Optimal. Leaf size=200 \[ -\frac{\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{5/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]
[Out]
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Rubi [A] time = 0.297264, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}-\frac{(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{5/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x)/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x]
[Out]
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Rubi in Sympy [A] time = 17.0718, size = 194, normalized size = 0.97 \[ \frac{f \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{b d} + \frac{3 \left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \log{\left (-1 + \frac{\sqrt [3]{d} \sqrt [3]{a + b x}}{\sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{2 b^{\frac{4}{3}} d^{\frac{5}{3}}} + \frac{\left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \log{\left (c + d x \right )}}{2 b^{\frac{4}{3}} d^{\frac{5}{3}}} + \frac{\sqrt{3} \left (- b d e + \frac{f \left (a d + 2 b c\right )}{3}\right ) \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2 \sqrt{3} \sqrt [3]{d} \sqrt [3]{a + b x}}{3 \sqrt [3]{b} \sqrt [3]{c + d x}} \right )}}{b^{\frac{4}{3}} d^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)/(b*x+a)**(1/3)/(d*x+c)**(2/3),x)
[Out]
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Mathematica [C] time = 0.222653, size = 99, normalized size = 0.5 \[ \frac{\sqrt [3]{c+d x} \left (\sqrt [3]{\frac{d (a+b x)}{a d-b c}} (-a d f-2 b c f+3 b d e) \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{b (c+d x)}{b c-a d}\right )+d f (a+b x)\right )}{b d^2 \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[(e + f*x)/((a + b*x)^(1/3)*(c + d*x)^(2/3)),x]
[Out]
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Maple [F] time = 0.042, size = 0, normalized size = 0. \[ \int{(fx+e){\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)/(b*x+a)^(1/3)/(d*x+c)^(2/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/((b*x + a)^(1/3)*(d*x + c)^(2/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231874, size = 360, normalized size = 1.8 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} f + \sqrt{3}{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \log \left (\frac{b d^{2} x + a d^{2} + \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} d + \left (b d^{2}\right )^{\frac{2}{3}}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{b x + a}\right ) - 2 \, \sqrt{3}{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \log \left (-\frac{b d x + a d - \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{b x + a}\right ) + 6 \,{\left (3 \, b d e -{\left (2 \, b c + a d\right )} f\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (b d^{2}\right )^{\frac{1}{3}}{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} + \sqrt{3}{\left (b d x + a d\right )}}{3 \,{\left (b d x + a d\right )}}\right )\right )}}{18 \, \left (b d^{2}\right )^{\frac{1}{3}} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/((b*x + a)^(1/3)*(d*x + c)^(2/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x+e)/(b*x+a)**(1/3)/(d*x+c)**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x + e}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)/((b*x + a)^(1/3)*(d*x + c)^(2/3)),x, algorithm="giac")
[Out]