3.2473 \(\int (d+e x) \left (a+b x+c x^2\right )^{4/3} \, dx\)

Optimal. Leaf size=539 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} (2 c d-b e)}{110 c^3}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} (2 c d-b e)}{44 c^2}+\frac{3 e \left (a+b x+c x^2\right )^{7/3}}{14 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(110*c^3) +
(3*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(44*c^2) + (3*e*(a + b*x
+ c*x^2)^(7/3))/(14*c) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(2*c*d - b*e
)*((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*2^(2
/3)*c^(10/3)*(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 = 1.20807, antiderivative size = 539, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{55\ 2^{2/3} c^{10/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} (2 c d-b e)}{110 c^3}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} (2 c d-b e)}{44 c^2}+\frac{3 e \left (a+b x+c x^2\right )^{7/3}}{14 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[(d + e*x)*(a + b*x + c*x^2)^(4/3),x]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(110*c^3) +
(3*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(44*c^2) + (3*e*(a + b*x
+ c*x^2)^(7/3))/(14*c) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(2*c*d - b*e
)*((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*2^(2
/3)*c^(10/3)*(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 = 60.6781, size = 622, normalized size = 1.15 \[ \frac{3 e \left (a + b x + c x^{2}\right )^{\frac{7}{3}}}{14 c} - \frac{3 \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{4}{3}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{44 c^{2} \left (b + 2 c x\right )} + \frac{3 \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt [3]{a + b x + c x^{2}} \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )} \sqrt{\left (b + 2 c x\right )^{2}}}{110 c^{3} \left (b + 2 c x\right )} - \frac{\sqrt [3]{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} c^{\frac{2}{3}} \left (a + b x + c x^{2}\right )^{\frac{2}{3}} - 2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{- 4 a c + b^{2}} \sqrt [3]{a + b x + c x^{2}} + \left (- 4 a c + b^{2}\right )^{\frac{2}{3}}}{\left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \sqrt [3]{- 4 a c + b^{2}}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} - \left (-1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}}{2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{110 c^{\frac{10}{3}} \sqrt{\frac{\sqrt [3]{- 4 a c + b^{2}} \left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \sqrt [3]{- 4 a c + b^{2}}\right )}{\left (2^{\frac{2}{3}} \sqrt [3]{c} \sqrt [3]{a + b x + c x^{2}} + \left (1 + \sqrt{3}\right ) \sqrt [3]{- 4 a c + b^{2}}\right )^{2}}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*x**2+b*x+a)**(4/3),x)

[Out]

3*e*(a + b*x + c*x**2)**(7/3)/(14*c) - 3*(b*e - 2*c*d)*(a + b*x + c*x**2)**(4/3)
*sqrt(-4*a*c + b**2 + c*(4*a + 4*b*x + 4*c*x**2))*sqrt((b + 2*c*x)**2)/(44*c**2*
(b + 2*c*x)) + 3*(-4*a*c + b**2)*(b*e - 2*c*d)*(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)/(110*c**3*(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*(b*e - 2*c*d)*(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 + s
qrt(3))*(-4*a*c + b**2)**(1/3))), -7 - 4*sqrt(3))/(110*c**(10/3)*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 = 1.02254, size = 273, normalized size = 0.51 \[ -\frac{3 \left (2 c (a+x (b+c x)) \left (-2 c^2 \left (55 a^2 e+2 a c x (91 d+55 e x)+5 c^2 x^3 (14 d+11 e x)\right )+b^2 c (91 a e-c x (14 d+5 e x))-2 b c^2 \left (a (91 d+19 e x)+15 c x^2 (7 d+5 e x)\right )-14 b^4 e+7 b^3 c (4 d+e x)\right )+7 \sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{2/3} (b e-2 c d) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )\right )}{3080 c^4 (a+x (b+c x))^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(a + b*x + c*x^2)^(4/3),x]

[Out]

(-3*(2*c*(a + x*(b + c*x))*(-14*b^4*e + 7*b^3*c*(4*d + e*x) + b^2*c*(91*a*e - c*
x*(14*d + 5*e*x)) - 2*b*c^2*(15*c*x^2*(7*d + 5*e*x) + a*(91*d + 19*e*x)) - 2*c^2
*(55*a^2*e + 5*c^2*x^3*(14*d + 11*e*x) + 2*a*c*x*(91*d + 55*e*x))) + 7*2^(1/3)*(
b^2 - 4*a*c)^2*(-2*c*d + b*e)*(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 + S
qrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]))/(3080*c^4*(a + x*(b + c*x))^(
2/3))

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+b*x+a)^(4/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c e x^{3} +{\left (c d + b e\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d),x, algorithm="fricas")

[Out]

integral((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*(c*x^2 + b*x + a)^(1/
3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(c*x**2+b*x+a)**(4/3),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}{\left (e x + d\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)*(e*x + d), x)