\(\int (2-5 x) x^{5/2} (2+5 x+3 x^2)^{3/2} \, dx\) [189]

Optimal result

Integrand size = 25, antiderivative size = 252 \[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {497824 \sqrt {x} (2+3 x)}{32837805 \sqrt {2+5 x+3 x^2}}+\frac {61736 \sqrt {x} \sqrt {2+5 x+3 x^2}}{2189187}-\frac {3688 x^{3/2} \sqrt {2+5 x+3 x^2}}{93555}+\frac {13004 x^{5/2} \sqrt {2+5 x+3 x^2}}{243243}+\frac {4 x^{7/2} (2803+2484 x) \sqrt {2+5 x+3 x^2}}{11583}+\frac {2}{39} (1-13 x) x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac {497824 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{32837805 \sqrt {1+x} \sqrt {2+3 x}}-\frac {61736 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2189187 \sqrt {2+5 x+3 x^2}} \] Output:

-497824/32837805*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+61736/2189187*x^(1/2) 
*(3*x^2+5*x+2)^(1/2)-3688/93555*x^(3/2)*(3*x^2+5*x+2)^(1/2)+13004/243243*x 
^(5/2)*(3*x^2+5*x+2)^(1/2)+4/11583*x^(7/2)*(2803+2484*x)*(3*x^2+5*x+2)^(1/ 
2)+2/39*(1-13*x)*x^(7/2)*(3*x^2+5*x+2)^(3/2)+497824/32837805*2^(1/2)*(3*x^ 
2+5*x+2)^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))/(1+x)^(1/2)/(2 
+3*x)^(1/2)-61736/2189187*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJacobiA 
M(arctan(x^(1/2)),1/2*I*2^(1/2))/(3*x^2+5*x+2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.24 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.73 \[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-497824 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (497824+318520 x-273876 x^2+91620 x^3-37601118 x^4-83323080 x^5+69664455 x^6+337486905 x^7+320800095 x^8+98513415 x^9+214108 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )\right )}{32837805 \sqrt {x} \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

((-497824*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*Ar 
cSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - 2*(497824 + 318520*x - 273876*x^2 + 91620 
*x^3 - 37601118*x^4 - 83323080*x^5 + 69664455*x^6 + 337486905*x^7 + 320800 
095*x^8 + 98513415*x^9 + (214108*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x] 
*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2]))/(32837805*Sqrt[x]* 
Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {1236, 27, 1236, 27, 1236, 25, 1231, 27, 1231, 25, 1240, 1503, 1413, 1456}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (2-5 x) x^{5/2} \left (3 x^2+5 x+2\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{45} \int 5 x^{3/2} (34 x+5) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \int x^{3/2} (34 x+5) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{9} \left (\frac {2}{39} \int -\frac {1}{2} \sqrt {x} (1165 x+204) \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}-\frac {1}{39} \int \sqrt {x} (1165 x+204) \left (3 x^2+5 x+2\right )^{3/2}dx\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (-\frac {2}{33} \int -\frac {(14109 x+1165) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {x}}dx-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \int \frac {(14109 x+1165) \left (3 x^2+5 x+2\right )^{3/2}}{\sqrt {x}}dx-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{63} \int \frac {3 (22823 x+5090) \sqrt {3 x^2+5 x+2}}{2 \sqrt {x}}dx\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{21} \int \frac {(22823 x+5090) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (\frac {2}{45} \int -\frac {31114 x+38585}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2}{45} \sqrt {x} (205407 x+190465) \sqrt {3 x^2+5 x+2}\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (-\frac {2}{45} \int \frac {31114 x+38585}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (205407 x+190465)\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (-\frac {4}{45} \int \frac {31114 x+38585}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (205407 x+190465)\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (-\frac {4}{45} \left (38585 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+31114 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (205407 x+190465)\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (-\frac {4}{45} \left (31114 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {38585 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (205407 x+190465)\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2}{9} \left (\frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{21} \left (-\frac {4}{45} \left (\frac {38585 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+31114 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2}{45} \sqrt {x} \sqrt {3 x^2+5 x+2} (205407 x+190465)\right )+\frac {2}{21} \sqrt {x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {2330}{33} \sqrt {x} \left (3 x^2+5 x+2\right )^{5/2}\right )+\frac {68}{39} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}\)

