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

Optimal. Leaf size=166 \[ \frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{2 e \left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+b x+a}\right )}{2 d e-b f^2}\right )}{2 e (n+1) \left (2 d e-b f^2\right )^2}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]

[Out]

(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(1 + n)/(2*e*(1 + n)) + (f^2*(4*a*e^
2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(1 + n)*Hypergeometric2
F1[2, 1 + n, 2 + n, (2*e*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]))/(2*d*e - b
*f^2)])/(2*e*(2*d*e - b*f^2)^2*(1 + n))

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Rubi [A]  time = 0.386172, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{f^2 \left (4 a e^2-b^2 f^2\right ) \left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{2 e \left (d+e x+f \sqrt{\frac{e^2 x^2}{f^2}+b x+a}\right )}{2 d e-b f^2}\right )}{2 e (n+1) \left (2 d e-b f^2\right )^2}+\frac{\left (f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{2 e (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(1 + n)/(2*e*(1 + n)) + (f^2*(4*a*e^
2 - b^2*f^2)*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(1 + n)*Hypergeometric2
F1[2, 1 + n, 2 + n, (2*e*(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]))/(2*d*e - b
*f^2)])/(2*e*(2*d*e - b*f^2)^2*(1 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.20409, size = 0, normalized size = 0. \[ \int \left (d+e x+f \sqrt{a+b x+\frac{e^2 x^2}{f^2}}\right )^n \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \[ \int \left ( d+ex+f\sqrt{a+bx+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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