3.2604 \(\int \frac{(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{15/2}} \, dx\)

Optimal. Leaf size=234 \[ -\frac{9421 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-5083*Sqrt[2 + 5*x + 3*x^2])/(247500*Sqrt[3 + 2*x]) + ((21492 + 17833*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 +
2*x)^(5/2)) + ((73 - 33*x)*(2 + 5*x + 3*x^2)^(3/2))/(6930*(3 + 2*x)^(9/2)) + ((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2
))/(11*(3 + 2*x)^(13/2)) + (5083*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(165000*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (9421*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
31000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.161592, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}-\frac{9421 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

(-5083*Sqrt[2 + 5*x + 3*x^2])/(247500*Sqrt[3 + 2*x]) + ((21492 + 17833*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 +
2*x)^(5/2)) + ((73 - 33*x)*(2 + 5*x + 3*x^2)^(3/2))/(6930*(3 + 2*x)^(9/2)) + ((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2
))/(11*(3 + 2*x)^(13/2)) + (5083*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(165000*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (9421*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
31000*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)^{15/2}} \, dx &=\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac{1}{286} \int \frac{(104+39 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx\\ &=\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac{\int \frac{(-4979-6357 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx}{60060}\\ &=\frac{(21492+17833 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac{\int \frac{319852+367419 x}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx}{9009000}\\ &=-\frac{5083 \sqrt{2+5 x+3 x^2}}{247500 \sqrt{3+2 x}}+\frac{(21492+17833 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac{\int \frac{\frac{1162941}{2}+\frac{1387659 x}{2}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{22522500}\\ &=-\frac{5083 \sqrt{2+5 x+3 x^2}}{247500 \sqrt{3+2 x}}+\frac{(21492+17833 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac{5083 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{330000}-\frac{9421 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{462000}\\ &=-\frac{5083 \sqrt{2+5 x+3 x^2}}{247500 \sqrt{3+2 x}}+\frac{(21492+17833 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac{\left (5083 \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 )}{165000 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (9421 \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 )}{231000 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{5083 \sqrt{2+5 x+3 x^2}}{247500 \sqrt{3+2 x}}+\frac{(21492+17833 x) \sqrt{2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac{(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac{(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac{5083 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{9421 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.527435, size = 232, normalized size = 0.99 \[ -\frac{8 \left (3 x^2+5 x+2\right ) \left (2277184 x^6+6409516 x^5+12953760 x^4+33648370 x^3+54318160 x^2+41339721 x+11865789\right )-4 (2 x+3)^6 \left (-7318 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+71162 \left (3 x^2+5 x+2\right )+35581 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{13860000 (2 x+3)^{13/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

-(8*(2 + 5*x + 3*x^2)*(11865789 + 41339721*x + 54318160*x^2 + 33648370*x^3 + 12953760*x^4 + 6409516*x^5 + 2277
184*x^6) - 4*(3 + 2*x)^6*(71162*(2 + 5*x + 3*x^2) + 35581*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] - 7318*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]))/(13860000*(3 + 2*x)
^(13/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.037, size = 668, 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)^(15/2),x)

[Out]

-1/34650000*(737536*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)+2277184*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)+6637824*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^5*(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^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*(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]*x^8+56006640*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)[phone]*x^7+33603984*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+8400996*(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))[phone]*x^5+7142077000*x^4+9258134060
*x^3+7018645840*x^2+2840167740*x+474631560)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(13/2)

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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{15}{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)^(15/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), 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}}{256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561}, 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)^(15/2),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)/(256*x^8 + 3072
*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561), 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)**(15/2),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{15}{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)^(15/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)