3.212 \(\int \frac{x^3}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=209 \[ \frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(-24*x)/(5005*d^3*e^3*(d^2 - e^2*x^2)^(5/2)) + d^2/(13*e^4*(d + e*x)^4*(d^2 - e^
2*x^2)^(5/2)) - (30*d)/(143*e^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) + 21/(143*e^4
*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) + 4/(1001*d*e^4*(d + e*x)*(d^2 - e^2*x^2)^(5
/2)) - (32*x)/(5005*d^5*e^3*(d^2 - e^2*x^2)^(3/2)) - (64*x)/(5005*d^7*e^3*Sqrt[d
^2 - e^2*x^2])

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Rubi [A]  time = 0.571417, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{d^2}{13 e^4 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{30 d}{143 e^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{21}{143 e^4 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{4}{1001 d e^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}-\frac{64 x}{5005 d^7 e^3 \sqrt{d^2-e^2 x^2}}-\frac{32 x}{5005 d^5 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{24 x}{5005 d^3 e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(-24*x)/(5005*d^3*e^3*(d^2 - e^2*x^2)^(5/2)) + d^2/(13*e^4*(d + e*x)^4*(d^2 - e^
2*x^2)^(5/2)) - (30*d)/(143*e^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2)) + 21/(143*e^4
*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) + 4/(1001*d*e^4*(d + e*x)*(d^2 - e^2*x^2)^(5
/2)) - (32*x)/(5005*d^5*e^3*(d^2 - e^2*x^2)^(3/2)) - (64*x)/(5005*d^7*e^3*Sqrt[d
^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 73.4009, size = 187, normalized size = 0.89 \[ \frac{d^{2}}{13 e^{4} \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{30 d}{143 e^{4} \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{21}{143 e^{4} \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{4}{1001 d e^{4} \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{24 x}{5005 d^{3} e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{32 x}{5005 d^{5} e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{64 x}{5005 d^{7} e^{3} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**2/(13*e**4*(d + e*x)**4*(d**2 - e**2*x**2)**(5/2)) - 30*d/(143*e**4*(d + e*x)
**3*(d**2 - e**2*x**2)**(5/2)) + 21/(143*e**4*(d + e*x)**2*(d**2 - e**2*x**2)**(
5/2)) + 4/(1001*d*e**4*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) - 24*x/(5005*d**3*e*
*3*(d**2 - e**2*x**2)**(5/2)) - 32*x/(5005*d**5*e**3*(d**2 - e**2*x**2)**(3/2))
- 64*x/(5005*d**7*e**3*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.110981, size = 137, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (90 d^9+360 d^8 e x+315 d^7 e^2 x^2-540 d^6 e^3 x^3+160 d^5 e^4 x^4+776 d^4 e^5 x^5+384 d^3 e^6 x^6-224 d^2 e^7 x^7-256 d e^8 x^8-64 e^9 x^9\right )}{5005 d^7 e^4 (d-e x)^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(90*d^9 + 360*d^8*e*x + 315*d^7*e^2*x^2 - 540*d^6*e^3*x^3 +
160*d^5*e^4*x^4 + 776*d^4*e^5*x^5 + 384*d^3*e^6*x^6 - 224*d^2*e^7*x^7 - 256*d*e
^8*x^8 - 64*e^9*x^9))/(5005*d^7*e^4*(d - e*x)^3*(d + e*x)^7)

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Maple [A]  time = 0.015, size = 132, normalized size = 0.6 \[{\frac{ \left ( -ex+d \right ) \left ( -64\,{e}^{9}{x}^{9}-256\,{e}^{8}{x}^{8}d-224\,{e}^{7}{x}^{7}{d}^{2}+384\,{e}^{6}{x}^{6}{d}^{3}+776\,{e}^{5}{x}^{5}{d}^{4}+160\,{x}^{4}{d}^{5}{e}^{4}-540\,{x}^{3}{d}^{6}{e}^{3}+315\,{x}^{2}{d}^{7}{e}^{2}+360\,{d}^{8}xe+90\,{d}^{9} \right ) }{5005\,{e}^{4}{d}^{7} \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/5005*(-e*x+d)*(-64*e^9*x^9-256*d*e^8*x^8-224*d^2*e^7*x^7+384*d^3*e^6*x^6+776*d
^4*e^5*x^5+160*d^5*e^4*x^4-540*d^6*e^3*x^3+315*d^7*e^2*x^2+360*d^8*e*x+90*d^9)/(
e*x+d)^3/d^7/e^4/(-e^2*x^2+d^2)^(7/2)

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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(x^3/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.52139, size = 961, normalized size = 4.6 \[ \frac{90 \, e^{14} x^{18} - 216 \, d e^{13} x^{17} - 5634 \, d^{2} e^{12} x^{16} - 9456 \, d^{3} e^{11} x^{15} + 43716 \, d^{4} e^{10} x^{14} + 121416 \, d^{5} e^{9} x^{13} - 79092 \, d^{6} e^{8} x^{12} - 491244 \, d^{7} e^{7} x^{11} - 154011 \, d^{8} e^{6} x^{10} + 901472 \, d^{9} e^{5} x^{9} + 692120 \, d^{10} e^{4} x^{8} - 777920 \, d^{11} e^{3} x^{7} - 816816 \, d^{12} e^{2} x^{6} + 256256 \, d^{13} e x^{5} + 320320 \, d^{14} x^{4} +{\left (64 \, e^{13} x^{17} + 1066 \, d e^{12} x^{16} + 904 \, d^{2} e^{11} x^{15} - 18184 \, d^{3} e^{10} x^{14} - 40816 \, d^{4} e^{9} x^{13} + 64220 \, d^{5} e^{8} x^{12} + 252148 \, d^{6} e^{7} x^{11} + 14157 \, d^{7} e^{6} x^{10} - 608608 \, d^{8} e^{5} x^{9} - 403832 \, d^{9} e^{4} x^{8} + 649792 \, d^{10} e^{3} x^{7} + 656656 \, d^{11} e^{2} x^{6} - 256256 \, d^{12} e x^{5} - 320320 \, d^{13} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5005 \,{\left (d^{7} e^{18} x^{18} + 4 \, d^{8} e^{17} x^{17} - 37 \, d^{9} e^{16} x^{16} - 168 \, d^{10} e^{15} x^{15} + 106 \, d^{11} e^{14} x^{14} + 1280 \, d^{12} e^{13} x^{13} + 846 \, d^{13} e^{12} x^{12} - 3704 \, d^{14} e^{11} x^{11} - 5011 \, d^{15} e^{10} x^{10} + 4284 \, d^{16} e^{9} x^{9} + 10519 \, d^{17} e^{8} x^{8} + 32 \, d^{18} e^{7} x^{7} - 10536 \, d^{19} e^{6} x^{6} - 4544 \, d^{20} e^{5} x^{5} + 4688 \, d^{21} e^{4} x^{4} + 3840 \, d^{22} e^{3} x^{3} - 320 \, d^{23} e^{2} x^{2} - 1024 \, d^{24} e x - 256 \, d^{25} +{\left (9 \, d^{8} e^{16} x^{16} + 36 \, d^{9} e^{15} x^{15} - 84 \, d^{10} e^{14} x^{14} - 516 \, d^{11} e^{13} x^{13} - 138 \, d^{12} e^{12} x^{12} + 2172 \, d^{13} e^{11} x^{11} + 2388 \, d^{14} e^{10} x^{10} - 3516 \, d^{15} e^{9} x^{9} - 6839 \, d^{16} e^{8} x^{8} + 1120 \, d^{17} e^{7} x^{7} + 8392 \, d^{18} e^{6} x^{6} + 3008 \, d^{19} e^{5} x^{5} - 4432 \, d^{20} e^{4} x^{4} - 3328 \, d^{21} e^{3} x^{3} + 448 \, d^{22} e^{2} x^{2} + 1024 \, d^{23} e x + 256 \, d^{24}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

1/5005*(90*e^14*x^18 - 216*d*e^13*x^17 - 5634*d^2*e^12*x^16 - 9456*d^3*e^11*x^15
+ 43716*d^4*e^10*x^14 + 121416*d^5*e^9*x^13 - 79092*d^6*e^8*x^12 - 491244*d^7*e
^7*x^11 - 154011*d^8*e^6*x^10 + 901472*d^9*e^5*x^9 + 692120*d^10*e^4*x^8 - 77792
0*d^11*e^3*x^7 - 816816*d^12*e^2*x^6 + 256256*d^13*e*x^5 + 320320*d^14*x^4 + (64
*e^13*x^17 + 1066*d*e^12*x^16 + 904*d^2*e^11*x^15 - 18184*d^3*e^10*x^14 - 40816*
d^4*e^9*x^13 + 64220*d^5*e^8*x^12 + 252148*d^6*e^7*x^11 + 14157*d^7*e^6*x^10 - 6
08608*d^8*e^5*x^9 - 403832*d^9*e^4*x^8 + 649792*d^10*e^3*x^7 + 656656*d^11*e^2*x
^6 - 256256*d^12*e*x^5 - 320320*d^13*x^4)*sqrt(-e^2*x^2 + d^2))/(d^7*e^18*x^18 +
4*d^8*e^17*x^17 - 37*d^9*e^16*x^16 - 168*d^10*e^15*x^15 + 106*d^11*e^14*x^14 +
1280*d^12*e^13*x^13 + 846*d^13*e^12*x^12 - 3704*d^14*e^11*x^11 - 5011*d^15*e^10*
x^10 + 4284*d^16*e^9*x^9 + 10519*d^17*e^8*x^8 + 32*d^18*e^7*x^7 - 10536*d^19*e^6
*x^6 - 4544*d^20*e^5*x^5 + 4688*d^21*e^4*x^4 + 3840*d^22*e^3*x^3 - 320*d^23*e^2*
x^2 - 1024*d^24*e*x - 256*d^25 + (9*d^8*e^16*x^16 + 36*d^9*e^15*x^15 - 84*d^10*e
^14*x^14 - 516*d^11*e^13*x^13 - 138*d^12*e^12*x^12 + 2172*d^13*e^11*x^11 + 2388*
d^14*e^10*x^10 - 3516*d^15*e^9*x^9 - 6839*d^16*e^8*x^8 + 1120*d^17*e^7*x^7 + 839
2*d^18*e^6*x^6 + 3008*d^19*e^5*x^5 - 4432*d^20*e^4*x^4 - 3328*d^21*e^3*x^3 + 448
*d^22*e^2*x^2 + 1024*d^23*e*x + 256*d^24)*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(x**3/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]