\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{7/2}} \, dx\) [2600]

Optimal result

Integrand size = 29, antiderivative size = 207 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {2333 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {2+5 x+3 x^2}}-\frac {12857 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

1/210*(2291+879*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(3/2)-1/35*(53+5*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(5/2)-12857/168
*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)+2333/40*Ellipti
cE(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)-1/140*(10763+3117*x)*(3
*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {826, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {12857 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {2333 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{40 \sqrt {3 x^2+5 x+2}}-\frac {(5 x+53) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{5/2}}+\frac {(879 x+2291) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{3/2}}-\frac {(3117 x+10763) \sqrt {3 x^2+5 x+2}}{140 \sqrt {2 x+3}} \]

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(7/2),x]

[Out]

-1/140*((10763 + 3117*x)*Sqrt[2 + 5*x + 3*x^2])/Sqrt[3 + 2*x] + ((2291 + 879*x)*(2 + 5*x + 3*x^2)^(3/2))/(210*
(3 + 2*x)^(3/2)) - ((53 + 5*x)*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^(5/2)) + (2333*Sqrt[3]*Sqrt[-2 - 5*x - 3
*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(40*Sqrt[2 + 5*x + 3*x^2]) - (12857*Sqrt[-2 - 5*x - 3*x^2]
*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(56*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 826

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 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 {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {1}{14} \int \frac {(-245-293 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{5/2}} \, dx \\ & = \frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {1}{140} \int \frac {(-7939-9351 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {1}{840} \int \frac {-124041-146979 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {6999}{80} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {12857}{112} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}-\frac {\left (12857 \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 )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (2333 \sqrt {3} \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 )}{40 \sqrt {2+5 x+3 x^2}} \\ & = -\frac {(10763+3117 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}+\frac {(2291+879 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{3/2}}-\frac {(53+5 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{5/2}}+\frac {2333 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {2+5 x+3 x^2}}-\frac {12857 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{56 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 31.60 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\frac {701136+2556580 x+3262382 x^2+1717690 x^3+335988 x^4+41220 x^5+12744 x^6-3240 x^7+48993 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-10422 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{840 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \]

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(7/2),x]

[Out]

(701136 + 2556580*x + 3262382*x^2 + 1717690*x^3 + 335988*x^4 + 41220*x^5 + 12744*x^6 - 3240*x^7 + 48993*Sqrt[5
]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]],
3/5] - 10422*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[
5/3]/Sqrt[3 + 2*x]], 3/5])/(840*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.48

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{256 \left (x +\frac {3}{2}\right )^{3}}+\frac {1169 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{384 \left (x +\frac {3}{2}\right )^{2}}-\frac {6889 \left (6 x^{2}+10 x +4\right )}{240 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {9 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{56}+\frac {909 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{560}-\frac {1559 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{280}-\frac {41347 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{4200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {2333 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{200 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(307\)
default	\(\frac {30584 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-65324 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{2} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-16200 x^{7}+91752 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-195972 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+63720 x^{6}+68814 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-146979 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+206100 x^{5}-4199220 x^{4}-18847630 x^{3}-30231440 x^{2}-21022270 x -5313060}{4200 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {5}{2}}}\)	\(311\)

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2),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)*(-65/256*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3
+1169/384*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2-6889/240*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-9/56*x^
2*(6*x^3+19*x^2+19*x+6)^(1/2)+909/560*x*(6*x^3+19*x^2+19*x+6)^(1/2)-1559/280*(6*x^3+19*x^2+19*x+6)^(1/2)-41347
/4200*(-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))-2333/200*(-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*Ellip
ticE(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 = 121, normalized size of antiderivative = 0.58 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\frac {62207 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 293958 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 12 \, {\left (540 \, x^{5} - 3024 \, x^{4} - 2190 \, x^{3} + 145640 \, x^{2} + 386981 \, x + 265653\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{5040 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2),x, algorithm="fricas")

[Out]

-1/5040*(62207*sqrt(6)*(8*x^3 + 36*x^2 + 54*x + 27)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 293958*sq
rt(6)*(8*x^3 + 36*x^2 + 54*x + 27)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/
18)) + 12*(540*x^5 - 3024*x^4 - 2190*x^3 + 145640*x^2 + 386981*x + 265653)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)
)/(8*x^3 + 36*x^2 + 54*x + 27)

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt {2 x + 3} + 36 x^{2} \sqrt {2 x + 3} + 54 x \sqrt {2 x + 3} + 27 \sqrt {2 x + 3}}\, dx \]

[In]

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(7/2),x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*s
qrt(2*x + 3)), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x
*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*
x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(8
*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(-15*x**4*s
qrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)),
x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3)
+ 27*sqrt(2*x + 3)), x)

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(7/2), x)

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{7/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{7/2}} \,d x \]

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(7/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(7/2), x)