\(\int \frac {(5-x) (3+2 x)^{9/2}}{(2+5 x+3 x^2)^{5/2}} \, dx\) [2621]

Optimal result

Integrand size = 29, antiderivative size = 202 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

-2/9*(3+2*x)^(7/2)*(121+139*x)/(3*x^2+5*x+2)^(3/2)+4/9*(3+2*x)^(3/2)*(2164+2571*x)/(3*x^2+5*x+2)^(1/2)-110516/
243*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)+148780/243*E
llipticF(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)-59512/81*(3+2*x)^
(1/2)*(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.08 (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 = {832, 846, 857, 732, 435, 430} \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {148780 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {110516 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2 (139 x+121) (2 x+3)^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {4 (2571 x+2164) (2 x+3)^{3/2}}{9 \sqrt {3 x^2+5 x+2}}-\frac {59512}{81} \sqrt {3 x^2+5 x+2} \sqrt {2 x+3} \]

[In]

Int[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(3 + 2*x)^(7/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (4*(3 + 2*x)^(3/2)*(2164 + 2571*x))/(9*Sqrt[2
+ 5*x + 3*x^2]) - (59512*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/81 - (110516*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[A
rcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(81*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (148780*Sqrt[-2 - 5*x - 3*x^2]*Ellipti
cF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(81*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 832

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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

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 {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {2}{9} \int \frac {(3+2 x)^{5/2} (4+411 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}+\frac {4}{27} \int \frac {(-18243-22317 x) \sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {8}{243} \int \frac {-34269-\frac {82887 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {55258}{81} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {74390}{81} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {\left (110516 \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 )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (148780 \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 )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {2 (3+2 x)^{7/2} (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x)^{3/2} (2164+2571 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {59512}{81} \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {110516 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {148780 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{81 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 31.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 \left (3 (3+2 x) \left (-85285-330053 x-411640 x^2-166566 x^3+144 x^4\right )+2 \left (2+5 x+3 x^2\right ) \left (55258 \left (2+5 x+3 x^2\right )+27629 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-5312 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )\right )}{243 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \]

[In]

Integrate[((5 - x)*(3 + 2*x)^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(3*(3 + 2*x)*(-85285 - 330053*x - 411640*x^2 - 166566*x^3 + 144*x^4) + 2*(2 + 5*x + 3*x^2)*(55258*(2 + 5*x
+ 3*x^2) + 27629*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[S
qrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 5312*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])))/(243*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.24

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {23194}{2187}-\frac {24166 x}{2187}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (9+6 x \right ) \left (-\frac {133822}{729}-\frac {55682 x}{243}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {32 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{81}+\frac {91384 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {110516 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1215 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(250\)
default	\(-\frac {2 \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (86094 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-165774 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+143490 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-276290 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+57396 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-110516 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+12960 x^{5}-14971500 x^{4}-59534010 x^{3}-85276170 x^{2}-52232805 x -11513475\right )}{3645 \left (1+x \right ) \left (2 x^{2}+5 x +3\right ) \left (2+3 x \right )^{2}}\)	\(325\)

[In]

int((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),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)*((-23194/2187-24166/2187*x)*(6*x^3+19*x^2+19*x
+6)^(1/2)/(x^2+5/3*x+2/3)^2-2*(9+6*x)*(-133822/729-55682/243*x)/((x^2+5/3*x+2/3)*(9+6*x))^(1/2)-32/81*(6*x^3+1
9*x^2+19*x+6)^(1/2)+91384/1215*(-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)*Elli
pticF(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))+110516/1215*(-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 = 131, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (113723 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 497322 \, \sqrt {6} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 27 \, {\left (144 \, x^{4} - 166566 \, x^{3} - 411640 \, x^{2} - 330053 \, x - 85285\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{2187 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

2/2187*(113723*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 4
97322*sqrt(6)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27,
-28/729, x + 19/18)) - 27*(144*x^4 - 166566*x^3 - 411640*x^2 - 330053*x - 85285)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*
x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((5-x)*(3+2*x)**(9/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")

[Out]

-integrate((2*x + 3)^(9/2)*(x - 5)/(3*x^2 + 5*x + 2)^(5/2), x)

Giac [F]

\[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\int { -\frac {{\left (2 \, x + 3\right )}^{\frac {9}{2}} {\left (x - 5\right )}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((5-x)*(3+2*x)^(9/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

integrate(-(2*x + 3)^(9/2)*(x - 5)/(3*x^2 + 5*x + 2)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) (3+2 x)^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^{9/2}\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]

[In]

int(-((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^(5/2),x)

[Out]

-int(((2*x + 3)^(9/2)*(x - 5))/(5*x + 3*x^2 + 2)^(5/2), x)