Optimal. Leaf size=68 \[ -\frac{(d-e x) (d+e x)^{m+1} \, _2F_1\left (1,m-5;m-\frac{3}{2};\frac{d+e x}{2 d}\right )}{d e (5-2 m) \left (d^2-e^2 x^2\right )^{7/2}} \]
[Out]
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Rubi [A] time = 0.162846, antiderivative size = 83, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{5}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 24.3192, size = 83, normalized size = 1.22 \[ \frac{\left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- m - \frac{1}{2}} \left (d + e x\right )^{m + \frac{1}{2}} \sqrt{d^{2} - e^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} - m + \frac{7}{2}, - \frac{5}{2} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{40 d^{4} e \left (d - e x\right )^{3} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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Mathematica [A] time = 0.106014, size = 91, normalized size = 1.34 \[ \frac{2^{m-\frac{5}{2}} (d+e x)^m \left (\frac{e x}{d}+1\right )^{\frac{1}{2}-m} \, _2F_1\left (-\frac{5}{2},\frac{7}{2}-m;-\frac{3}{2};\frac{d-e x}{2 d}\right )}{5 d^3 e (d-e x)^2 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{m} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(-e^2*x^2+d^2)^(7/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (e x + d\right )}^{m}}{{\left (e^{6} x^{6} - 3 \, d^{2} e^{4} x^{4} + 3 \, d^{4} e^{2} x^{2} - d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")
[Out]
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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((e*x+d)**m/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")
[Out]