\(\int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx\) [181]

Optimal result

Integrand size = 25, antiderivative size = 201 \[ \int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx=\frac {2360 \sqrt {x} (2+3 x)}{5103 \sqrt {2+5 x+3 x^2}}-\frac {668 \sqrt {x} \sqrt {2+5 x+3 x^2}}{1701}+\frac {80}{189} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {2}{189} (29-105 x) x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {2360 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5103 \sqrt {1+x} \sqrt {2+3 x}}+\frac {668 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{1701 \sqrt {2+5 x+3 x^2}} \] Output:

2360/5103*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)-668/1701*x^(1/2)*(3*x^2+5*x+ 
2)^(1/2)+80/189*x^(3/2)*(3*x^2+5*x+2)^(1/2)+2/189*(29-105*x)*x^(5/2)*(3*x^ 
2+5*x+2)^(1/2)-2360/5103*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)+668/1701*2^(1/2)*(1+x)^( 
1/2)*(2+3*x)^(1/2)*InverseJacobiAM(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 = 21.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82 \[ \int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx=\frac {4720+7792 x+1380 x^2+7920 x^3+2970 x^4-23652 x^5-17010 x^6+2360 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-356 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{5103 \sqrt {x} \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

(4720 + 7792*x + 1380*x^2 + 7920*x^3 + 2970*x^4 - 23652*x^5 - 17010*x^6 + 
(2360*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSin 
h[Sqrt[2/3]/Sqrt[x]], 3/2] - (356*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x 
]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(5103*Sqrt[x]*Sqrt 
[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1236, 27, 1236, 27, 1231, 27, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{27} \int 3 \sqrt {x} (34 x+5) \sqrt {3 x^2+5 x+2}dx-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \int \sqrt {x} (34 x+5) \sqrt {3 x^2+5 x+2}dx-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{9} \left (\frac {2}{21} \int -\frac {(575 x+68) \sqrt {3 x^2+5 x+2}}{2 \sqrt {x}}dx+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{21} \int \frac {(575 x+68) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {2}{45} \int \frac {5 (295 x+167)}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {2}{9} \int \frac {295 x+167}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{9} \int \frac {295 x+167}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{9} \left (167 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+295 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{9} \left (295 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {167 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{9} \left (\frac {167 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+295 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2}{9} \sqrt {x} (1035 x+779) \sqrt {3 x^2+5 x+2}\right )+\frac {68}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )-\frac {10}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

(-10*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 + (2*((68*Sqrt[x]*(2 + 5*x + 3*x^ 
2)^(3/2))/21 + ((-2*Sqrt[x]*(779 + 1035*x)*Sqrt[2 + 5*x + 3*x^2])/9 + (4*( 
295*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt 
[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3* 
x^2])) + (167*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], - 
1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/9)/21))/9
 

Defintions of rubi rules used

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 = 1.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {2 \left (25515 x^{6}+35478 x^{5}+768 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-590 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-4455 x^{4}-11880 x^{3}+8550 x^{2}+6012 x \right )}{15309 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(127\)
risch	\(-\frac {2 \left (945 x^{3}-261 x^{2}-360 x +334\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{1701}-\frac {\left (-\frac {668 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1180 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) x}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(193\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {10 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}+\frac {58 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}+\frac {80 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}-\frac {668 \sqrt {3 x^{3}+5 x^{2}+2 x}}{1701}+\frac {668 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {1180 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{5103 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(238\)

Input:

int((2-5*x)*x^(3/2)*(3*x^2+5*x+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15309/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(25515*x^6+35478*x^5+768*(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))- 
590*(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))-4455*x^4-11880*x^3+8550*x^2+6012*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.29 \[ \int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2}{1701} \, {\left (945 \, x^{3} - 261 \, x^{2} - 360 \, x + 334\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {32}{6561} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {2360}{5103} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \] Input:

integrate((2-5*x)*x^(3/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")
 

Output:

-2/1701*(945*x^3 - 261*x^2 - 360*x + 334)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) + 
32/6561*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) - 2360/5103*sq 
rt(3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x 
+ 5/9))
 

Sympy [F]

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

integrate((2-5*x)*x**(3/2)*(3*x**2+5*x+2)**(1/2),x)
 

Output:

-Integral(-2*x**(3/2)*sqrt(3*x**2 + 5*x + 2), x) - Integral(5*x**(5/2)*sqr 
t(3*x**2 + 5*x + 2), x)
 

Maxima [F]

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

integrate((2-5*x)*x^(3/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((2-5*x)*x^(3/2)*(3*x^2+5*x+2)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(-x^(3/2)*(5*x - 2)*(5*x + 3*x^2 + 2)^(1/2),x)
 

Output:

-int(x^(3/2)*(5*x - 2)*(5*x + 3*x^2 + 2)^(1/2), x)
 

Reduce [F]

\[ \int (2-5 x) x^{3/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{9}+\frac {58 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{189}+\frac {80 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{189}-\frac {16 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{63}-\frac {118 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{3 x^{2}+5 x +2}d x \right )}{189}+\frac {16 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{3}+5 x^{2}+2 x}d x \right )}{63} \] Input:

int((2-5*x)*x^(3/2)*(3*x^2+5*x+2)^(1/2),x)
 

Output:

(2*( - 105*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**3 + 29*sqrt(x)*sqrt(3*x**2 + 
5*x + 2)*x**2 + 40*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x - 24*sqrt(x)*sqrt(3*x* 
*2 + 5*x + 2) - 59*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(3*x**2 + 5*x + 
2),x) + 24*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2))/(3*x**3 + 5*x**2 + 2*x),x) 
))/189