Optimal. Leaf size=107 \[ \frac{\left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{2 e (1-n)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
[Out]
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Rubi [A] time = 0.234673, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ \frac{\left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{2 e (1-n)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x + f \sqrt{\frac{a f^{2} + e x \left (2 d + e x\right )}{f^{2}}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d+e*x+f*((a*f**2+e*x*(e*x+2*d))/f**2)**(1/2))**n,x)
[Out]
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Mathematica [A] time = 0.0833649, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right )^n \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(d + e*x + f*Sqrt[(a*f^2 + e*x*(2*d + e*x))/f^2])^n,x]
[Out]
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Maple [F] time = 0.014, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{{\frac{a{f}^{2}+ex \left ( ex+2\,d \right ) }{{f}^{2}}}} \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d+e*x+f*((a*f^2+e*x*(e*x+2*d))/f^2)^(1/2))^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + f \sqrt{\frac{a f^{2} +{\left (e x + 2 \, d\right )} e x}{f^{2}}} + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + f*sqrt((a*f^2 + (e*x + 2*d)*e*x)/f^2) + d)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.302633, size = 108, normalized size = 1.01 \[ \frac{{\left (f n \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} - e x - d\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e n^{2} - e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + f*sqrt((a*f^2 + (e*x + 2*d)*e*x)/f^2) + d)^n,x, algorithm="fricas")
[Out]
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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*f**2+e*x*(e*x+2*d))/f**2)**(1/2))**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + f \sqrt{\frac{a f^{2} +{\left (e x + 2 \, d\right )} e x}{f^{2}}} + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + f*sqrt((a*f^2 + (e*x + 2*d)*e*x)/f^2) + d)^n,x, algorithm="giac")
[Out]