3.943 \(\int (d+e x)^m \sqrt{d^2-e^2 x^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{(d-e x) \sqrt{d^2-e^2 x^2} (d+e x)^{m+1} \, _2F_1\left (1,m+3;m+\frac{5}{2};\frac{d+e x}{2 d}\right )}{d e (2 m+3)} \]

[Out]

((d - e*x)*(d + e*x)^(1 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[1, 3 + m, 5/2
+ m, (d + e*x)/(2*d)])/(d*e*(3 + 2*m))

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Rubi [A]  time = 0.143598, antiderivative size = 83, normalized size of antiderivative = 1.24, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2^{m+\frac{3}{2}} \left (d^2-e^2 x^2\right )^{3/2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{3}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )}{3 d e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*Sqrt[d^2 - e^2*x^2],x]

[Out]

-(2^(3/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-3/2 - m)*(d^2 - e^2*x^2)^(3/2)*Hyperge
ometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)])/(3*d*e)

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Rubi in Sympy [A]  time = 23.3686, size = 80, normalized size = 1.19 \[ - \frac{2 \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d - e x\right ) \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m - \frac{1}{2}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{3 e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(-e**2*x**2+d**2)**(1/2),x)

[Out]

-2*((d/2 + e*x/2)/d)**(-m - 1/2)*(d - e*x)*(d + e*x)**(m + 1/2)*sqrt(d**2 - e**2
*x**2)*hyper((-m - 1/2, 3/2), (5/2,), (d/2 - e*x/2)/d)/(3*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.0991224, size = 86, normalized size = 1.28 \[ -\frac{2^{m+\frac{3}{2}} (d-e x) \sqrt{d^2-e^2 x^2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{d-e x}{2 d}\right )}{3 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^m*Sqrt[d^2 - e^2*x^2],x]

[Out]

-(2^(3/2 + m)*(d - e*x)*(d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*Sqrt[d^2 - e^2*x^2]
*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (d - e*x)/(2*d)])/(3*e)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x)

[Out]

int((e*x+d)^m*(-e^2*x^2+d^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m,x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m,x, algorithm="fricas")

[Out]

integral(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m,x, algorithm="giac")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)*(e*x + d)^m, x)