3.2995 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx\)

Optimal. Leaf size=256 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac{(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{2/3} (d e-c f)^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.310589, antiderivative size = 256, 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 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac{(b c-a d) \log (e+f x)}{6 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} (d e-c f)^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{2/3} (d e-c f)^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.1614, size = 219, normalized size = 0.86 \[ - \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{\left (e + f x\right ) \left (c f - d e\right )} + \frac{\left (a d - b c\right ) \log{\left (e + f x \right )}}{6 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{4}{3}}} - \frac{\left (a d - b c\right ) \log{\left (- \sqrt [3]{a + b x} + \frac{\sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{\sqrt [3]{c f - d e}} \right )}}{2 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{c + d x} \sqrt [3]{a f - b e}}{3 \sqrt [3]{a + b x} \sqrt [3]{c f - d e}} \right )}}{3 \left (a f - b e\right )^{\frac{2}{3}} \left (c f - d e\right )^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.1572, size = 124, normalized size = 0.48 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{\, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(c f-d e) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{2/3}}+\frac{1}{d e-c f}\right )}{e+f x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{2}}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231159, size = 896, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError