3.347 \(\int \left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right ) \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx\)

Optimal. Leaf size=239 \[ \frac{\left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac{3 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac{3 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]

[Out]

((d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))
/(8*e*f^2*(3 - n)) - (3*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e
^2*x^2)/f^2])^(-1 + n))/(8*e*f^2*(1 - n)) - (3*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a
+ (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(8*e*f^2*(1 + n)) + (d + e*x + f*Sqr
t[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n)/(8*e*f^2*(3 + n))

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Rubi [A]  time = 0.482507, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac{3 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac{3 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]

[Out]

((d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))
/(8*e*f^2*(3 - n)) - (3*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e
^2*x^2)/f^2])^(-1 + n))/(8*e*f^2*(1 - n)) - (3*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a
+ (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(8*e*f^2*(1 + n)) + (d + e*x + f*Sqr
t[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n)/(8*e*f^2*(3 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}} \frac{e^{3} x^{n} \left (a f^{2} - d^{2} + x^{2}\right )^{3}}{x^{4}}\, dx}{8 e^{4} f^{2}} \]

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)*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

Integral(e**3*x**n*(a*f**2 - d**2 + x**2)**3/x**4, (x, d + e*x + f*sqrt(a + 2*d*
e*x/f**2 + e**2*x**2/f**2)))/(8*e**4*f**2)

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Mathematica [A]  time = 0.227906, size = 0, normalized size = 0. \[ \int \left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right ) \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^n,x]

[Out]

Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^
2 + (e^2*x^2)/f^2])^n, x]

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \[ \int \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]

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)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)

[Out]

int((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]

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)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="maxima")

[Out]

integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/
f^2)*f + d)^n, x)

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Fricas [A]  time = 0.327124, size = 323, normalized size = 1.35 \[ -\frac{{\left (3 \, a d f^{2} n^{2} - 9 \, a d f^{2} + 3 \,{\left (e^{3} n^{2} - e^{3}\right )} x^{3} + 6 \, d^{3} + 9 \,{\left (d e^{2} n^{2} - d e^{2}\right )} x^{2} - 3 \,{\left (3 \, a e f^{2} -{\left (a e f^{2} + 2 \, d^{2} e\right )} n^{2}\right )} x -{\left (a f^{3} n^{3} +{\left (e^{2} f n^{3} - e^{2} f n\right )} x^{2} -{\left (7 \, a f^{3} - 6 \, d^{2} f\right )} n + 2 \,{\left (d e f n^{3} - d e f n\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{2} n^{4} - 10 \, e f^{2} n^{2} + 9 \, e f^{2}} \]

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)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="fricas")

[Out]

-(3*a*d*f^2*n^2 - 9*a*d*f^2 + 3*(e^3*n^2 - e^3)*x^3 + 6*d^3 + 9*(d*e^2*n^2 - d*e
^2)*x^2 - 3*(3*a*e*f^2 - (a*e*f^2 + 2*d^2*e)*n^2)*x - (a*f^3*n^3 + (e^2*f*n^3 -
e^2*f*n)*x^2 - (7*a*f^3 - 6*d^2*f)*n + 2*(d*e*f*n^3 - d*e*f*n)*x)*sqrt((e^2*x^2
+ a*f^2 + 2*d*e*x)/f^2))*(e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2) + d)^n/(
e*f^2*n^4 - 10*e*f^2*n^2 + 9*e*f^2)

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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)*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \]

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)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n,x, algorithm="giac")

[Out]

integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/
f^2)*f + d)^n, x)