Optimal. Leaf size=365 \[ \frac{\left (d^2-a f^2\right )^5 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{32 e f^4 (5-n)}-\frac{5 \left (d^2-a f^2\right )^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{32 e f^4 (3-n)}+\frac{5 \left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{16 e f^4 (1-n)}+\frac{5 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{16 e f^4 (n+1)}-\frac{5 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{32 e f^4 (n+3)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{32 e f^4 (n+5)} \]
[Out]
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Rubi [A] time = 0.863627, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{\left (d^2-a f^2\right )^5 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{32 e f^4 (5-n)}-\frac{5 \left (d^2-a f^2\right )^4 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{32 e f^4 (3-n)}+\frac{5 \left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{16 e f^4 (1-n)}+\frac{5 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{16 e f^4 (n+1)}-\frac{5 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{32 e f^4 (n+3)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{32 e f^4 (n+5)} \]
Antiderivative was successfully verified.
[In] Int[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,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((a+2*d*e*x/f**2+e**2*x**2/f**2)**2*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)
[Out]
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Mathematica [A] time = 0.646068, size = 0, normalized size = 0. \[ \int \left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right )^2 \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]
[Out]
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Maple [F] time = 0.124, size = 0, normalized size = 0. \[ \int \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{2} \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+2*d*e*x/f^2+e^2*x^2/f^2)^2*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}^{2}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)^2*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.332434, size = 883, normalized size = 2.42 \[ -\frac{{\left (5 \, a^{2} d f^{4} n^{4} + 225 \, a^{2} d f^{4} - 300 \, a d^{3} f^{2} + 5 \,{\left (e^{5} n^{4} - 10 \, e^{5} n^{2} + 9 \, e^{5}\right )} x^{5} + 120 \, d^{5} + 25 \,{\left (d e^{4} n^{4} - 10 \, d e^{4} n^{2} + 9 \, d e^{4}\right )} x^{4} + 10 \,{\left (15 \, a e^{3} f^{2} + 30 \, d^{2} e^{3} +{\left (a e^{3} f^{2} + 4 \, d^{2} e^{3}\right )} n^{4} - 2 \,{\left (8 \, a e^{3} f^{2} + 17 \, d^{2} e^{3}\right )} n^{2}\right )} x^{3} - 10 \,{\left (11 \, a^{2} d f^{4} - 6 \, a d^{3} f^{2}\right )} n^{2} + 10 \,{\left (45 \, a d e^{2} f^{2} +{\left (3 \, a d e^{2} f^{2} + 2 \, d^{3} e^{2}\right )} n^{4} - 2 \,{\left (24 \, a d e^{2} f^{2} + d^{3} e^{2}\right )} n^{2}\right )} x^{2} + 5 \,{\left (45 \, a^{2} e f^{4} +{\left (a^{2} e f^{4} + 4 \, a d^{2} e f^{2}\right )} n^{4} - 2 \,{\left (11 \, a^{2} e f^{4} + 26 \, a d^{2} e f^{2} - 12 \, d^{4} e\right )} n^{2}\right )} x -{\left (a^{2} f^{5} n^{5} +{\left (e^{4} f n^{5} - 10 \, e^{4} f n^{3} + 9 \, e^{4} f n\right )} x^{4} - 10 \,{\left (3 \, a^{2} f^{5} - 2 \, a d^{2} f^{3}\right )} n^{3} + 4 \,{\left (d e^{3} f n^{5} - 10 \, d e^{3} f n^{3} + 9 \, d e^{3} f n\right )} x^{3} + 2 \,{\left ({\left (a e^{2} f^{3} + 2 \, d^{2} e^{2} f\right )} n^{5} - 10 \,{\left (2 \, a e^{2} f^{3} + d^{2} e^{2} f\right )} n^{3} +{\left (19 \, a e^{2} f^{3} + 8 \, d^{2} e^{2} f\right )} n\right )} x^{2} +{\left (149 \, a^{2} f^{5} - 260 \, a d^{2} f^{3} + 120 \, d^{4} f\right )} n + 4 \,{\left (a d e f^{3} n^{5} - 10 \,{\left (2 \, a d e f^{3} - d^{3} e f\right )} n^{3} +{\left (19 \, a d e f^{3} - 10 \, d^{3} e f\right )} n\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{4} n^{6} - 35 \, e f^{4} n^{4} + 259 \, e f^{4} n^{2} - 225 \, e f^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)^2*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + 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((a+2*d*e*x/f**2+e**2*x**2/f**2)**2*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}^{2}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)^2*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="giac")
[Out]