3.3003 \(\int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2} \, dx\)

Optimal. Leaf size=293 \[ -\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{(e+f x) (b e-a f) (d e-c f)}+\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{4/3} (d e-c f)^{5/3}} \]

[Out]

-((f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x))) - ((3
*b*d*e - b*c*f - 2*a*d*f)*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3
))/(Sqrt[3]*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))])/(Sqrt[3]*(b*e - a*f)^(4/3)*(d*e
- c*f)^(5/3)) + ((3*b*d*e - b*c*f - 2*a*d*f)*Log[e + f*x])/(6*(b*e - a*f)^(4/3)
*(d*e - c*f)^(5/3)) - ((3*b*d*e - b*c*f - 2*a*d*f)*Log[((d*e - c*f)^(1/3)*(a + b
*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(4/3)*(d*e - c*f
)^(5/3))

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Rubi [A]  time = 0.559468, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{(e+f x) (b e-a f) (d e-c f)}+\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{4/3} (d e-c f)^{5/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^2),x]

[Out]

-((f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x))) - ((3
*b*d*e - b*c*f - 2*a*d*f)*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3
))/(Sqrt[3]*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))])/(Sqrt[3]*(b*e - a*f)^(4/3)*(d*e
- c*f)^(5/3)) + ((3*b*d*e - b*c*f - 2*a*d*f)*Log[e + f*x])/(6*(b*e - a*f)^(4/3)
*(d*e - c*f)^(5/3)) - ((3*b*d*e - b*c*f - 2*a*d*f)*Log[((d*e - c*f)^(1/3)*(a + b
*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(4/3)*(d*e - c*f
)^(5/3))

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Rubi in Sympy [A]  time = 47.5826, size = 264, normalized size = 0.9 \[ - \frac{f \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{\left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (e + f x \right )}}{6 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} - \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (\frac{\sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{\sqrt [3]{a f - b e}} - \sqrt [3]{c + d x} \right )}}{2 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 a d f + b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{3 \sqrt [3]{c + d x} \sqrt [3]{a f - b e}} + \frac{\sqrt{3}}{3} \right )}}{3 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e)**2,x)

[Out]

-f*(a + b*x)**(2/3)*(c + d*x)**(1/3)/((e + f*x)*(a*f - b*e)*(c*f - d*e)) + (2*a*
d*f + b*c*f - 3*b*d*e)*log(e + f*x)/(6*(a*f - b*e)**(4/3)*(c*f - d*e)**(5/3)) -
(2*a*d*f + b*c*f - 3*b*d*e)*log((a + b*x)**(1/3)*(c*f - d*e)**(1/3)/(a*f - b*e)*
*(1/3) - (c + d*x)**(1/3))/(2*(a*f - b*e)**(4/3)*(c*f - d*e)**(5/3)) - sqrt(3)*(
2*a*d*f + b*c*f - 3*b*d*e)*atan(2*sqrt(3)*(a + b*x)**(1/3)*(c*f - d*e)**(1/3)/(3
*(c + d*x)**(1/3)*(a*f - b*e)**(1/3)) + sqrt(3)/3)/(3*(a*f - b*e)**(4/3)*(c*f -
d*e)**(5/3))

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Mathematica [C]  time = 0.823068, size = 171, normalized size = 0.58 \[ \frac{(a+b x)^{2/3} \left (\frac{(-2 a d f-b c f+3 b d e) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (d e-c f)}+\frac{2 f (c+d x)}{(e+f x) (c f-d e)}\right )}{2 (c+d x)^{2/3} (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^2),x]

[Out]

((a + b*x)^(2/3)*((2*f*(c + d*x))/((-(d*e) + c*f)*(e + f*x)) + ((3*b*d*e - b*c*f
- 2*a*d*f)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^(2/3)*Hypergeometr
ic2F1[2/3, 2/3, 5/3, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))])/((b*e
- a*f)*(d*e - c*f))))/(2*(b*e - a*f)*(c + d*x)^(2/3))

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{2}}{\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.243797, size = 1022, normalized size = 3.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(6*sqrt(3)*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*
c^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3)*f + sqrt(3)*(3*b*d*e
^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)*log((a*d^2*e^2 - 2*a
*c*d*e*f + a*c^2*f^2 - (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^
2 + 2*a*c*d)*e*f^2)^(1/3)*(d*e - c*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (b*d^2*e
^2 - 2*b*c*d*e*f + b*c^2*f^2)*x + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^
2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b*x + a))
- 2*sqrt(3)*(3*b*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)
*log((a*d*e - a*c*f + (b*d*e - b*c*f)*x + (-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a
*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(b
*x + a)) + 6*(3*b*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*f^2)*x)
*arctan(-1/3*(2*sqrt(3)*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c
^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - sqrt(3)*(a*d*e - a*
c*f + (b*d*e - b*c*f)*x))/(a*d*e - a*c*f + (b*d*e - b*c*f)*x)))/((-b*d^2*e^3 + a
*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*(b*d*e^3 + a
*c*e*f^2 - (b*c + a*d)*e^2*f + (b*d*e^2*f + a*c*f^3 - (b*c + a*d)*e*f^2)*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(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="giac")

[Out]

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