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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]