Optimal. Leaf size=33 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
[Out]
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Rubi [A] time = 0.0412624, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(d+e x)^5}{5 d e \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 5.31678, size = 24, normalized size = 0.73 \[ \frac{\left (d + e x\right )^{5}}{5 d e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0446283, size = 41, normalized size = 1.24 \[ \frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{5 d e (d-e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(d^2 - e^2*x^2)^(7/2),x]
[Out]
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Maple [A] time = 0.009, size = 36, normalized size = 1.1 \[{\frac{ \left ( ex+d \right ) ^{6} \left ( -ex+d \right ) }{5\,de} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(-e^2*x^2+d^2)^(7/2),x)
[Out]
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Maxima [A] time = 0.722244, size = 200, normalized size = 6.06 \[ \frac{e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{5 \, d e^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d^{2} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, d^{3} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{d^{4}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221507, size = 203, normalized size = 6.15 \[ \frac{2 \,{\left (e^{4} x^{5} + 5 \, d^{2} e^{2} x^{3} - 10 \, d^{4} x + 10 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x\right )}}{5 \,{\left (d e^{5} x^{5} - 5 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} + 5 \, d^{4} e^{2} x^{2} - 10 \, d^{5} e x + 4 \, d^{6} +{\left (d e^{4} x^{4} - 7 \, d^{3} e^{2} x^{2} + 10 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**5/(-e**2*x**2+d**2)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.230482, size = 103, normalized size = 3.12 \[ -\frac{{\left (d^{4} e^{\left (-1\right )} +{\left (5 \, d^{3} +{\left (10 \, d^{2} e +{\left (x{\left (\frac{x e^{4}}{d} + 5 \, e^{3}\right )} + 10 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")
[Out]