\(\int (5-x) (3+2 x)^{5/2} (2+5 x+3 x^2)^{3/2} \, dx\) [2584]

Optimal result

Integrand size = 29, antiderivative size = 256 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1015187 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

202/351*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(5/2)-2/45*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(5/2)+1/243243*(534271+629153*x)*(3
*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)+13318/11583*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2)-207851/18764460*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)+1015187/26270244*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)-1/21891870*(6006884+7817373*x)*
(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {846, 828, 857, 732, 435, 430} \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {1015187 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {207851 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {13318 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac {\sqrt {2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac {\sqrt {2 x+3} (7817373 x+6006884) \sqrt {3 x^2+5 x+2}}{21891870} \]

[In]

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

[Out]

-1/21891870*(Sqrt[3 + 2*x]*(6006884 + 7817373*x)*Sqrt[2 + 5*x + 3*x^2]) + (Sqrt[3 + 2*x]*(534271 + 629153*x)*(
2 + 5*x + 3*x^2)^(3/2))/243243 + (13318*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/11583 + (202*(3 + 2*x)^(3/2)*(2
+ 5*x + 3*x^2)^(5/2))/351 - (2*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/45 - (207851*Sqrt[-2 - 5*x - 3*x^2]*E
llipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(6254820*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (1015187*Sqrt[-2 - 5*x
- 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(8756748*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 828

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

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}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{45} \int (3+2 x)^{3/2} \left (385+\frac {505 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx \\ & = \frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {4 \int \sqrt {3+2 x} \left (\frac {46155}{4}+\frac {33295 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{1755} \\ & = \frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {8 \int \frac {\left (242380+\frac {1348185 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{57915} \\ & = \frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {4 \int \frac {\left (\frac {17298645}{8}+\frac {13028955 x}{8}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{3648645} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2 \int \frac {\frac {1333245}{2}-\frac {21824355 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{164189025} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{12509640}+\frac {1015187 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{17513496} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (207851 \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 )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (1015187 \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 )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = -\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1015187 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 31.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.85 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {2 \sqrt {3+2 x} \left (-5523159638-44206631441 x-150475882830 x^2-283276026729 x^3-319887585072 x^4-217661096106 x^5-82311172272 x^6-11776907520 x^7+1907623872 x^8+630485856 x^9\right )+1454957 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+1590604 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{131351220 (3+2 x) \sqrt {2+5 x+3 x^2}} \]

[In]

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

[Out]

-1/131351220*(2*Sqrt[3 + 2*x]*(-5523159638 - 44206631441*x - 150475882830*x^2 - 283276026729*x^3 - 31988758507
2*x^4 - 217661096106*x^5 - 82311172272*x^6 - 11776907520*x^7 + 1907623872*x^8 + 630485856*x^9) + 1454957*Sqrt[
5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/
5] + 1590604*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/
Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (18914575680 x^{9}[phone] x^{8}-353307225600 x^{7}-2469335168160 x^{6}-6529832883180 x^{5}+5431467 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-1454957 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-9596627552160 x^{4}-8498280801870 x^{3}-4514407431030 x^{2}-1326417186780 x -165782086560\right )}{1970268300 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(166\)
risch	\(-\frac {\left (35026992 x^{6}-4939704 x^{5}-749549052 x^{4}-2218655502 x^{3}-2688028992 x^{2}-1484149221 x -307003864\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{21891870}-\frac {\left (-\frac {88883 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{164189025 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {207851 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{93822300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(223\)
elliptic	\(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {44 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{195}+\frac {73442 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2145}+\frac {5869459 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{57915}+\frac {13575904 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{110565}+\frac {164905469 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2432430}+\frac {153501932 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{10945935}-\frac {88883 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{164189025 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {207851 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{93822300 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\)	\(336\)

[In]

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

[Out]

-1/1970268300*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(18914575680*x^9+57228716160*x^8-353307225600*x^7-246933516816
0*x^6-6529832883180*x^5+5431467*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticF(1/5*(-20-30*x)
^(1/2),1/2*10^(1/2))-1454957*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticE(1/5*(-20-30*x)^(1
/2),1/2*10^(1/2))-9596627552160*x^4-8498280801870*x^3-4514407431030*x^2-1326417186780*x-165782086560)/(6*x^3+1
9*x^2+19*x+6)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.30 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{21891870} \, {\left (35026992 \, x^{6} - 4939704 \, x^{5} - 749549052 \, x^{4} - 2218655502 \, x^{3} - 2688028992 \, x^{2} - 1484149221 \, x - 307003864\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {34043759}{2364321960} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {207851}{18764460} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]

[In]

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

[Out]

-1/21891870*(35026992*x^6 - 4939704*x^5 - 749549052*x^4 - 2218655502*x^3 - 2688028992*x^2 - 1484149221*x - 307
003864)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 34043759/2364321960*sqrt(6)*weierstrassPInverse(19/27, -28/729,
x + 19/18) + 207851/18764460*sqrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 1
9/18))

Sympy [F]

\[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=- \int \left (- 90 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 327 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]

[In]

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

[Out]

-Integral(-90*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-327*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2),
x) - Integral(-406*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-185*x**3*sqrt(2*x + 3)*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(-4*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(12*x**5*sqrt(2*x + 3)
*sqrt(3*x**2 + 5*x + 2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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