\(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{15/2}} \, dx\) [2604]

Optimal result

Integrand size = 29, antiderivative size = 234 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {9421 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

1/6930*(73-33*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2)+1/11*(8+9*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2)+5083/495000
*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)-9421/693000*Ell
ipticF(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)+1/346500*(21492+178
33*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)-5083/247500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {824, 848, 857, 732, 435, 430} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {9421 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {5083 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac {(17833 x+21492) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac {5083 \sqrt {3 x^2+5 x+2}}{247500 \sqrt {2 x+3}} \]

[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 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 {(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 ) \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 )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (9421 \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 )}{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] (verified)

Time = 31.42 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {8 \left (2+5 x+3 x^2\right ) \left (11865789+41339721 x+54318160 x^2+33648370 x^3+12953760 x^4+6409516 x^5+2277184 x^6\right )-4 (3+2 x)^6 \left (71162 \left (2+5 x+3 x^2\right )+35581 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-7318 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )\right )}{13860000 (3+2 x)^{13/2} \sqrt {2+5 x+3 x^2}} \]

[In]

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

[Out]

-1/13860000*(8*(2 + 5*x + 3*x^2)*(11865789 + 41339721*x + 54318160*x^2 + 33648370*x^3 + 12953760*x^4 + 6409516
*x^5 + 2277184*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]))/((3 + 2*
x)^(13/2)*Sqrt[2 + 5*x + 3*x^2])

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.46

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {25 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4096 \left (x +\frac {3}{2}\right )^{7}}+\frac {1105 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{22528 \left (x +\frac {3}{2}\right )^{6}}-\frac {3929 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{25344 \left (x +\frac {3}{2}\right )^{5}}+\frac {487 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2016 \left (x +\frac {3}{2}\right )^{4}}-\frac {278249 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1478400 \left (x +\frac {3}{2}\right )^{3}}+\frac {704257 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{11088000 \left (x +\frac {3}{2}\right )^{2}}-\frac {5083 \left (6 x^{2}+10 x +4\right )}{495000 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {29819 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{17325000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {5083 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{2475000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(342\)
default	\(\frac {1106304 F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) \sqrt {15}\, x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}-2277184 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{6} \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {3+2 x}+9956736 \sqrt {15}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}-20494656 \sqrt {15}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-20-30 x}\, \sqrt {3+3 x}[phone] \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}-76854960 \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}[phone] \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}-153709920 \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}-204946560 x^{8}[phone] \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}-172923660 \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}-918434040 x^{7}[phone] 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}-103754196 \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}-2263896840 x^{6}[phone] \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 )-25938549 \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 )-5355988260 x^{5}-10713115500 x^{4}-13887201090 x^{3}-10527968760 x^{2}-4260251610 x -711947340}{51975000 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {13}{2}}}\)	\(668\)

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),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)*(-25/4096*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^
7+1105/22528*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^6-3929/25344*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^5+487/2016*(
6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^4-278249/1478400*(6*x^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^3+704257/11088000*(6*x
^3+19*x^2+19*x+6)^(1/2)/(x+3/2)^2-5083/495000*(6*x^2+10*x+4)/((x+3/2)*(6*x^2+10*x+4))^(1/2)-29819/17325000*(-2
0-30*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))-5083/2475000*(-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 = 0.12 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\frac {139297 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 640458 \, \sqrt {6} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (2277184 \, x^{6} + 6409516 \, x^{5} + 12953760 \, x^{4} + 33648370 \, x^{3} + 54318160 \, x^{2} + 41339721 \, x + 11865789\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{62370000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x, algorithm="fricas")

[Out]

-1/62370000*(139297*sqrt(6)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 218
7)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 640458*sqrt(6)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4
+ 22680*x^3 + 20412*x^2 + 10206*x + 2187)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729,
x + 19/18)) + 36*(2277184*x^6 + 6409516*x^5 + 12953760*x^4 + 33648370*x^3 + 54318160*x^2 + 41339721*x + 118657
89)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 +
10206*x + 2187)

Sympy [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{15/2}} \,d x \]

[In]

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

[Out]

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