + 3*x^2)^(
3/2))/(34650*(3 + 2*x)^(7/2)) + ((114 + 115*x)*(2 + 5*x + 3*x^2)^(5/2))/(99*(3 + 2*x)^(11/2)) - (107857*Sqrt[-
2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(33000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (198109
*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(46200*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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)^{13/2}} \, dx &=\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac{1}{198} \int \frac{(326+321 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{9/2}} \, dx\\ &=\frac{(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}+\frac{\int \frac{(-31975-37437 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{5/2}} \, dx}{23100}\\ &=\frac{(1301762+948443 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac{(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac{\int \frac{1911678+2264997 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{693000}\\ &=\frac{(1301762+948443 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac{(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac{107857 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{66000}+\frac{198109 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{92400}\\ &=\frac{(1301762+948443 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac{(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac{\left (107857 \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 )}{33000 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (198109 \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 )}{46200 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{(1301762+948443 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{3/2}}+\frac{(24161+18699 x) \left (2+5 x+3 x^2\right )^{3/2}}{34650 (3+2 x)^{7/2}}+\frac{(114+115 x) \left (2+5 x+3 x^2\right )^{5/2}}{99 (3+2 x)^{11/2}}-\frac{107857 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{33000 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{198109 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{46200 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.463988, size = 227, normalized size = 1.1 \[ -\frac{4 (2 x+3)^5 \left (-160672 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+1509998 \left (3 x^2+5 x+2\right )+754999 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-8 \left (3 x^2+5 x+2\right ) \left (21041468 x^5+140915480 x^4+387989550 x^3+544712540 x^2+387631385 x+111387702\right )}{2772000 (2 x+3)^{11/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]

[Out]

-(-8*(2 + 5*x + 3*x^2)*(111387702 + 387631385*x + 544712540*x^2 + 387989550*x^3 + 140915480*x^4 + 21041468*x^5
) + 4*(3 + 2*x)^5*(1509998*(2 + 5*x + 3*x^2) + 754999*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] - 160672*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]))/(2772000*(3 + 2*x)^(1
1/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.02, size = 575, normalized size = 2.8 \begin{align*}{\frac{1}{6930000} \left ( 24159968\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{5}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+7537472\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{5}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,{x}^{7}[phone]\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}[phone]\,{x}^{6}[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]\,\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]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+4455508080 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{11}{2}}}} \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)^(13/2),x)

[Out]

1/6930000*(24159968*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20
-30*x)^(1/2)+7537472*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-2
0-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*
(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2
)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^3*(3+2*x)^(1/2)*(-2-2*x)^(
1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^3*(3+2*x)^(1/2)*(-2-2*x
)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-
2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(
-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*x^7+611549190*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x
*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*
x*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*x^6+183464757*(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]*(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]*x^5+77118326600*x^4+93248719100*x^3+6
7234902220*x^2+26644025600*x+4455508080)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(11/2)

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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{13}{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)^(13/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), 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}}{128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187}, 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)^(13/2),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)/(128*x^7 + 1344
*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(13/2),x)

[Out]

Timed out

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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{13}{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)^(13/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), x)