3.1409 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx\)

Optimal. Leaf size=99 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [3]{d (b+2 c x)}} \]

[Out]

(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)*Hypergeometric2F1[-4/3, -1/6, 5/6, (b +
2*c*x)^2/(b^2 - 4*a*c)])/(8*c^2*d*(d*(b + 2*c*x))^(1/3)*(1 - (b + 2*c*x)^2/(b^2
- 4*a*c))^(1/3))

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Rubi [A]  time = 0.326831, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [3]{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)*Hypergeometric2F1[-4/3, -1/6, 5/6, (b +
2*c*x)^2/(b^2 - 4*a*c)])/(8*c^2*d*(d*(b + 2*c*x))^(1/3)*(1 - (b + 2*c*x)^2/(b^2
- 4*a*c))^(1/3))

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Rubi in Sympy [A]  time = 31.3493, size = 104, normalized size = 1.05 \[ \frac{3 \left (- a c + \frac{b^{2}}{4}\right ) \sqrt [3]{a - \frac{b^{2}}{4 c} + \frac{\left (b + 2 c x\right )^{2}}{4 c}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{1}{6} \\ \frac{5}{6} \end{matrix}\middle |{- \frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}}} \right )}}{2 c^{2} d \sqrt [3]{b d + 2 c d x} \sqrt [3]{\frac{\left (b + 2 c x\right )^{2}}{4 a c - b^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-a*c + b**2/4)*(a - b**2/(4*c) + (b + 2*c*x)**2/(4*c))**(1/3)*hyper((-4/3, -1
/6), (5/6,), -(b + 2*c*x)**2/(4*a*c - b**2))/(2*c**2*d*(b*d + 2*c*d*x)**(1/3)*((
b + 2*c*x)**2/(4*a*c - b**2) + 1)**(1/3))

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Mathematica [A]  time = 0.433188, size = 149, normalized size = 1.51 \[ \frac{3 \left (5 c (a+x (b+c x)) \left (c \left (c x^2-7 a\right )+2 b^2+b c x\right )-2 \sqrt [3]{2} \left (b^2-4 a c\right ) (b+2 c x)^2 \left (-\frac{c (a+x (b+c x))}{b^2-4 a c}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )}{70 c^3 d (a+x (b+c x))^{2/3} \sqrt [3]{d (b+2 c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(5*c*(a + x*(b + c*x))*(2*b^2 + b*c*x + c*(-7*a + c*x^2)) - 2*2^(1/3)*(b^2 -
4*a*c)*(b + 2*c*x)^2*(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(2/3)*Hypergeometr
icPFQ[{2/3, 5/6}, {11/6}, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(70*c^3*d*(d*(b + 2*c*x
))^(1/3)*(a + x*(b + c*x))^(2/3))

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \[ \int{1 \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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