\(\int \frac {1}{(a+b x+c x^2)^{2/3}} \, dx\) [688]

Optimal result

Integrand size = 14, antiderivative size = 85 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\frac {\sqrt [3]{2} (b+2 c x) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c \left (a+b x+c x^2\right )^{2/3}} \] Output:

2^(1/3)*(2*c*x+b)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(2/3)*hypergeom([1/2, 2/ 
3],[3/2],(2*c*x+b)^2/(-4*a*c+b^2))/c/(c*x^2+b*x+a)^(2/3)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\frac {\sqrt [3]{2} (b+2 c x) \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c (a+x (b+c x))^{2/3}} \] Input:

Integrate[(a + b*x + c*x^2)^(-2/3),x]
 

Output:

(2^(1/3)*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(2/3)*Hypergeo 
metric2F1[1/2, 2/3, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c*(a + x*(b + c*x) 
)^(2/3))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(451\) vs. \(2(85)=170\).

Time = 0.35 (sec) , antiderivative size = 451, normalized size of antiderivative = 5.31, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1095, 759}

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.

\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx\)

\(\Big \downarrow \) 1095

\(\displaystyle \frac {3 \sqrt {(b+2 c x)^2} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{b+2 c x}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\sqrt [3]{2} 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {(b+2 c x)^2} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{c} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\)

Input:

Int[(a + b*x + c*x^2)^(-2/3),x]
 

Output:

(2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*Sqrt[(b + 2*c*x)^2]*((b^2 - 4*a*c)^(1/3 
) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2 
^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^( 
2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3) 
*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b^2 
- 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*( 
b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqr 
t[3]])/(c^(1/3)*(b + 2*c*x)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) 
 + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^ 
(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*Sqrt[b^2 - 4*a*c + 4*c 
*(a + b*x + c*x^2)])
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 1095

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[3*p]
 

Maple [F]

\[\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {2}{3}}}d x\]

Input:

int(1/(c*x^2+b*x+a)^(2/3),x)
 

Output:

int(1/(c*x^2+b*x+a)^(2/3),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(c*x^2+b*x+a)^(2/3),x, algorithm="fricas")
 

Output:

integral((c*x^2 + b*x + a)^(-2/3), x)
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (a + b x + c x^{2}\right )^{\frac {2}{3}}}\, dx \] Input:

integrate(1/(c*x**2+b*x+a)**(2/3),x)
 

Output:

Integral((a + b*x + c*x**2)**(-2/3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(c*x^2+b*x+a)^(2/3),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(-2/3), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(1/(c*x^2+b*x+a)^(2/3),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(-2/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^{2/3}} \,d x \] Input:

int(1/(a + b*x + c*x^2)^(2/3),x)
 

Output:

int(1/(a + b*x + c*x^2)^(2/3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {2}{3}}}d x \] Input:

int(1/(c*x^2+b*x+a)^(2/3),x)
 

Output:

int(1/(a + b*x + c*x**2)**(2/3),x)