3.2605 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{17/2}} \, dx\)

Optimal. Leaf size=261 \[ \frac{594851 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{75075000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac{(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{64350 (2 x+3)^{11/2}}-\frac{(328339 x+386846) \sqrt{3 x^2+5 x+2}}{7507500 (2 x+3)^{7/2}}+\frac{335723 \sqrt{3 x^2+5 x+2}}{80437500 \sqrt{2 x+3}}+\frac{594851 \sqrt{3 x^2+5 x+2}}{112612500 (2 x+3)^{3/2}}-\frac{335723 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{53625000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(594851*Sqrt[2 + 5*x + 3*x^2])/(112612500*(3 + 2*x)^(3/2)) + (335723*Sqrt[2 + 5*x + 3*x^2])/(80437500*Sqrt[3 +
2*x]) - ((386846 + 328339*x)*Sqrt[2 + 5*x + 3*x^2])/(7507500*(3 + 2*x)^(7/2)) - ((8901 + 8399*x)*(2 + 5*x + 3
*x^2)^(3/2))/(64350*(3 + 2*x)^(11/2)) + ((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)) - (33572
3*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(53625000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
) + (594851*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(75075000*Sqrt[3]*Sqrt[2 + 5*
x + 3*x^2])

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Rubi [A]  time = 0.187698, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {810, 834, 843, 718, 424, 419} \[ \frac{(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac{(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{64350 (2 x+3)^{11/2}}-\frac{(328339 x+386846) \sqrt{3 x^2+5 x+2}}{7507500 (2 x+3)^{7/2}}+\frac{335723 \sqrt{3 x^2+5 x+2}}{80437500 \sqrt{2 x+3}}+\frac{594851 \sqrt{3 x^2+5 x+2}}{112612500 (2 x+3)^{3/2}}+\frac{594851 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{75075000 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{335723 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{53625000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

(594851*Sqrt[2 + 5*x + 3*x^2])/(112612500*(3 + 2*x)^(3/2)) + (335723*Sqrt[2 + 5*x + 3*x^2])/(80437500*Sqrt[3 +
2*x]) - ((386846 + 328339*x)*Sqrt[2 + 5*x + 3*x^2])/(7507500*(3 + 2*x)^(7/2)) - ((8901 + 8399*x)*(2 + 5*x + 3
*x^2)^(3/2))/(64350*(3 + 2*x)^(11/2)) + ((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)) - (33572
3*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(53625000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
) + (594851*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(75075000*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (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)^{17/2}} \, dx &=\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{1}{390} \int \frac{(-118-243 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx\\ &=-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}+\frac{\int \frac{(20841+22347 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx}{128700}\\ &=-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{\int \frac{-1257990-1433511 x}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx}{45045000}\\ &=\frac{594851 \sqrt{2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}+\frac{\int \frac{\frac{4505397}{2}+\frac{5353659 x}{2}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{337837500}\\ &=\frac{594851 \sqrt{2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac{335723 \sqrt{2+5 x+3 x^2}}{80437500 \sqrt{3+2 x}}-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{\int \frac{4585419+\frac{21150549 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{844593750}\\ &=\frac{594851 \sqrt{2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac{335723 \sqrt{2+5 x+3 x^2}}{80437500 \sqrt{3+2 x}}-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{335723 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{107250000}+\frac{594851 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{150150000}\\ &=\frac{594851 \sqrt{2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac{335723 \sqrt{2+5 x+3 x^2}}{80437500 \sqrt{3+2 x}}-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{\left (335723 \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 )}{53625000 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (594851 \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 )}{75075000 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{594851 \sqrt{2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac{335723 \sqrt{2+5 x+3 x^2}}{80437500 \sqrt{3+2 x}}-\frac{(386846+328339 x) \sqrt{2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac{(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac{(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac{335723 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{53625000 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{594851 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{75075000 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.54396, size = 237, normalized size = 0.91 \[ -\frac{2 (2 x+3)^7 \left (-1131016 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+9400244 \left (3 x^2+5 x+2\right )+4700122 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-8 \left (3 x^2+5 x+2\right ) \left (300807808 x^7+3348834304 x^6+17742950508 x^5+46830142120 x^4+67557035830 x^3+55283449932 x^2+24502214271 x+4641518352\right )}{4504500000 (2 x+3)^{15/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

-(-8*(2 + 5*x + 3*x^2)*(4641518352 + 24502214271*x + 55283449932*x^2 + 67557035830*x^3 + 46830142120*x^4 + 177
42950508*x^5 + 3348834304*x^6 + 300807808*x^7) + 2*(3 + 2*x)^7*(9400244*(2 + 5*x + 3*x^2) + 4700122*Sqrt[5]*Sq
rt[(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
] - 1131016*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]))/(4504500000*(3 + 2*x)^(15/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.04, size = 761, normalized size = 2.9 \begin{align*} \text{result too large to display} \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)^(17/2),x)

[Out]

1/11261250000*(185660734080+1444240406040*x+838916736*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^6
*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*(3+2*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))
*x^6*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*(3+2*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^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*(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^7*(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^7*(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^5*(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)*Elliptic
E(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)*Ell
ipticF(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*(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^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]
00*x^2+9446154382120*x^5+4718056950160*x^6+1411492773200*x^7+11945916263720*x^4+9700759282660*x^3+231010839040
*x^8+18048468480*x^9+1365112278*(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)))/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(15/2)

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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{17}{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)^(17/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), 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}}{512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683}, 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)^(17/2),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)/(512*x^9 + 6912
*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683), 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)**(17/2),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{17}{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)^(17/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)