3.1589 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=171 \[ \frac{(b c-a d) \log (a+b x)}{6 b^{2/3} d^{4/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} d^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d^{4/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{d} \]

[Out]

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

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Rubi [A]  time = 0.147105, antiderivative size = 171, 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 \[ \frac{(b c-a d) \log (a+b x)}{6 b^{2/3} d^{4/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} d^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d^{4/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5358, size = 160, normalized size = 0.94 \[ \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{d} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{6 b^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\left (a d - b c\right ) \log{\left (\frac{\sqrt [3]{b} \sqrt [3]{c + d x}}{\sqrt [3]{d} \sqrt [3]{a + b x}} - 1 \right )}}{2 b^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{b} \sqrt [3]{c + d x}}{3 \sqrt [3]{d} \sqrt [3]{a + b x}} + \frac{\sqrt{3}}{3} \right )}}{3 b^{\frac{2}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.180656, size = 76, normalized size = 0.44 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{\, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [3]{\frac{d (a+b x)}{a d-b c}}}+2\right )}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3)*(2 + Hypergeometric2F1[2/3, 2/3, 5/3, (b*(c + d
*x))/(b*c - a*d)]/((d*(a + b*x))/(-(b*c) + a*d))^(1/3)))/(2*d)

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{1\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218595, size = 327, normalized size = 1.91 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (b c - a d\right )} \log \left (\frac{b^{2} d x + b^{2} c - \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b + \left (-b^{2} d\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{d x + c}\right ) - 2 \, \sqrt{3}{\left (b c - a d\right )} \log \left (\frac{b d x + b c + \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{d x + c}\right ) - 6 \,{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} - \sqrt{3}{\left (b d x + b c\right )}}{3 \,{\left (b d x + b c\right )}}\right ) + 6 \, \sqrt{3} \left (-b^{2} d\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}\right )}}{18 \, \left (-b^{2} d\right )^{\frac{1}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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