3.802 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx\)

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]

-(d^2 - e^2*x^2)^(9/2)/(13*d*e*(d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(143*d^
2*e*(d + e*x)^10) - (2*(d^2 - e^2*x^2)^(9/2))/(1287*d^3*e*(d + e*x)^9)

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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]

-(d^2 - e^2*x^2)^(9/2)/(13*d*e*(d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(143*d^
2*e*(d + e*x)^10) - (2*(d^2 - e^2*x^2)^(9/2))/(1287*d^3*e*(d + e*x)^9)

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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]

-(d**2 - e**2*x**2)**(9/2)/(13*d*e*(d + e*x)**11) - 2*(d**2 - e**2*x**2)**(9/2)/
(143*d**2*e*(d + e*x)**10) - 2*(d**2 - e**2*x**2)**(9/2)/(1287*d**3*e*(d + e*x)*
*9)

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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]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(119*d^2 + 22*d*e*x + 2*e^2*x^2))/(1287*d^3*e*
(d + e*x)^7)

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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]

-1/1287*(-e*x+d)*(2*e^2*x^2+22*d*e*x+119*d^2)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^10/d^
3/e

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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]

Exception raised: ValueError

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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]

-1/1287*(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*(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)*sqrt(-e^2*x^2 + d^2))/(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^1
0*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 - (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 - 6
4*d^15)*sqrt(-e^2*x^2 + d^2))

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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]

Timed 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]

Done