3.834 \(\int \frac{1}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(8*x)/(99*d^6*(d^2 - e^2*x^2)^(3/2)) - 1/(11*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(3/
2)) - 7/(99*d^2*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^3*e*(d + e*x)^2*(
d^2 - e^2*x^2)^(3/2)) - 2/(33*d^4*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(9
9*d^8*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.221239, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{7}{99 d^2 e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{11 d e (d+e x)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{99 d^8 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{99 d^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^4 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{33 d^3 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(99*d^6*(d^2 - e^2*x^2)^(3/2)) - 1/(11*d*e*(d + e*x)^4*(d^2 - e^2*x^2)^(3/
2)) - 7/(99*d^2*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2)) - 2/(33*d^3*e*(d + e*x)^2*(
d^2 - e^2*x^2)^(3/2)) - 2/(33*d^4*e*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(9
9*d^8*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 24.7539, size = 155, normalized size = 0.86 \[ - \frac{1}{11 d e \left (d + e x\right )^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{7}{99 d^{2} e \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2}{33 d^{3} e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2}{33 d^{4} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{8 x}{99 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{16 x}{99 d^{8} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)

[Out]

-1/(11*d*e*(d + e*x)**4*(d**2 - e**2*x**2)**(3/2)) - 7/(99*d**2*e*(d + e*x)**3*(
d**2 - e**2*x**2)**(3/2)) - 2/(33*d**3*e*(d + e*x)**2*(d**2 - e**2*x**2)**(3/2))
- 2/(33*d**4*e*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) + 8*x/(99*d**6*(d**2 - e**2
*x**2)**(3/2)) + 16*x/(99*d**8*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0910204, size = 115, normalized size = 0.64 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (28 d^7+13 d^6 e x-72 d^5 e^2 x^2-122 d^4 e^3 x^3-32 d^3 e^4 x^4+72 d^2 e^5 x^5+64 d e^6 x^6+16 e^7 x^7\right )}{99 d^8 e (d-e x)^2 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(28*d^7 + 13*d^6*e*x - 72*d^5*e^2*x^2 - 122*d^4*e^3*x^3 -
32*d^3*e^4*x^4 + 72*d^2*e^5*x^5 + 64*d*e^6*x^6 + 16*e^7*x^7))/(99*d^8*e*(d - e*x
)^2*(d + e*x)^6)

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Maple [A]  time = 0.015, size = 110, normalized size = 0.6 \[ -{\frac{ \left ( -ex+d \right ) \left ( 16\,{e}^{7}{x}^{7}+64\,{e}^{6}{x}^{6}d+72\,{e}^{5}{x}^{5}{d}^{2}-32\,{e}^{4}{x}^{4}{d}^{3}-122\,{e}^{3}{x}^{3}{d}^{4}-72\,{e}^{2}{x}^{2}{d}^{5}+13\,x{d}^{6}e+28\,{d}^{7} \right ) }{99\,e{d}^{8} \left ( ex+d \right ) ^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/99*(-e*x+d)*(16*e^7*x^7+64*d*e^6*x^6+72*d^2*e^5*x^5-32*d^3*e^4*x^4-122*d^4*e^
3*x^3-72*d^5*e^2*x^2+13*d^6*e*x+28*d^7)/(e*x+d)^3/d^8/e/(-e^2*x^2+d^2)^(5/2)

