Optimal. Leaf size=100 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \]
[Out]
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Rubi [A] time = 0.117651, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \]
Antiderivative was successfully verified.
[In] Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]
[Out]
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Rubi in Sympy [A] time = 12.7614, size = 83, normalized size = 0.83 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{13 d e \left (d + e x\right )^{11}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{143 d^{2} e \left (d + e x\right )^{10}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{1287 d^{3} e \left (d + e x\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**11,x)
[Out]
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Mathematica [A] time = 0.0710785, size = 60, normalized size = 0.6 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (119 d^2+22 d e x+2 e^2 x^2\right )}{1287 d^3 e (d+e x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]
[Out]
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Maple [A] time = 0.01, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,{e}^{2}{x}^{2}+22\,dex+119\,{d}^{2} \right ) \left ( -ex+d \right ) }{1287\, \left ( ex+d \right ) ^{10}{d}^{3}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^11,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.309674, size = 749, normalized size = 7.49 \[ -\frac{121 \, e^{12} x^{13} + 1547 \, d e^{11} x^{12} + 4550 \, d^{2} e^{10} x^{11} - 5434 \, d^{3} e^{9} x^{10} - 44759 \, d^{4} e^{8} x^{9} - 77649 \, d^{5} e^{7} x^{8} - 25740 \, d^{6} e^{6} x^{7} + 192192 \, d^{7} e^{5} x^{6} + 248820 \, d^{8} e^{4} x^{5} - 34320 \, d^{9} e^{3} x^{4} - 109824 \, d^{10} e^{2} x^{3} - 82368 \, d^{11} e x^{2} - 82368 \, d^{12} x - 13 \,{\left (9 \, e^{11} x^{12} - 2 \, d e^{10} x^{11} - 418 \, d^{2} e^{9} x^{10} - 1628 \, d^{3} e^{8} x^{9} - 1551 \, d^{4} e^{7} x^{8} + 2442 \, d^{5} e^{6} x^{7} + 11088 \, d^{6} e^{5} x^{6} + 12540 \, d^{7} e^{4} x^{5} - 5808 \, d^{8} e^{3} x^{4} - 11616 \, d^{9} e^{2} x^{3} - 6336 \, d^{10} e x^{2} - 6336 \, d^{11} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1287 \,{\left (d^{3} e^{13} x^{13} + 13 \, d^{4} e^{12} x^{12} + 39 \, d^{5} e^{11} x^{11} - 39 \, d^{6} e^{10} x^{10} - 403 \, d^{7} e^{9} x^{9} - 689 \, d^{8} e^{8} x^{8} + 13 \, d^{9} e^{7} x^{7} + 1443 \, d^{10} e^{6} x^{6} + 1742 \, d^{11} e^{5} x^{5} + 312 \, d^{12} e^{4} x^{4} - 1040 \, d^{13} e^{3} x^{3} - 1040 \, d^{14} e^{2} x^{2} - 416 \, d^{15} e x - 64 \, d^{16} -{\left (d^{3} e^{12} x^{12} - 45 \, d^{5} e^{10} x^{10} - 182 \, d^{6} e^{9} x^{9} - 193 \, d^{7} e^{8} x^{8} + 364 \, d^{8} e^{7} x^{7} + 1189 \, d^{9} e^{6} x^{6} + 1066 \, d^{10} e^{5} x^{5} - 232 \, d^{11} e^{4} x^{4} - 1248 \, d^{12} e^{3} x^{3} - 1072 \, d^{13} e^{2} x^{2} - 416 \, d^{14} e x - 64 \, d^{15}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^11,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**2*x**2+d**2)**(7/2)/(e*x+d)**11,x)
[Out]
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GIAC/XCAS [A] time = 1.44388, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^11,x, algorithm="giac")
[Out]