Optimal. Leaf size=41 \[ \frac{f \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.38793, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.052 \[ \frac{f \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 96.1647, size = 37, normalized size = 0.9 \[ \frac{f \left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n}}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n/(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.198977, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n}{\sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n/Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2],x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.109, size = 0, normalized size = 0. \[ \int{1 \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}{\frac{1}{\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n/(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.850073, size = 47, normalized size = 1.15 \[ \frac{{\left (e x + d + \sqrt{e^{2} x^{2} + a f^{2} + 2 \, d e x}\right )}^{n} f}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n/sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.324129, size = 55, normalized size = 1.34 \[ \frac{{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f}{e n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n/sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n/(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}}{\sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n/sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2),x, algorithm="giac")
[Out]