\(\int \frac {(2-5 x) (2+5 x+3 x^2)^{3/2}}{x^{9/2}} \, dx\) [196]

Optimal result

Integrand size = 25, antiderivative size = 183 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx=-\frac {633 \sqrt {x} (2+3 x)}{7 \sqrt {2+5 x+3 x^2}}+\frac {3 (22+133 x) \sqrt {2+5 x+3 x^2}}{7 x^{3/2}}-\frac {4 (1-2 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 x^{7/2}}+\frac {633 \sqrt {2} \sqrt {2+5 x+3 x^2} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{7 \sqrt {1+x} \sqrt {2+3 x}}-\frac {783 \sqrt {2} \sqrt {1+x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{7 \sqrt {2+5 x+3 x^2}} \] Output:

-633/7*x^(1/2)*(2+3*x)/(3*x^2+5*x+2)^(1/2)+3/7*(22+133*x)*(3*x^2+5*x+2)^(1 
/2)/x^(3/2)-4/7*(1-2*x)*(3*x^2+5*x+2)^(3/2)/x^(7/2)+633/7*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)-783/7*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 = 21.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx=\frac {-2 \left (8+24 x-72 x^2-19 x^3+384 x^4+315 x^5\right )-633 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-150 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{9/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{7 x^{7/2} \sqrt {2+5 x+3 x^2}} \] Input:

Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(9/2),x]
 

Output:

(-2*(8 + 24*x - 72*x^2 - 19*x^3 + 384*x^4 + 315*x^5) - (633*I)*Sqrt[2]*Sqr 
t[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(9/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]] 
, 3/2] - (150*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(9/2)*EllipticF[ 
I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(7*x^(7/2)*Sqrt[2 + 5*x + 3*x^2])
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1229, 27, 1229, 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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {3}{35} \int \frac {15 (13 x+11) \sqrt {3 x^2+5 x+2}}{x^{5/2}}dx-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {9}{7} \int \frac {(13 x+11) \sqrt {3 x^2+5 x+2}}{x^{5/2}}dx-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {9}{7} \left (-\frac {1}{3} \int -\frac {3 (211 x+174)}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {\sqrt {3 x^2+5 x+2} (133 x+22)}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {9}{7} \left (\frac {1}{2} \int \frac {211 x+174}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {(133 x+22) \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 1240

\(\displaystyle -\frac {9}{7} \left (\int \frac {211 x+174}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {(133 x+22) \sqrt {3 x^2+5 x+2}}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 1503

\(\displaystyle -\frac {9}{7} \left (174 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+211 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {\sqrt {3 x^2+5 x+2} (133 x+22)}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 1413

\(\displaystyle -\frac {9}{7} \left (211 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {87 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {\sqrt {3 x^2+5 x+2} (133 x+22)}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

\(\Big \downarrow \) 1456

\(\displaystyle -\frac {9}{7} \left (\frac {87 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+211 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {\sqrt {3 x^2+5 x+2} (133 x+22)}{3 x^{3/2}}\right )-\frac {4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}\)

Input:

Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(9/2),x]
 

Output:

(-4*(1 - 2*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*x^(7/2)) - (9*(-1/3*((22 + 133*x 
)*Sqrt[2 + 5*x + 3*x^2])/x^(3/2) + 211*((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])) + (87*Sqrt[2]*(1 + x)*Sqrt[(2 + 
3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/7
 

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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(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] - Simp[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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
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 = 1.00 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {111 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{3}-211 \operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, x^{3}+2538 x^{5}+4794 x^{4}+2608 x^{3}+288 x^{2}-96 x -32}{14 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {7}{2}}}\)	\(129\)
elliptic	\(\frac {\sqrt {\left (3 x^{2}+5 x +2\right ) x}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{4}}-\frac {4 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{3}}+\frac {94 \sqrt {3 x^{3}+5 x^{2}+2 x}}{7 x^{2}}+\frac {\frac {1269}{7} x^{2}+\frac {2115}{7} x +\frac {846}{7}}{\sqrt {\left (3 x^{2}+5 x +2\right ) x}}-\frac {261 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{7 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {633 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\operatorname {EllipticF}\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{14 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(248\)

Input:

int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2),x,method=_RETURNVERBOSE)
 

Output:

1/14*(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))*x^3-211*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*(6*x+ 
4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*x^3+2538*x^5+4794*x^4+2608*x^3+2 
88*x^2-96*x-32)/(3*x^2+5*x+2)^(1/2)/x^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.38 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx=-\frac {511 \, \sqrt {3} x^{4} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 1899 \, \sqrt {3} x^{4} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 3 \, {\left (423 \, x^{3} + 94 \, x^{2} - 4 \, x - 8\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{21 \, x^{4}} \] Input:

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2),x, algorithm="fricas")
 

Output:

-1/21*(511*sqrt(3)*x^4*weierstrassPInverse(28/27, 80/729, x + 5/9) - 1899* 
sqrt(3)*x^4*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/7 
29, x + 5/9)) - 3*(423*x^3 + 94*x^2 - 4*x - 8)*sqrt(3*x^2 + 5*x + 2)*sqrt( 
x))/x^4
 

Sympy [F]

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

integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)/x**(9/2),x)
 

Output:

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

Maxima [F]

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

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2),x, algorithm="maxima")
 

Output:

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(9/2), x)
 

Giac [F]

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

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2),x, algorithm="giac")
 

Output:

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{9/2}} \,d x \] Input:

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(9/2),x)
 

Output:

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(9/2), x)
 

Reduce [F]

\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx=\frac {-630 \sqrt {3 x^{2}+5 x +2}\, x^{3}+1848 \sqrt {3 x^{2}+5 x +2}\, x^{2}-1330 \sqrt {3 x^{2}+5 x +2}\, x -24 \sqrt {3 x^{2}+5 x +2}-6590 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{5}+5 \sqrt {x}\, x^{4}+2 \sqrt {x}\, x^{3}}d x \right ) x^{3}-3295 \sqrt {x}\, \left (\int \frac {\sqrt {3 x^{2}+5 x +2}}{3 \sqrt {x}\, x^{4}+5 \sqrt {x}\, x^{3}+2 \sqrt {x}\, x^{2}}d x \right ) x^{3}-5187 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{3 x^{5}+5 x^{4}+2 x^{3}}d x \right ) x^{3}}{21 \sqrt {x}\, x^{3}} \] Input:

int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(9/2),x)
 

Output:

( - 630*sqrt(3*x**2 + 5*x + 2)*x**3 + 1848*sqrt(3*x**2 + 5*x + 2)*x**2 - 1 
330*sqrt(3*x**2 + 5*x + 2)*x - 24*sqrt(3*x**2 + 5*x + 2) - 6590*sqrt(x)*in 
t(sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)*x**5 + 5*sqrt(x)*x**4 + 2*sqrt(x)*x**3 
),x)*x**3 - 3295*sqrt(x)*int(sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)*x**4 + 5*sq 
rt(x)*x**3 + 2*sqrt(x)*x**2),x)*x**3 - 5187*sqrt(x)*int((sqrt(x)*sqrt(3*x* 
*2 + 5*x + 2))/(3*x**5 + 5*x**4 + 2*x**3),x)*x**3)/(21*sqrt(x)*x**3)