3.2602 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{11/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{32513 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac{(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac{(7877 x+27213) \sqrt{3 x^2+5 x+2}}{2100 \sqrt{2 x+3}}+\frac{17699 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((27213 + 7877*x)*Sqrt[2 + 5*x + 3*x^2])/(2100*Sqrt[3 + 2*x]) + ((14311 + 10729*x)*(2 + 5*x + 3*x^2)^(3/2))/(
3150*(3 + 2*x)^(5/2)) + ((124 + 113*x)*(2 + 5*x + 3*x^2)^(5/2))/(63*(3 + 2*x)^(9/2)) + (17699*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(600*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (32513*Sqrt[-2 - 5*
x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(840*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.128924, antiderivative size = 207, 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 = {810, 812, 843, 718, 424, 419} \[ \frac{(113 x+124) \left (3 x^2+5 x+2\right )^{5/2}}{63 (2 x+3)^{9/2}}+\frac{(10729 x+14311) \left (3 x^2+5 x+2\right )^{3/2}}{3150 (2 x+3)^{5/2}}-\frac{(7877 x+27213) \sqrt{3 x^2+5 x+2}}{2100 \sqrt{2 x+3}}-\frac{32513 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{17699 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(11/2),x]

[Out]

-((27213 + 7877*x)*Sqrt[2 + 5*x + 3*x^2])/(2100*Sqrt[3 + 2*x]) + ((14311 + 10729*x)*(2 + 5*x + 3*x^2)^(3/2))/(
3150*(3 + 2*x)^(5/2)) + ((124 + 113*x)*(2 + 5*x + 3*x^2)^(5/2))/(63*(3 + 2*x)^(9/2)) + (17699*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(600*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (32513*Sqrt[-2 - 5*
x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(840*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 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 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)^{11/2}} \, dx &=\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac{1}{126} \int \frac{(548+603 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx\\ &=\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{\int \frac{(-60147-70893 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx}{6300}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}-\frac{\int \frac{-941013-1115037 x}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{37800}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{17699 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{1200}-\frac{32513 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1680}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{\left (17699 \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 )}{600 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (32513 \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 )}{840 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{(27213+7877 x) \sqrt{2+5 x+3 x^2}}{2100 \sqrt{3+2 x}}+\frac{(14311+10729 x) \left (2+5 x+3 x^2\right )^{3/2}}{3150 (3+2 x)^{5/2}}+\frac{(124+113 x) \left (2+5 x+3 x^2\right )^{5/2}}{63 (3+2 x)^{9/2}}+\frac{17699 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{600 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{32513 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{840 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.440175, size = 227, normalized size = 1.1 \[ -\frac{8 \left (3 x^2+5 x+2\right ) \left (18900 x^5+1579088 x^4+8510198 x^3+17853828 x^2+16893773 x+6047963\right )-4 (2 x+3)^4 \left (-26354 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+247786 \left (3 x^2+5 x+2\right )+123893 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{50400 (2 x+3)^{9/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(11/2),x]

[Out]

-(8*(2 + 5*x + 3*x^2)*(6047963 + 16893773*x + 17853828*x^2 + 8510198*x^3 + 1579088*x^4 + 18900*x^5) - 4*(3 + 2
*x)^4*(247786*(2 + 5*x + 3*x^2) + 123893*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] - 26354*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*S
qrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(50400*(3 + 2*x)^(9/2)*Sqrt[2 + 5*x
+ 3*x^2])

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Maple [B]  time = 0.02, size = 487, normalized size = 2.4 \begin{align*} -{\frac{1}{126000} \left ( 618752\,\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}+1982288\,\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}+3712512\,\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}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+8353152\,\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]\,\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}+1134000\,{x}^{7}+8353152\,\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]\,\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]\,{x}^{6}+3132432\,\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}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+241918520 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{9}{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)^(11/2),x)

[Out]

-1/126000*(618752*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)*(-20-3
0*x)^(1/2)+1982288*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)+3712512*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^3*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-
20-30*x)^(1/2)+8353152*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]*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)+1134000*x^7+8353152*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]*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]*x^6+3132432*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*Ellip
ticF(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)*Ellipti
cE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))[phone]*x^5+1985413000*x^4+3139417100*x^3+2766408200*x^2+1280547220*x+2
41918520)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(9/2)

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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{11}{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)^(11/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(11/2), 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}}{64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729}, 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)^(11/2),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)/(64*x^6 + 576*x
^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729), 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)**(11/2),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{11}{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)^(11/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(11/2), x)