\(\int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx\) [1059]

Optimal result

Integrand size = 25, antiderivative size = 154 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {136 \sqrt {x} (2+3 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {136 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}} \]

[Out]

136/27*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-136/27*(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)+10/9*(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)-10/9*x^(1/2)*(3*x^2+5*x+2)^(1
/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {846, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {10 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}}-\frac {136 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {3 x^2+5 x+2}}+\frac {136 \sqrt {x} (3 x+2)}{27 \sqrt {3 x^2+5 x+2}}-\frac {10}{9} \sqrt {x} \sqrt {3 x^2+5 x+2} \]

[In]

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

[Out]

(136*Sqrt[x]*(2 + 3*x))/(27*Sqrt[2 + 5*x + 3*x^2]) - (10*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/9 - (136*Sqrt[2]*(1 +
x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(27*Sqrt[2 + 5*x + 3*x^2]) + (10*Sqrt[2]*(1 + x)*
Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2])

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 853

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0]

Rule 1114

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]))*Elli
pticF[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 1150

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 1203

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {2}{9} \int \frac {5+34 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {4}{9} \text {Subst}\left (\int \frac {5+34 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {20}{9} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {136}{9} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {136 \sqrt {x} (2+3 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {10}{9} \sqrt {x} \sqrt {2+5 x+3 x^2}-\frac {136 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{27 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=\frac {272+620 x+258 x^2-90 x^3+136 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-106 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{27 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]

[In]

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

[Out]

(272 + 620*x + 258*x^2 - 90*x^3 + (136*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] - (106*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]
/Sqrt[x]], 3/2])/(27*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {2 \left (87 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-34 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+135 x^{3}+225 x^{2}+90 x \right )}{81 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(112\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 \sqrt {3 x^{3}+5 x^{2}+2 x}}{9}+\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {68 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(177\)
risch	\(-\frac {10 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{9}-\frac {\left (-\frac {10 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {68 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{27 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(178\)

[In]

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

[Out]

-2/81/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(87*(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))-34*(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))+135*x^3
+225*x^2+90*x)

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.28 \[ \int \frac {(2-5 x) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx=-\frac {500}{243} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {136}{27} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - \frac {10}{9} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} \]

[In]

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

[Out]

-500/243*sqrt(3)*weierstrassPInverse(28/27, 80/729, x + 5/9) - 136/27*sqrt(3)*weierstrassZeta(28/27, 80/729, w
eierstrassPInverse(28/27, 80/729, x + 5/9)) - 10/9*sqrt(3*x^2 + 5*x + 2)*sqrt(x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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