3.945 \(\int \frac{(d+e x)^m}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=80 \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.161125, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/(d^2 - e^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.4533, size = 80, normalized size = 1. \[ \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m + \frac{3}{2}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{2 d^{2} e \left (d - e x\right ) \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(-e**2*x**2+d**2)**(3/2),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 + 3/2, -1/2), (1/2,), (d/2 - e*x/2)/d)/(2*d**2*e*(d - e*x)*sqrt(d + e*x))

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Mathematica [A]  time = 0.0789308, size = 80, normalized size = 1. \[ \frac{2^{m-\frac{1}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{d-e x}{2 d}\right )}{d e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^m/(d^2 - e^2*x^2)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(-e^2*x^2+d^2)^(3/2),x)

[Out]

int((e*x+d)^m/(-e^2*x^2+d^2)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral((d + e*x)**m/(-(-d + e*x)*(d + e*x))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(3/2), x)