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

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

[Out]

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

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Rubi [A]  time = 0.147382, antiderivative size = 81, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2^{m+\frac{1}{2}} \sqrt{d^2-e^2 x^2} (d+e x)^m \left (\frac{e x}{d}+1\right )^{-m-\frac{1}{2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right )}{d e} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(1/2 + m)*(d + e*x)^m*(1 + (e*x)/d)^(-1/2 - m)*Sqrt[d^2 - e^2*x^2]*Hypergeo
metric2F1[1/2, 1/2 - m, 3/2, (d - e*x)/(2*d)])/(d*e))

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Rubi in Sympy [A]  time = 24.758, size = 71, normalized size = 1.06 \[ - \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m + \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{d e \sqrt{d + e x}} \]

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]

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

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Mathematica [A]  time = 0.122218, size = 84, normalized size = 1.25 \[ -\frac{2^{m+\frac{1}{2}} (d-e x) (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{d-e x}{2 d}\right )}{e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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