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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(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.309366, size = 795, normalized size = 4.39 \[ -\frac{28 \, e^{13} x^{14} - 1008 \, d^{2} e^{11} x^{12} - 2408 \, d^{3} e^{10} x^{11} + 3080 \, d^{4} e^{9} x^{10} + 14630 \, d^{5} e^{8} x^{9} + 5544 \, d^{6} e^{7} x^{8} - 27291 \, d^{7} e^{6} x^{7} - 28776 \, d^{8} e^{5} x^{6} + 14520 \, d^{9} e^{4} x^{5} + 33792 \, d^{10} e^{3} x^{4} + 6864 \, d^{11} e^{2} x^{3} - 12672 \, d^{12} e x^{2} - 6336 \, d^{13} x +{\left (16 \, e^{12} x^{13} + 260 \, d e^{11} x^{12} + 472 \, d^{2} e^{10} x^{11} - 2156 \, d^{3} e^{9} x^{10} - 6842 \, d^{4} e^{8} x^{9} + 132 \, d^{5} e^{7} x^{8} + 19437 \, d^{6} e^{6} x^{7} + 16632 \, d^{7} e^{5} x^{6} - 15576 \, d^{8} e^{4} x^{5} - 27456 \, d^{9} e^{3} x^{4} - 3696 \, d^{10} e^{2} x^{3} + 12672 \, d^{11} e x^{2} + 6336 \, d^{12} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{99 \,{\left (d^{8} e^{14} x^{14} + 4 \, d^{9} e^{13} x^{13} - 20 \, d^{10} e^{12} x^{12} - 100 \, d^{11} e^{11} x^{11} - 26 \, d^{12} e^{10} x^{10} + 412 \, d^{13} e^{9} x^{9} + 500 \, d^{14} e^{8} x^{8} - 476 \, d^{15} e^{7} x^{7} - 1151 \, d^{16} e^{6} x^{6} - 160 \, d^{17} e^{5} x^{5} + 936 \, d^{18} e^{4} x^{4} + 576 \, d^{19} e^{3} x^{3} - 176 \, d^{20} e^{2} x^{2} - 256 \, d^{21} e x - 64 \, d^{22} +{\left (7 \, d^{9} e^{12} x^{12} + 28 \, d^{10} e^{11} x^{11} - 21 \, d^{11} e^{10} x^{10} - 224 \, d^{12} e^{9} x^{9} - 203 \, d^{13} e^{8} x^{8} + 420 \, d^{14} e^{7} x^{7} + 769 \, d^{15} e^{6} x^{6} - 32 \, d^{16} e^{5} x^{5} - 824 \, d^{17} e^{4} x^{4} - 448 \, d^{18} e^{3} x^{3} + 208 \, d^{19} e^{2} x^{2} + 256 \, d^{20} e x + 64 \, d^{21}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-1/99*(28*e^13*x^14 - 1008*d^2*e^11*x^12 - 2408*d^3*e^10*x^11 + 3080*d^4*e^9*x^1
0 + 14630*d^5*e^8*x^9 + 5544*d^6*e^7*x^8 - 27291*d^7*e^6*x^7 - 28776*d^8*e^5*x^6
+ 14520*d^9*e^4*x^5 + 33792*d^10*e^3*x^4 + 6864*d^11*e^2*x^3 - 12672*d^12*e*x^2
- 6336*d^13*x + (16*e^12*x^13 + 260*d*e^11*x^12 + 472*d^2*e^10*x^11 - 2156*d^3*
e^9*x^10 - 6842*d^4*e^8*x^9 + 132*d^5*e^7*x^8 + 19437*d^6*e^6*x^7 + 16632*d^7*e^
5*x^6 - 15576*d^8*e^4*x^5 - 27456*d^9*e^3*x^4 - 3696*d^10*e^2*x^3 + 12672*d^11*e
*x^2 + 6336*d^12*x)*sqrt(-e^2*x^2 + d^2))/(d^8*e^14*x^14 + 4*d^9*e^13*x^13 - 20*
d^10*e^12*x^12 - 100*d^11*e^11*x^11 - 26*d^12*e^10*x^10 + 412*d^13*e^9*x^9 + 500
*d^14*e^8*x^8 - 476*d^15*e^7*x^7 - 1151*d^16*e^6*x^6 - 160*d^17*e^5*x^5 + 936*d^
18*e^4*x^4 + 576*d^19*e^3*x^3 - 176*d^20*e^2*x^2 - 256*d^21*e*x - 64*d^22 + (7*d
^9*e^12*x^12 + 28*d^10*e^11*x^11 - 21*d^11*e^10*x^10 - 224*d^12*e^9*x^9 - 203*d^
13*e^8*x^8 + 420*d^14*e^7*x^7 + 769*d^15*e^6*x^6 - 32*d^16*e^5*x^5 - 824*d^17*e^
4*x^4 - 448*d^18*e^3*x^3 + 208*d^19*e^2*x^2 + 256*d^20*e*x + 64*d^21)*sqrt(-e^2*
x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

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