Optimal. Leaf size=93 \[ \frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \]
[Out]
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Rubi [A] time = 0.839321, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}} \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n \sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \]
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*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2],x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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*g+2*d*e*g*x/f**2+e**2*g*x**2/f**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.133682, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n}{\sqrt{a g+\frac{2 d e g x}{f^2}+\frac{e^2 g x^2}{f^2}}} \, dx \]
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*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2],x]
[Out]
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Maple [F] time = 0.11, size = 0, normalized size = 0. \[ \int{1 \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}{\frac{1}{\sqrt{ag+2\,{\frac{degx}{{f}^{2}}}+{\frac{{e}^{2}g{x}^{2}}{{f}^{2}}}}}}}\, dx \]
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*g+2*d*e*g*x/f^2+e^2*g*x^2/f^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.811837, size = 51, normalized size = 0.55 \[ \frac{{\left (e x + d + \sqrt{e^{2} x^{2} + a f^{2} + 2 \, d e x}\right )}^{n} f}{e \sqrt{g} n} \]
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*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.311372, size = 158, normalized size = 1.7 \[ \frac{{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f^{3} \sqrt{\frac{e^{2} g x^{2} + a f^{2} g + 2 \, d e g x}{f^{2}}} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}}{e^{3} g n x^{2} + a e f^{2} g n + 2 \, d e^{2} g n x} \]
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*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n}}{\sqrt{g \left (a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}\right )}}\, dx \]
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*g+2*d*e*g*x/f**2+e**2*g*x**2/f**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}}{\sqrt{\frac{e^{2} g x^{2}}{f^{2}} + a g + \frac{2 \, d e g x}{f^{2}}}}\,{d x} \]
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*g*x^2/f^2 + a*g + 2*d*e*g*x/f^2),x, algorithm="giac")
[Out]