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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]