3.2474 \(\int \left (a+b x+c x^2\right )^{4/3} \, dx\)

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]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(55*c^2) + (3*(b + 2*c*x)
*(a + b*x + c*x^2)^(4/3))/(22*c) + (2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a
*c)^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*Sqrt[3]])/(55*
c^(7/3)*(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])

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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]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(55*c^2) + (3*(b + 2*c*x)
*(a + b*x + c*x^2)^(4/3))/(22*c) + (2^(1/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a
*c)^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*Sqrt[3]])/(55*
c^(7/3)*(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])

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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]

3*(a + b*x + c*x**2)**(4/3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqr
t((b + 2*c*x)**2)/(22*c*(b + 2*c*x)) - 3*(-4*a*c + b**2)*(a + b*x + c*x**2)**(1/
3)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/(55*c**
2*(b + 2*c*x)) + 2**(1/3)*3**(3/4)*sqrt((2*2**(1/3)*c**(2/3)*(a + b*x + c*x**2)*
*(2/3) - 2**(2/3)*c**(1/3)*(-4*a*c + b**2)**(1/3)*(a + b*x + c*x**2)**(1/3) + (-
4*a*c + b**2)**(2/3))/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3
))*(-4*a*c + b**2)**(1/3))**2)*sqrt(sqrt(3) + 2)*(-4*a*c + b**2)**2*(2**(2/3)*c*
*(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3))*sqrt((b + 2*c*x)**2)*
elliptic_f(asin((2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) - (-1 + sqrt(3))*(-
4*a*c + b**2)**(1/3))/(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3
))*(-4*a*c + b**2)**(1/3))), -7 - 4*sqrt(3))/(55*c**(7/3)*sqrt((-4*a*c + b**2)**
(1/3)*(2**(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (-4*a*c + b**2)**(1/3))/(2*
*(2/3)*c**(1/3)*(a + b*x + c*x**2)**(1/3) + (1 + sqrt(3))*(-4*a*c + b**2)**(1/3)
)**2)*(b + 2*c*x)*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2)))

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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]

(3*(2*c*(b + 2*c*x)*(a + x*(b + c*x))*(-2*b^2 + 5*b*c*x + c*(13*a + 5*c*x^2)) +
2^(1/3)*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[b^2 - 4*a*c]
+ 2*c*x)/Sqrt[b^2 - 4*a*c])^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, (-b + Sqrt[b^
2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(220*c^3*(a + x*(b + c*x))^(2/3))

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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]

int((c*x^2+b*x+a)^(4/3),x)

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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]

integrate((c*x^2 + b*x + a)^(4/3), x)

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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]

integral((c*x^2 + b*x + a)^(4/3), x)

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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]

Integral((a + b*x + c*x**2)**(4/3), x)

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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]

integrate((c*x^2 + b*x + a)^(4/3), x)