3.803 \(\int \frac{\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}} \]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(15*d*e*(d + e*x)^12) - (d^2 - e^2*x^2)^(9/2)/(65*d^2*e*(
d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(715*d^3*e*(d + e*x)^10) - (2*(d^2 - e^
2*x^2)^(9/2))/(6435*d^4*e*(d + e*x)^9)

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Rubi [A]  time = 0.170211, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (d^2-e^2 x^2\right )^{9/2}}{65 d^2 e (d+e x)^{11}}-\frac{\left (d^2-e^2 x^2\right )^{9/2}}{15 d e (d+e x)^{12}}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{6435 d^4 e (d+e x)^9}-\frac{2 \left (d^2-e^2 x^2\right )^{9/2}}{715 d^3 e (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-(d^2 - e^2*x^2)^(9/2)/(15*d*e*(d + e*x)^12) - (d^2 - e^2*x^2)^(9/2)/(65*d^2*e*(
d + e*x)^11) - (2*(d^2 - e^2*x^2)^(9/2))/(715*d^3*e*(d + e*x)^10) - (2*(d^2 - e^
2*x^2)^(9/2))/(6435*d^4*e*(d + e*x)^9)

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Rubi in Sympy [A]  time = 17.8811, size = 110, normalized size = 0.83 \[ - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{15 d e \left (d + e x\right )^{12}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{65 d^{2} e \left (d + e x\right )^{11}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{715 d^{3} e \left (d + e x\right )^{10}} - \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{9}{2}}}{6435 d^{4} e \left (d + e x\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**12,x)

[Out]

-(d**2 - e**2*x**2)**(9/2)/(15*d*e*(d + e*x)**12) - (d**2 - e**2*x**2)**(9/2)/(6
5*d**2*e*(d + e*x)**11) - 2*(d**2 - e**2*x**2)**(9/2)/(715*d**3*e*(d + e*x)**10)
- 2*(d**2 - e**2*x**2)**(9/2)/(6435*d**4*e*(d + e*x)**9)

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Mathematica [A]  time = 0.0719533, size = 71, normalized size = 0.53 \[ -\frac{(d-e x)^4 \sqrt{d^2-e^2 x^2} \left (548 d^3+141 d^2 e x+24 d e^2 x^2+2 e^3 x^3\right )}{6435 d^4 e (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^12,x]

[Out]

-((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(548*d^3 + 141*d^2*e*x + 24*d*e^2*x^2 + 2*e^3*
x^3))/(6435*d^4*e*(d + e*x)^8)

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Maple [A]  time = 0.011, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,{e}^{3}{x}^{3}+24\,{e}^{2}{x}^{2}d+141\,x{d}^{2}e+548\,{d}^{3} \right ) \left ( -ex+d \right ) }{6435\, \left ( ex+d \right ) ^{11}{d}^{4}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^12,x)

[Out]

