Optimal. Leaf size=490 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{55 c^{7/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{55 c^2}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.980496, antiderivative size = 490, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{55 c^{7/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{55 c^2}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{22 c} \]
Warning: Unable to verify antiderivative.
[In] Int[(a + b*x + c*x^2)^(4/3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 43.0956, size = 575, normalized size = 1.17 \[ \frac{3 \left (a + b x + c x^{2}\right )^{\frac{4}{3}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{22 c \left (b + 2 c x\right )} - \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt [3]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{55 c^{2} \left (b + 2 c x\right )} + \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} c^{\frac{2}{3}} \left (a + b x + c x^{2}\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{- 4 a c + b^{2}} \sqrt [3]{a + b x + c x^{2}} + \left (- 4 a c + b^{2}\right )^{\frac{2}{3}}}{\left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 4 a c + b^{2}\right )^{2} \left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \sqrt [3]{- 4 a c + b^{2}}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}}{2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{55 c^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{- 4 a c + b^{2}} \left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \sqrt [3]{- 4 a c + b^{2}}\right )}{\left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}\right )^{2}}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(4/3),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.498509, size = 179, normalized size = 0.37 \[ \frac{3 \left (\sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )+2 c (b+2 c x) (a+x (b+c x)) \left (c \left (13 a+5 c x^2\right )-2 b^2+5 b c x\right )\right )}{220 c^3 (a+x (b+c x))^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(4/3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.197, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(4/3),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x + c x^{2}\right )^{\frac{4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(4/3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(4/3),x, algorithm="giac")
[Out]