\(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^{11/2}} \, dx\) [2592]

Optimal result

Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

1/45*(44+51*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2)+23/22500*EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2
-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+7/4500*EllipticF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(
1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)-1/2250*(189+211*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)-23/11250*(3*x^2+5*x+2)^(
1/2)/(3+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {824, 848, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {7 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {23 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(51 x+44) \left (3 x^2+5 x+2\right )^{3/2}}{45 (2 x+3)^{9/2}}-\frac {(211 x+189) \sqrt {3 x^2+5 x+2}}{2250 (2 x+3)^{5/2}}-\frac {23 \sqrt {3 x^2+5 x+2}}{11250 \sqrt {2 x+3}} \]

[In]

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

[Out]

(-23*Sqrt[2 + 5*x + 3*x^2])/(11250*Sqrt[3 + 2*x]) - ((189 + 211*x)*Sqrt[2 + 5*x + 3*x^2])/(2250*(3 + 2*x)^(5/2
)) + ((44 + 51*x)*(2 + 5*x + 3*x^2)^(3/2))/(45*(3 + 2*x)^(9/2)) + (23*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[
Sqrt[3]*Sqrt[1 + x]], -2/3])/(7500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (7*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1500*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 848

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 857

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {1}{210} \int \frac {(28-21 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx \\ & = -\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\int \frac {301+147 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{31500} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}-\frac {\int \frac {-546-\frac {483 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{78750} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{15000}+\frac {7 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3000} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {\left (23 \sqrt {-2-5 x-3 x^2}\right ) \text {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 )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (7 \sqrt {-2-5 x-3 x^2}\right ) \text {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 )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {23 \sqrt {2+5 x+3 x^2}}{11250 \sqrt {3+2 x}}-\frac {(189+211 x) \sqrt {2+5 x+3 x^2}}{2250 (3+2 x)^{5/2}}+\frac {(44+51 x) \left (2+5 x+3 x^2\right )^{3/2}}{45 (3+2 x)^{9/2}}+\frac {23 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{7500 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {7 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1500 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 31.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {53980+373610 x+998860 x^2+1297210 x^3+822160 x^4+204180 x^5+23 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-44 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{11/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{22500 (3+2 x)^{9/2} \sqrt {2+5 x+3 x^2}} \]

[In]

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

[Out]

(53980 + 373610*x + 998860*x^2 + 1297210*x^3 + 822160*x^4 + 204180*x^5 + 23*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3
+ 2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 44*Sqrt[5]*Sqrt[(1
+ x)/(3 + 2*x)]*(3 + 2*x)^(11/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(2
2500*(3 + 2*x)^(9/2)*Sqrt[2 + 5*x + 3*x^2])

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.46

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2304 \left (x +\frac {3}{2}\right )^{5}}+\frac {155 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1152 \left (x +\frac {3}{2}\right )^{4}}-\frac {971 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4800 \left (x +\frac {3}{2}\right )^{3}}+\frac {3403 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{36000 \left (x +\frac {3}{2}\right )^{2}}-\frac {23 \left (6 x^{2}+10 x +4\right )}{22500 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {13 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{28125 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {23 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{112500 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(294\)
default	\(-\frac {1392 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+368 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{4} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+8352 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+2208 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{3} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+18792 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+4968 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+18792 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+4968 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+33120 x^{6}+7047 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+1863 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-2808780 x^{5}-11532000 x^{4}-18133350 x^{3}-13771950 x^{2}-5026620 x -697920}{337500 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {9}{2}}}\)	\(482\)

[In]

int((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x,method=_RETURNVERBOSE)

[Out]

((3+2*x)*(3*x^2+5*x+2))^(1/2)/(3+2*x)^(1/2)/(3*x^2+5*x+2)^(1/2)*(-65/2304*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^
5+155/1152*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4-971/4800*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3+3403/36000*(6*
x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2-23/22500*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-13/28125*(-20-30*x)^
(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-2
3/112500*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*(45+30*x)^(1/2)/(6*x^3+19*x^2+19*x+6)^(1/2)*(1/3*EllipticE(1/5*(-20-30
*x)^(1/2),1/2*10^(1/2))-EllipticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\frac {499 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 414 \, \sqrt {6} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (368 \, x^{4} - 31822 \, x^{3} - 75342 \, x^{2} - 54697 \, x - 11632\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{405000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="fricas")

[Out]

1/405000*(499*sqrt(6)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*weierstrassPInverse(19/27, -28/729
, x + 19/18) - 414*sqrt(6)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*weierstrassZeta(19/27, -28/72
9, weierstrassPInverse(19/27, -28/729, x + 19/18)) - 36*(368*x^4 - 31822*x^3 - 75342*x^2 - 54697*x - 11632)*sq
rt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} \sqrt {2 x + 3} + 240 x^{4} \sqrt {2 x + 3} + 720 x^{3} \sqrt {2 x + 3} + 1080 x^{2} \sqrt {2 x + 3} + 810 x \sqrt {2 x + 3} + 243 \sqrt {2 x + 3}}\, dx \]

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x + 3)
+ 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt(2*x + 3) + 243*sqrt(2*x + 3)), x) - Integral(-23*x*sqrt(3*x**2 + 5*x +
2)/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x*
sqrt(2*x + 3) + 243*sqrt(2*x + 3)), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(32*x**5*sqrt(2*x + 3) + 240
*x**4*sqrt(2*x + 3) + 720*x**3*sqrt(2*x + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt(2*x + 3) + 243*sqrt(2*x +
3)), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(32*x**5*sqrt(2*x + 3) + 240*x**4*sqrt(2*x + 3) + 720*x**3*sq
rt(2*x + 3) + 1080*x**2*sqrt(2*x + 3) + 810*x*sqrt(2*x + 3) + 243*sqrt(2*x + 3)), x)

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^{11/2}} \,d x \]

[In]

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

[Out]

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