3.11.58 \(\int \frac {(2-5 x) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx\) [1058]

3.11.58.1 Optimal result

Integrand size = 25, antiderivative size = 177 \[ \int \frac {(2-5 x) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {412 \sqrt {x} (2+3 x)}{81 \sqrt {2+5 x+3 x^2}}+\frac {52}{27} \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {2}{3} x^{3/2} \sqrt {2+5 x+3 x^2}+\frac {412 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{81 \sqrt {2+5 x+3 x^2}}-\frac {52 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{27 \sqrt {2+5 x+3 x^2}} \]

output

-412/81*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+412/81*(1+x)^(3/2)*(1/(1+x))^( 
1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^ 
(1/2)/(3*x^2+5*x+2)^(1/2)-52/27*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1 
/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2) 
^(1/2)-2/3*x^(3/2)*(3*x^2+5*x+2)^(1/2)+52/27*x^(1/2)*(3*x^2+5*x+2)^(1/2)
 

3.11.58.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.89 \[ \int \frac {(2-5 x) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {-2 \left (412+874 x+282 x^2-99 x^3+81 x^4\right )-412 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+256 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{81 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]

input

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

output

(-2*(412 + 874*x + 282*x^2 - 99*x^3 + 81*x^4) - (412*I)*Sqrt[2]*Sqrt[1 + x 
^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] 
+ (256*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSi 
nh[Sqrt[2/3]/Sqrt[x]], 3/2])/(81*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])
 

3.11.58.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1236, 27, 1236, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{15} \int \frac {5 \sqrt {x} (13 x+3)}{\sqrt {3 x^2+5 x+2}}dx-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \int \frac {\sqrt {x} (13 x+3)}{\sqrt {3 x^2+5 x+2}}dx-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2}{3} \left (\frac {2}{9} \int -\frac {103 x+26}{2 \sqrt {x} \sqrt {3 x^2+5 x+2}}dx+\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}-\frac {1}{9} \int \frac {103 x+26}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1240

\(\displaystyle \frac {2}{3} \left (\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}-\frac {2}{9} \int \frac {103 x+26}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1503

\(\displaystyle \frac {2}{3} \left (\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}-\frac {2}{9} \left (26 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+103 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1413

\(\displaystyle \frac {2}{3} \left (\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}-\frac {2}{9} \left (103 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {13 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

\(\Big \downarrow \) 1456

\(\displaystyle \frac {2}{3} \left (\frac {26}{9} \sqrt {x} \sqrt {3 x^2+5 x+2}-\frac {2}{9} \left (\frac {13 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}+103 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )\right )-\frac {2}{3} x^{3/2} \sqrt {3 x^2+5 x+2}\)

input

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

output

(-2*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/3 + (2*((26*Sqrt[x]*Sqrt[2 + 5*x + 3*x^ 
2])/9 - (2*(103*((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])) + (13*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF 
[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]))/9))/3
 

3.11.58.3.1 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 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]
 

3.11.58.4 Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\frac {154 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{81}-\frac {206 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{243}-2 x^{4}+\frac {22 x^{3}}{9}+\frac {224 x^{2}}{27}+\frac {104 x}{27}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(117\)
risch	\(-\frac {2 \left (-26+9 x \right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{27}-\frac {\left (\frac {52 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{81 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {206 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{81 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(183\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {2 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{3}+\frac {52 \sqrt {3 x^{3}+5 x^{2}+2 x}}{27}-\frac {52 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{81 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {206 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{81 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(196\)

input

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

output

2/243/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(231*(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))-103*(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))-243*x^4 
+297*x^3+1008*x^2+468*x)
 

3.11.58.5 Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.27 \[ \int \frac {(2-5 x) x^{3/2}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {2}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (9 \, x - 26\right )} \sqrt {x} + \frac {1124}{729} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {412}{81} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]

input

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

output

-2/27*sqrt(3*x^2 + 5*x + 2)*(9*x - 26)*sqrt(x) + 1124/729*sqrt(3)*weierstr 
assPInverse(28/27, 80/729, x + 5/9) + 412/81*sqrt(3)*weierstrassZeta(28/27 
, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))
 

3.11.58.6 Sympy [F]

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

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)
 

3.11.58.7 Maxima [F]

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

input

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

output

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

3.11.58.8 Giac [F]

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

input

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

output

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

3.11.58.9 Mupad [F(-1)]

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

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)