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

Optimal. Leaf size=41 \[ \frac{f \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n} \]

[Out]

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

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Rubi [A]  time = 0.668488, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{f \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^n}{e n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 109.276, size = 37, normalized size = 0.9 \[ \frac{f \left (d + e x + f \sqrt{a + \frac{2 d e x}{f^{2}} + \frac{e^{2} x^{2}}{f^{2}}}\right )^{n}}{e n} \]

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

[Out]

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

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Mathematica [A]  time = 0.123001, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x+f \sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}\right )^n}{\sqrt{\frac{a f^2+e x (2 d+e x)}{f^2}}} \, dx \]

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

[Out]

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

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Maple [F]  time = 0.111, size = 0, normalized size = 0. \[ \int{1 \left ( d+ex+f\sqrt{{\frac{a{f}^{2}+ex \left ( ex+2\,d \right ) }{{f}^{2}}}} \right ) ^{n}{\frac{1}{\sqrt{{\frac{a{f}^{2}+ex \left ( ex+2\,d \right ) }{{f}^{2}}}}}}}\, dx \]

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

[Out]

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

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Maxima [A]  time = 0.851402, size = 47, normalized size = 1.15 \[ \frac{{\left (e x + d + \sqrt{e^{2} x^{2} + a f^{2} + 2 \, d e x}\right )}^{n} f}{e n} \]

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

[Out]

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

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Fricas [A]  time = 0.312753, size = 55, normalized size = 1.34 \[ \frac{{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n} f}{e n} \]

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

[Out]

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

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

[Out]

Timed out

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

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

[Out]

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