3.1577 \(\int \frac{\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{2/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b} \]

[Out]

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

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Rubi [A]  time = 0.147964, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{2/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7766, size = 160, normalized size = 0.93 \[ \frac{\left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{b} + \frac{\left (a d - b c\right ) \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{2 b^{\frac{4}{3}} d^{\frac{2}{3}}} + \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{6 b^{\frac{4}{3}} d^{\frac{2}{3}}} + \frac{\sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{3 b^{\frac{4}{3}} d^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.151228, size = 90, normalized size = 0.52 \[ \frac{\sqrt [3]{c+d x} \left ((b c-a d) \sqrt [3]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x)\right )}{b d \sqrt [3]{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{dx+c}{\frac{1}{\sqrt [3]{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220981, size = 319, normalized size = 1.85 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (b c - a d\right )} \log \left (\frac{b d^{2} x + a d^{2} + \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \, \sqrt{3}{\left (b c - a d\right )} \log \left (-\frac{b d x + a d - \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) + 6 \,{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} + \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right ) + 6 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}\right )}}{18 \, \left (b d^{2}\right )^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{c + d x}}{\sqrt [3]{a + b x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/3)/(b*x+a)**(1/3),x)

[Out]

Integral((c + d*x)**(1/3)/(a + b*x)**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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