Optimal. Leaf size=293 \[ -\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{(e+f x) (b e-a f) (d e-c f)}+\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{4/3} (d e-c f)^{5/3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.559468, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{(e+f x) (b e-a f) (d e-c f)}+\frac{\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 (b e-a f)^{4/3} (d e-c f)^{5/3}}-\frac{(-2 a d f-b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} (b e-a f)^{4/3} (d e-c f)^{5/3}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 47.5826, size = 264, normalized size = 0.9 \[ - \frac{f \left (a + b x\right )^{\frac{2}{3}} \sqrt [3]{c + d x}}{\left (e + f x\right ) \left (a f - b e\right ) \left (c f - d e\right )} + \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (e + f x \right )}}{6 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} - \frac{\left (2 a d f + b c f - 3 b d e\right ) \log{\left (\frac{\sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{\sqrt [3]{a f - b e}} - \sqrt [3]{c + d x} \right )}}{2 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} - \frac{\sqrt{3} \left (2 a d f + b c f - 3 b d e\right ) \operatorname{atan}{\left (\frac{2 \sqrt{3} \sqrt [3]{a + b x} \sqrt [3]{c f - d e}}{3 \sqrt [3]{c + d x} \sqrt [3]{a f - b e}} + \frac{\sqrt{3}}{3} \right )}}{3 \left (a f - b e\right )^{\frac{4}{3}} \left (c f - d e\right )^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.823068, size = 171, normalized size = 0.58 \[ \frac{(a+b x)^{2/3} \left (\frac{(-2 a d f-b c f+3 b d e) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (d e-c f)}+\frac{2 f (c+d x)}{(e+f x) (c f-d e)}\right )}{2 (c+d x)^{2/3} (b e-a f)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( fx+e \right ) ^{2}}{\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.243797, size = 1022, normalized size = 3.49 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^2),x, algorithm="giac")
[Out]