Input:

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

Output:

(-2*x^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/9 + (2*((68*x^(3/2)*(2 + 5*x + 3*x^2) 
^(5/2))/39 + ((-2330*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/33 + (2*((2*Sqrt[x]* 
(27010 + 32921*x)*(2 + 5*x + 3*x^2)^(3/2))/21 + ((-2*Sqrt[x]*(190465 + 205 
407*x)*Sqrt[2 + 5*x + 3*x^2])/45 - (4*(31114*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[ 
2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[Arc 
Tan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (38585*(1 + x)*Sqrt[(2 + 
 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3 
*x^2])))/45)/21))/33)/39))/9
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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] - Simp[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] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1236

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

rule 1240

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), 
x_Symbol] :> Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, 
 Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 1413

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b 
^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + 
(b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF 
[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; 
 FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1456

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 
])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q 
)*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan 
[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ 
{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
 

rule 1503

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d   Int[1/Sqrt[a + b*x^2 + c*x^4] 
, x], x] + Simp[e   Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) 
/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.56

method	result	size
default	\(-\frac {2 \left (295540245 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}-249969240 x^{5}+89652 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+124456 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-112803354 x^{4}+274860 x^{3}-3061836 x^{2}-2778120 x \right )}{98513415 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(142\)
risch	\(-\frac {2 \left (10945935 x^{6}[phone] x^{5}+1199205 x^{4}-5859000 x^{3}-292590 x^{2}+215748 x -154340\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{10945935}-\frac {\left (\frac {61736 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {248912 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{32837805 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) x}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(208\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-2 x^{6} \sqrt {3 x^{3}+5 x^{2}+2 x}-\frac {124 x^{5} \sqrt {3 x^{3}+5 x^{2}+2 x}}{39}-\frac {94 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{429}+\frac {12400 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11583}+\frac {13004 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{243243}-\frac {3688 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{93555}+\frac {61736 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2189187}-\frac {61736 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {248912 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{32837805 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(301\)

Input:

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

Output:

-2/98513415/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(295540245*x^9+962400285*x^8+10124 
60715*x^7+208993365*x^6-249969240*x^5+89652*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^ 
(1/2)*(-x)^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))+124456*(6*x+4)^(1/ 
2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2)) 
-112803354*x^4+274860*x^3-3061836*x^2-2778120*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.29 \[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {2}{10945935} \, {\left (10945935 \, x^{6} + 17401230 \, x^{5} + 1199205 \, x^{4} - 5859000 \, x^{3} - 292590 \, x^{2} + 215748 \, x - 154340\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {87632}{8444007} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {497824}{32837805} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \] Input:

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

Output:

-2/10945935*(10945935*x^6 + 17401230*x^5 + 1199205*x^4 - 5859000*x^3 - 292 
590*x^2 + 215748*x - 154340)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 87632/8444007 
*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) + 497824/32837805*sqr 
t(3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 
 5/9))
 

Sympy [F]

\[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=- \int \left (- 4 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac {11}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \] Input:

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

Output:

-Integral(-4*x**(5/2)*sqrt(3*x**2 + 5*x + 2), x) - Integral(19*x**(9/2)*sq 
rt(3*x**2 + 5*x + 2), x) - Integral(15*x**(11/2)*sqrt(3*x**2 + 5*x + 2), x 
)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx=-2 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{6}-\frac {124 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{5}}{39}-\frac {94 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{4}}{429}+\frac {12400 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{11583}+\frac {13004 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{243243}-\frac {3688 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{93555}+\frac {3688 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{155925}+\frac {124456 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{3 x^{2}+5 x +2}d x \right )}{6081075}-\frac {3688 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{3}+5 x^{2}+2 x}d x \right )}{155925} \] Input:

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

Output:

(2*( - 6081075*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**6 - 9667350*sqrt(x)*sqrt( 
3*x**2 + 5*x + 2)*x**5 - 666225*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**4 + 3255 
000*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**3 + 162550*sqrt(x)*sqrt(3*x**2 + 5*x 
 + 2)*x**2 - 119860*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x + 71916*sqrt(x)*sqrt( 
3*x**2 + 5*x + 2) + 62228*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(3*x**2 + 
 5*x + 2),x) - 71916*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2))/(3*x**3 + 5*x**2 
 + 2*x),x)))/6081075