3.3024 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx\)

Optimal. Leaf size=301 \[ -\frac{3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac{4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}}+\frac{4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (d e-c f)^{7/3}} \]

[Out]

(-3*(a + b*x)^(4/3))/((d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)) + (4*(b*e - a*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)^2*(e + f*x)) + (4*(b*c - a*d)*(b*e -
a*f)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d
*e - c*f)^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*(d*e - c*f)^(7/3)) - (2*(b*c - a*d)*
(b*e - a*f)^(1/3)*Log[e + f*x])/(3*(d*e - c*f)^(7/3)) + (2*(b*c - a*d)*(b*e - a*
f)^(1/3)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^
(1/3)])/(d*e - c*f)^(7/3)

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Rubi [A]  time = 0.547817, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}+\frac{4 \sqrt [3]{a+b x} (c+d x)^{2/3} (b e-a f)}{(e+f x) (d e-c f)^2}-\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac{2 (b c-a d) \sqrt [3]{b e-a f} \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{(d e-c f)^{7/3}}+\frac{4 (b c-a d) \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (d e-c f)^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(a + b*x)^(4/3))/((d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)) + (4*(b*e - a*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)^2*(e + f*x)) + (4*(b*c - a*d)*(b*e -
a*f)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d
*e - c*f)^(1/3)*(a + b*x)^(1/3))])/(Sqrt[3]*(d*e - c*f)^(7/3)) - (2*(b*c - a*d)*
(b*e - a*f)^(1/3)*Log[e + f*x])/(3*(d*e - c*f)^(7/3)) + (2*(b*c - a*d)*(b*e - a*
f)^(1/3)*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^
(1/3)])/(d*e - c*f)^(7/3)

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Rubi in Sympy [A]  time = 44.57, size = 257, normalized size = 0.85 \[ - \frac{\left (a + b x\right )^{\frac{4}{3}}}{\sqrt [3]{c + d x} \left (e + f x\right ) \left (c f - d e\right )} - \frac{4 \sqrt [3]{a + b x} \left (a d - b c\right )}{\sqrt [3]{c + d x} \left (c f - d e\right )^{2}} + \frac{2 \left (a d - b c\right ) \sqrt [3]{a f - b e} \log{\left (e + f x \right )}}{3 \left (c f - d e\right )^{\frac{7}{3}}} - \frac{2 \left (a d - b c\right ) \sqrt [3]{a f - b e} \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{\left (c f - d e\right )^{\frac{7}{3}}} - \frac{4 \sqrt{3} \left (a d - b c\right ) \sqrt [3]{a f - b e} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{3 \left (c f - d e\right )^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(4/3)/((c + d*x)**(1/3)*(e + f*x)*(c*f - d*e)) - 4*(a + b*x)**(1/3)*
(a*d - b*c)/((c + d*x)**(1/3)*(c*f - d*e)**2) + 2*(a*d - b*c)*(a*f - b*e)**(1/3)
*log(e + f*x)/(3*(c*f - d*e)**(7/3)) - 2*(a*d - b*c)*(a*f - b*e)**(1/3)*log(-(a
+ b*x)**(1/3) + (c + d*x)**(1/3)*(a*f - b*e)**(1/3)/(c*f - d*e)**(1/3))/(c*f - d
*e)**(7/3) - 4*sqrt(3)*(a*d - b*c)*(a*f - b*e)**(1/3)*atan(sqrt(3)/3 + 2*sqrt(3)
*(c + d*x)**(1/3)*(a*f - b*e)**(1/3)/(3*(a + b*x)**(1/3)*(c*f - d*e)**(1/3)))/(3
*(c*f - d*e)**(7/3))

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Mathematica [C]  time = 1.10472, size = 160, normalized size = 0.53 \[ \frac{\sqrt [3]{a+b x} \left (-4 (e+f x) (b c-a d) \sqrt [3]{\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )-a (c f+3 d e+4 d f x)+b (4 c e+3 c f x+d e x)\right )}{\sqrt [3]{c+d x} (e+f x) (d e-c f)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(b*(4*c*e + d*e*x + 3*c*f*x) - a*(3*d*e + c*f + 4*d*f*x) - 4*(b
*c - a*d)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(1/3)*(e + f*x)*Hype
rgeometric2F1[1/3, 1/3, 4/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))]
))/((d*e - c*f)^2*(c + d*x)^(1/3)*(e + f*x))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{2}} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.268119, size = 864, normalized size = 2.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(3)*(2*sqrt(3)*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*
d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*log(((d*x + c)*(
-(b*e - a*f)/(d*e - c*f))^(2/3) - (b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(b*e - a*f)/
(d*e - c*f))^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3))/(d*x + c)) - 4*sqrt(3)*((b
*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*d^2)*e + (b*c^2 - a*c*d)*f
)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*log(((d*x + c)*(-(b*e - a*f)/(d*e - c*f))^
(1/3) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/(d*x + c)) - 3*sqrt(3)*(a*c*f - (4*b*c
- 3*a*d)*e - (b*d*e + (3*b*c - 4*a*d)*f)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - 12
*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*d^2)*e + (b*c^2 - a*c*
d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*arctan(-1/3*(sqrt(3)*(d*x + c)*(-(b*e
- a*f)/(d*e - c*f))^(1/3) - 2*sqrt(3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/((d*x + c
)*(-(b*e - a*f)/(d*e - c*f))^(1/3))))/(c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2 + (
d^3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*x^2 + (d^3*e^3 - c*d^2*e^2*f - c^2*d*e*f^
2 + c^3*f^3)*x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError