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

Optimal. Leaf size=121 \[ \frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]

[Out]

(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(1 + n)/(2*e*(1 + n)) + (a*f^2*(d + e*x +
f*Sqrt[a + (e^2*x^2)/f^2])^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (d + e*x +
f*Sqrt[a + (e^2*x^2)/f^2])/d])/(2*d^2*e*(1 + n))

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Rubi [A]  time = 0.234842, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+a}}{d}\right )}{2 d^2 e (n+1)}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(1 + n)/(2*e*(1 + n)) + (a*f^2*(d + e*x +
f*Sqrt[a + (e^2*x^2)/f^2])^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (d + e*x +
f*Sqrt[a + (e^2*x^2)/f^2])/d])/(2*d^2*e*(1 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x+f*(a+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

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

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Mathematica [A]  time = 0.123827, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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