\(\int (a+b x+c x^2)^{2/3} \, dx\) [146]

Optimal result

Integrand size = 14, antiderivative size = 88 \[ \int \left (a+b x+c x^2\right )^{2/3} \, dx=\frac {(b+2 c x) \left (a+b x+c x^2\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 \sqrt [3]{2} c \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{2/3}} \] Output:

1/8*(2*c*x+b)*(c*x^2+b*x+a)^(2/3)*hypergeom([-2/3, 1/2],[3/2],(2*c*x+b)^2/ 
(-4*a*c+b^2))*2^(2/3)/c/(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(2/3)
 

Mathematica [A] (verified)

Time = 10.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99 \[ \int \left (a+b x+c x^2\right )^{2/3} \, dx=\frac {(b+2 c x) (a+x (b+c x))^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 \sqrt [3]{2} c \left (\frac {c (a+x (b+c x))}{-b^2+4 a c}\right )^{2/3}} \] Input:

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

Output:

((b + 2*c*x)*(a + x*(b + c*x))^(2/3)*Hypergeometric2F1[-2/3, 1/2, 3/2, (b 
+ 2*c*x)^2/(b^2 - 4*a*c)])/(4*2^(1/3)*c*((c*(a + x*(b + c*x)))/(-b^2 + 4*a 
*c))^(2/3))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1059\) vs. \(2(88)=176\).

Time = 1.39 (sec) , antiderivative size = 1059, normalized size of antiderivative = 12.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1087, 1095, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{2/3}}{14 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt [3]{c x^2+b x+a}}dx}{7 c}\)

\(\Big \downarrow \) 1095

\(\displaystyle \frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{2/3}}{14 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {\sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{7 c (b+2 c x)}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{2/3}}{14 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}\right )}{7 c (b+2 c x)}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {3 (b+2 c x) \left (a+b x+c x^2\right )^{2/3}}{14 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{7 c (b+2 c x)}\)

\(\Big \downarrow \) 2416

\(\displaystyle \frac {3 (b+2 c x) \left (c x^2+b x+a\right )^{2/3}}{14 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\frac {\sqrt [3]{2} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\sqrt [3]{c} \left (\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 )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^2}} E\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 )}{2^{2/3} \sqrt [3]{c} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\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 )^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]{2} \sqrt [4]{3} c^{2/3} \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]{c x^2+b x+a}\right )}{\left (\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 )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}\right )}{7 c (b+2 c x)}\)

Input:

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

Output:

(3*(b + 2*c*x)*(a + b*x + c*x^2)^(2/3))/(14*c) - (3*(b^2 - 4*a*c)*Sqrt[(b 
+ 2*c*x)^2]*(((2^(1/3)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])/(c^(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))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(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))*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]*EllipticE[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]])/(2^(2/3)*c^(1/3)*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)]))/(2^(2/3)*c^(1/3)) - ((1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(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))*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 + Sq 
rt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*E 
llipticF[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...
 

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 832

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

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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]
 

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b x+c x^2\right )^{2/3} \, dx=\frac {6 \left (c \,x^{2}+b x +a \right )^{\frac {2}{3}} a +3 \left (c \,x^{2}+b x +a \right )^{\frac {2}{3}} b x -8 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}}}d x \right ) a c +2 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}}}d x \right ) b^{2}}{7 b} \] Input:

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

Output:

(6*(a + b*x + c*x**2)**(2/3)*a + 3*(a + b*x + c*x**2)**(2/3)*b*x - 8*int(( 
(a + b*x + c*x**2)**(2/3)*x)/(a + b*x + c*x**2),x)*a*c + 2*int(((a + b*x + 
 c*x**2)**(2/3)*x)/(a + b*x + c*x**2),x)*b**2)/(7*b)