3.804 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(17*d*e*(d + e*x)^13) - (4*(d^2 - e^2*x^2)^(9/2))/(255*d^
2*e*(d + e*x)^12) - (4*(d^2 - e^2*x^2)^(9/2))/(1105*d^3*e*(d + e*x)^11) - (8*(d^
2 - e^2*x^2)^(9/2))/(12155*d^4*e*(d + e*x)^10) - (8*(d^2 - e^2*x^2)^(9/2))/(1093
95*d^5*e*(d + e*x)^9)

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Rubi [A]  time = 0.220895, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{255 d^2 e (d+e x)^{12}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{17 d e (d+e x)^{13}}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{109395 d^5 e (d+e x)^9}-\frac{8 \left (d^2-e^2 x^2\right )^{9/2}}{12155 d^4 e (d+e x)^{10}}-\frac{4 \left (d^2-e^2 x^2\right )^{9/2}}{1105 d^3 e (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(17*d*e*(d + e*x)^13) - (4*(d^2 - e^2*x^2)^(9/2))/(255*d^
2*e*(d + e*x)^12) - (4*(d^2 - e^2*x^2)^(9/2))/(1105*d^3*e*(d + e*x)^11) - (8*(d^
2 - e^2*x^2)^(9/2))/(12155*d^4*e*(d + e*x)^10) - (8*(d^2 - e^2*x^2)^(9/2))/(1093
95*d^5*e*(d + e*x)^9)

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Rubi in Sympy [A]  time = 24.198, size = 141, normalized size = 0.85 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{17 d e \left (d + e x\right )^{13}} - \frac{4 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{255 d^{2} e \left (d + e x\right )^{12}} - \frac{4 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{1105 d^{3} e \left (d + e x\right )^{11}} - \frac{8 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{12155 d^{4} e \left (d + e x\right )^{10}} - \frac{8 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{109395 d^{5} e \left (d + e x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**13,x)

[Out]

-(d**2 - e**2*x**2)**(9/2)/(17*d*e*(d + e*x)**13) - 4*(d**2 - e**2*x**2)**(9/2)/
(255*d**2*e*(d + e*x)**12) - 4*(d**2 - e**2*x**2)**(9/2)/(1105*d**3*e*(d + e*x)*
*11) - 8*(d**2 - e**2*x**2)**(9/2)/(12155*d**4*e*(d + e*x)**10) - 8*(d**2 - e**2
*x**2)**(9/2)/(109395*d**5*e*(d + e*x)**9)

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Mathematica [A]  time = 0.0782493, size = 82, normalized size = 0.49 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (8627 d^4+2756 d^3 e x+660 d^2 e^2 x^2+104 d e^3 x^3+8 e^4 x^4\right )}{109395 d^5 e (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^13,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(8627*d^4 + 2756*d^3*e*x + 660*d^2*e^2*x^2 + 1
04*d*e^3*x^3 + 8*e^4*x^4))/(109395*d^5*e*(d + e*x)^9)

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Maple [A]  time = 0.012, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( 8\,{e}^{4}{x}^{4}+104\,{e}^{3}{x}^{3}d+660\,{e}^{2}{x}^{2}{d}^{2}+2756\,x{d}^{3}e+8627\,{d}^{4} \right ) \left ( -ex+d \right ) }{109395\, \left ( ex+d \right ) ^{12}{d}^{5}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^13,x)

[Out]

-1/109395*(-e*x+d)*(8*e^4*x^4+104*d*e^3*x^3+660*d^2*e^2*x^2+2756*d^3*e*x+8627*d^
4)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^12/d^5/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)^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.486576, size = 987, normalized size = 5.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^13,x, algorithm="fricas")

[Out]

-1/109395*(8635*e^16*x^17 + 146659*d*e^15*x^16 + 586024*d^2*e^14*x^15 - 589900*d
^3*e^13*x^14 - 9684135*d^4*e^12*x^13 - 23769213*d^5*e^11*x^12 - 6165458*d^6*e^10
*x^11 + 73693334*d^7*e^9*x^10 + 136121414*d^8*e^8*x^9 + 57410496*d^9*e^7*x^8 - 1
33957824*d^10*e^6*x^7 - 249478944*d^11*e^5*x^6 - 115287744*d^12*e^4*x^5 + 886828
80*d^13*e^3*x^4 + 102685440*d^14*e^2*x^3 + 56010240*d^15*e*x^2 + 28005120*d^16*x
- 17*(507*e^15*x^16 - 8*d*e^14*x^15 - 38600*d^2*e^13*x^14 - 207020*d^3*e^12*x^1
3 - 313105*d^4*e^11*x^12 + 623740*d^5*e^10*x^11 + 3051334*d^6*e^9*x^10 + 3862144
*d^7*e^8*x^9 - 974688*d^8*e^7*x^8 - 8490768*d^9*e^6*x^7 - 10831392*d^10*e^5*x^6
- 3143712*d^11*e^4*x^5 + 6864000*d^12*e^3*x^4 + 6864000*d^13*e^2*x^3 + 3294720*d
^14*e*x^2 + 1647360*d^15*x)*sqrt(-e^2*x^2 + d^2))/(d^5*e^17*x^17 + 17*d^6*e^16*x
^16 + 68*d^7*e^15*x^15 - 68*d^8*e^14*x^14 - 1122*d^9*e^13*x^13 - 2754*d^10*e^12*
x^12 - 748*d^11*e^11*x^11 + 8500*d^12*e^10*x^10 + 16201*d^13*e^9*x^9 + 6409*d^14
*e^8*x^8 - 16864*d^15*e^7*x^7 - 27064*d^16*e^6*x^6 - 12512*d^17*e^5*x^5 + 7344*d
^18*e^4*x^4 + 13056*d^19*e^3*x^3 + 7616*d^20*e^2*x^2 + 2176*d^21*e*x + 256*d^22
- (d^5*e^16*x^16 - 76*d^7*e^14*x^14 - 408*d^8*e^13*x^13 - 618*d^9*e^12*x^12 + 12
24*d^10*e^11*x^11 + 5996*d^11*e^10*x^10 + 7752*d^12*e^9*x^9 - 1919*d^13*e^8*x^8
- 17544*d^14*e^7*x^7 - 20456*d^15*e^6*x^6 - 5168*d^16*e^5*x^5 + 11248*d^17*e^4*x
^4 + 14144*d^18*e^3*x^3 + 7744*d^19*e^2*x^2 + 2176*d^20*e*x + 256*d^21)*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)**13,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.84685, 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)^13,x, algorithm="giac")

[Out]

Done