3.2476 \(\int \frac{\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=189 \[ \frac{12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} F_1\left (-\frac{5}{3};-\frac{4}{3},-\frac{4}{3};-\frac{2}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e (d+e x) \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}} \]

[Out]

(12*2^(2/3)*(a + b*x + c*x^2)^(4/3)*AppellF1[-5/3, -4/3, -4/3, -2/3, (2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c)/
(2*(d + e*x))])/(5*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3)*(
(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3)*(d + e*x))

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Rubi [A]  time = 0.645289, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} F_1\left (-\frac{5}{3};-\frac{4}{3},-\frac{4}{3};-\frac{2}{3};\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e (d+e x) \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(12*2^(2/3)*(a + b*x + c*x^2)^(4/3)*AppellF1[-5/3, -4/3, -4/3, -2/3, (2*c*d - (b
- Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c)/
(2*(d + e*x))])/(5*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3)*(
(e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3)*(d + e*x))

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Rubi in Sympy [A]  time = 30.4244, size = 156, normalized size = 0.83 \[ \frac{12 \cdot 2^{\frac{2}{3}} \left (a + b x + c x^{2}\right )^{\frac{4}{3}} \operatorname{appellf_{1}}{\left (- \frac{5}{3},- \frac{4}{3},- \frac{4}{3},- \frac{2}{3},\frac{c d - \frac{e \left (b - \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )},\frac{c d - \frac{e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )} \right )}}{5 e \left (\frac{e \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{4}{3}} \left (\frac{e \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{c \left (d + e x\right )}\right )^{\frac{4}{3}} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*2**(2/3)*(a + b*x + c*x**2)**(4/3)*appellf1(-5/3, -4/3, -4/3, -2/3, (c*d - e*
(b - sqrt(-4*a*c + b**2))/2)/(c*(d + e*x)), (c*d - e*(b + sqrt(-4*a*c + b**2))/2
)/(c*(d + e*x)))/(5*e*(e*(b + 2*c*x - sqrt(-4*a*c + b**2))/(c*(d + e*x)))**(4/3)
*(e*(b + 2*c*x + sqrt(-4*a*c + b**2))/(c*(d + e*x)))**(4/3)*(d + e*x))

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Mathematica [A]  time = 1.78399, size = 0, normalized size = 0. \[ \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.222, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( ex+d \right ) ^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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