\(\int \frac {(2-5 x) (2+5 x+3 x^2)^{3/2}}{\sqrt {x}} \, dx\) [192]

Optimal result

Integrand size = 25, antiderivative size = 183 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx=\frac {860 \sqrt {x} (2+3 x)}{243 \sqrt {2+5 x+3 x^2}}+\frac {4}{81} \sqrt {x} (82+45 x) \sqrt {2+5 x+3 x^2}-\frac {2}{9} \sqrt {x} (1+5 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {860 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{243 \sqrt {1+x} \sqrt {2+3 x}}+\frac {356 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{81 \sqrt {2+5 x+3 x^2}} \] Output:

860/243*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+4/81*x^(1/2)*(82+45*x)*(3*x^2+ 
5*x+2)^(1/2)-2/9*x^(1/2)*(1+5*x)*(3*x^2+5*x+2)^(3/2)-860/243*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)+356/81*2^(1/2)*(1+x)^(1/2)*(2+3*x)^(1/2)*InverseJacobiAM(arcta 
n(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 = 21.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx=\frac {1720+6052 x+6420 x^2-1746 x^3-9990 x^4-8586 x^5-2430 x^6+860 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+208 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{243 \sqrt {x} \sqrt {2+5 x+3 x^2}} \] Input:

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

Output:

(1720 + 6052*x + 6420*x^2 - 1746*x^3 - 9990*x^4 - 8586*x^5 - 2430*x^6 + (8 
60*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[S 
qrt[2/3]/Sqrt[x]], 3/2] + (208*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x 
^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(243*Sqrt[x]*Sqrt[2 + 
 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1231, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {2}{63} \int -\frac {7 (25 x+19) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \int \frac {(25 x+19) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2}{9} \left (\frac {2}{9} \sqrt {x} (45 x+82) \sqrt {3 x^2+5 x+2}-\frac {2}{45} \int -\frac {5 (215 x+178)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} \left (\frac {1}{9} \int \frac {215 x+178}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {2}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} (45 x+82)\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2}{9} \left (\frac {2}{9} \int \frac {215 x+178}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {2}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} (45 x+82)\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2}{9} \left (\frac {2}{9} \left (178 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+215 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )+\frac {2}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} (45 x+82)\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2}{9} \left (\frac {2}{9} \left (215 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {89 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )+\frac {2}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} (45 x+82)\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2}{9} \left (\frac {2}{9} \left (\frac {89 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+215 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )+\frac {2}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} (45 x+82)\right )-\frac {2}{9} \sqrt {x} (5 x+1) \left (3 x^2+5 x+2\right )^{3/2}\)

Input:

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

Output:

(-2*Sqrt[x]*(1 + 5*x)*(2 + 5*x + 3*x^2)^(3/2))/9 + (2*((2*Sqrt[x]*(82 + 45 
*x)*Sqrt[2 + 5*x + 3*x^2])/9 + (2*(215*((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[Sq 
rt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (89*Sqrt[2]*(1 + x)*Sqrt[(2 + 
3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/9) 
)/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 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 = 1.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {2 \left (3645 x^{6}+12879 x^{5}+111 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-215 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+14985 x^{4}+2619 x^{3}-5760 x^{2}-2628 x \right )}{729 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}}\)	\(127\)
risch	\(-\frac {2 \left (135 x^{3}+252 x^{2}+45 x -146\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{81}-\frac {\left (-\frac {356 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {430 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {\left (3 x^{2}+5 x +2\right ) x}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(193\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {10 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{3}-\frac {56 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}-\frac {10 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}+\frac {292 \sqrt {3 x^{3}+5 x^{2}+2 x}}{81}+\frac {356 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {430 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{243 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(238\)

Input:

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

Output:

-2/729/(3*x^2+5*x+2)^(1/2)/x^(1/2)*(3645*x^6+12879*x^5+111*(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))-215 
*(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))+14985*x^4+2619*x^3-5760*x^2-2628*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.32 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx=-\frac {2}{81} \, {\left (135 \, x^{3} + 252 \, x^{2} + 45 \, x - 146\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} + \frac {2108}{2187} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {860}{243} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \] Input:

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

Output:

-2/81*(135*x^3 + 252*x^2 + 45*x - 146)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) + 210 
8/2187*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) - 860/243*sqrt( 
3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5 
/9))
 

Sympy [F]

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

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

Output:

-Integral(-4*sqrt(3*x**2 + 5*x + 2)/sqrt(x), x) - Integral(19*x**(3/2)*sqr 
t(3*x**2 + 5*x + 2), x) - Integral(15*x**(5/2)*sqrt(3*x**2 + 5*x + 2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {x}} \, dx=-\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{3}}{3}-\frac {56 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x^{2}}{9}-\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{9}+\frac {14 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3}-\frac {43 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}\, x}{3 x^{2}+5 x +2}d x \right )}{9}+\frac {10 \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{3}+5 x^{2}+2 x}d x \right )}{3} \] Input:

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

Output:

( - 30*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x**3 - 56*sqrt(x)*sqrt(3*x**2 + 5*x 
+ 2)*x**2 - 10*sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x + 42*sqrt(x)*sqrt(3*x**2 + 
 5*x + 2) - 43*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2)*x)/(3*x**2 + 5*x + 2),x 
) + 30*int((sqrt(x)*sqrt(3*x**2 + 5*x + 2))/(3*x**3 + 5*x**2 + 2*x),x))/9