-1/6435*(-e*x+d)*(2*e^3*x^3+24*d*e^2*x^2+141*d^2*e*x+548*d^3)*(-e^2*x^2+d^2)^(7/
2)/(e*x+d)^11/d^4/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)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.370511, size = 865, normalized size = 6.5 \[ -\frac{546 \, e^{14} x^{15} - 30 \, d e^{13} x^{14} - 32985 \, d^{2} e^{12} x^{13} - 153335 \, d^{3} e^{11} x^{12} - 177840 \, d^{4} e^{10} x^{11} + 471900 \, d^{5} e^{9} x^{10} + 1588015 \, d^{6} e^{8} x^{9} + 1512225 \, d^{7} e^{7} x^{8} - 875160 \, d^{8} e^{6} x^{7} - 4135560 \, d^{9} e^{5} x^{6} - 3171168 \, d^{10} e^{4} x^{5} + 1132560 \, d^{11} e^{3} x^{4} + 1921920 \, d^{12} e^{2} x^{3} + 1235520 \, d^{13} e x^{2} + 823680 \, d^{14} x +{\left (550 \, e^{13} x^{14} + 8220 \, d e^{12} x^{13} + 28925 \, d^{2} e^{11} x^{12} - 25220 \, d^{3} e^{10} x^{11} - 368940 \, d^{4} e^{9} x^{10} - 787072 \, d^{5} e^{8} x^{9} - 255255 \, d^{6} e^{7} x^{8} + 1482624 \, d^{7} e^{6} x^{7} + 3105960 \, d^{8} e^{5} x^{6} + 1901328 \, d^{9} e^{4} x^{5} - 1750320 \, d^{10} e^{3} x^{4} - 2333760 \, d^{11} e^{2} x^{3} - 1235520 \, d^{12} e x^{2} - 823680 \, d^{13} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6435 \,{\left (d^{4} e^{15} x^{15} - 60 \, d^{6} e^{13} x^{13} - 280 \, d^{7} e^{12} x^{12} - 330 \, d^{8} e^{11} x^{11} + 840 \, d^{9} e^{10} x^{10} + 3020 \, d^{10} e^{9} x^{9} + 2760 \, d^{11} e^{8} x^{8} - 2175 \, d^{12} e^{7} x^{7} - 6920 \, d^{13} e^{6} x^{6} - 5208 \, d^{14} e^{5} x^{5} + 720 \, d^{15} e^{4} x^{4} + 3920 \, d^{16} e^{3} x^{3} + 2880 \, d^{17} e^{2} x^{2} + 960 \, d^{18} e x + 128 \, d^{19} +{\left (d^{4} e^{14} x^{14} + 15 \, d^{5} e^{13} x^{13} + 53 \, d^{6} e^{12} x^{12} - 45 \, d^{7} e^{11} x^{11} - 669 \, d^{8} e^{10} x^{10} - 1467 \, d^{9} e^{9} x^{9} - 505 \, d^{10} e^{8} x^{8} + 3009 \, d^{11} e^{7} x^{7} + 5440 \, d^{12} e^{6} x^{6} + 2888 \, d^{13} e^{5} x^{5} - 2208 \, d^{14} e^{4} x^{4} - 4400 \, d^{15} e^{3} x^{3} - 2944 \, d^{16} e^{2} x^{2} - 960 \, d^{17} e x - 128 \, d^{18}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(7/2)/(e*x + d)^12,x, algorithm="fricas")

[Out]

-1/6435*(546*e^14*x^15 - 30*d*e^13*x^14 - 32985*d^2*e^12*x^13 - 153335*d^3*e^11*
x^12 - 177840*d^4*e^10*x^11 + 471900*d^5*e^9*x^10 + 1588015*d^6*e^8*x^9 + 151222
5*d^7*e^7*x^8 - 875160*d^8*e^6*x^7 - 4135560*d^9*e^5*x^6 - 3171168*d^10*e^4*x^5
+ 1132560*d^11*e^3*x^4 + 1921920*d^12*e^2*x^3 + 1235520*d^13*e*x^2 + 823680*d^14
*x + (550*e^13*x^14 + 8220*d*e^12*x^13 + 28925*d^2*e^11*x^12 - 25220*d^3*e^10*x^
11 - 368940*d^4*e^9*x^10 - 787072*d^5*e^8*x^9 - 255255*d^6*e^7*x^8 + 1482624*d^7
*e^6*x^7 + 3105960*d^8*e^5*x^6 + 1901328*d^9*e^4*x^5 - 1750320*d^10*e^3*x^4 - 23
33760*d^11*e^2*x^3 - 1235520*d^12*e*x^2 - 823680*d^13*x)*sqrt(-e^2*x^2 + d^2))/(
d^4*e^15*x^15 - 60*d^6*e^13*x^13 - 280*d^7*e^12*x^12 - 330*d^8*e^11*x^11 + 840*d
^9*e^10*x^10 + 3020*d^10*e^9*x^9 + 2760*d^11*e^8*x^8 - 2175*d^12*e^7*x^7 - 6920*
d^13*e^6*x^6 - 5208*d^14*e^5*x^5 + 720*d^15*e^4*x^4 + 3920*d^16*e^3*x^3 + 2880*d
^17*e^2*x^2 + 960*d^18*e*x + 128*d^19 + (d^4*e^14*x^14 + 15*d^5*e^13*x^13 + 53*d
^6*e^12*x^12 - 45*d^7*e^11*x^11 - 669*d^8*e^10*x^10 - 1467*d^9*e^9*x^9 - 505*d^1
0*e^8*x^8 + 3009*d^11*e^7*x^7 + 5440*d^12*e^6*x^6 + 2888*d^13*e^5*x^5 - 2208*d^1
4*e^4*x^4 - 4400*d^15*e^3*x^3 - 2944*d^16*e^2*x^2 - 960*d^17*e*x - 128*d^18)*sqr
t(-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)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.53753, 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)^12,x, algorithm="giac")

[Out]

Done