3.1603 \(\int \frac{(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=169 \[ \frac{(b c-a d) \log (c+d x)}{3 \sqrt [3]{b} d^{5/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{\sqrt [3]{b} d^{5/3}}+\frac{2 (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} d^{5/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{d} \]

[Out]

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

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Rubi [A]  time = 0.146693, antiderivative size = 169, 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 \[ \frac{(b c-a d) \log (c+d x)}{3 \sqrt [3]{b} d^{5/3}}+\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{\sqrt [3]{b} d^{5/3}}+\frac{2 (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b} d^{5/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9117, size = 160, normalized size = 0.95 \[ \frac{\left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{d} - \frac{\left (a d - b c\right ) \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{\sqrt [3]{b} d^{\frac{5}{3}}} - \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{3 \sqrt [3]{b} d^{\frac{5}{3}}} - \frac{2 \sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{3 \sqrt [3]{b} d^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(2/3)*(c + d*x)**(1/3)/d - (a*d - b*c)*log(-1 + d**(1/3)*(a + b*x)**(
1/3)/(b**(1/3)*(c + d*x)**(1/3)))/(b**(1/3)*d**(5/3)) - (a*d - b*c)*log(c + d*x)
/(3*b**(1/3)*d**(5/3)) - 2*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**(1/3)*d**(5/3))

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Mathematica [C]  time = 0.169299, size = 74, normalized size = 0.44 \[ \frac{(a+b x)^{2/3} \sqrt [3]{c+d x} \left (\frac{2 \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )}{\left (\frac{d (a+b x)}{a d-b c}\right )^{2/3}}+1\right )}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{2}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22859, size = 327, normalized size = 1.93 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (b c - a d\right )} \log \left (\frac{b d^{2} x + a d^{2} - \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (-b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \, \sqrt{3}{\left (b c - a d\right )} \log \left (\frac{b d x + a d + \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) - 6 \,{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} - \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right ) + 3 \, \sqrt{3} \left (-b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}\right )}}{9 \, \left (-b d^{2}\right )^{\frac{1}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*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)) + 3*sqr
t(3)*(-b*d^2)^(1/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/((-b*d^2)^(1/3)*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**(2/3)/(c + d*x)**(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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