3.833 \(\int \frac{1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(8*x)/(63*d^5*(d^2 - e^2*x^2)^(3/2)) - 1/(9*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2
)) - 2/(21*d^2*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 2/(21*d^3*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (16*x)/(63*d^7*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.16773, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2}{21 d^2 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{1}{9 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 x}{63 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{63 d^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2}{21 d^3 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*x)/(63*d^5*(d^2 - e^2*x^2)^(3/2)) - 1/(9*d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(3/2
)) - 2/(21*d^2*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 2/(21*d^3*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (16*x)/(63*d^7*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 18.2133, size = 126, normalized size = 0.85 \[ - \frac{1}{9 d e \left (d + e x\right )^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2}{21 d^{2} e \left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{2}{21 d^{3} e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{8 x}{63 d^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{16 x}{63 d^{7} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

-1/(9*d*e*(d + e*x)**3*(d**2 - e**2*x**2)**(3/2)) - 2/(21*d**2*e*(d + e*x)**2*(d
**2 - e**2*x**2)**(3/2)) - 2/(21*d**3*e*(d + e*x)*(d**2 - e**2*x**2)**(3/2)) + 8
*x/(63*d**5*(d**2 - e**2*x**2)**(3/2)) + 16*x/(63*d**7*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0822596, size = 104, normalized size = 0.7 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (19 d^6-6 d^5 e x-66 d^4 e^2 x^2-56 d^3 e^3 x^3+24 d^2 e^4 x^4+48 d e^5 x^5+16 e^6 x^6\right )}{63 d^7 e (d-e x)^2 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(19*d^6 - 6*d^5*e*x - 66*d^4*e^2*x^2 - 56*d^3*e^3*x^3 + 24
*d^2*e^4*x^4 + 48*d*e^5*x^5 + 16*e^6*x^6))/(63*d^7*e*(d - e*x)^2*(d + e*x)^5)

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Maple [A]  time = 0.013, size = 99, normalized size = 0.7 \[ -{\frac{ \left ( -ex+d \right ) \left ( 16\,{e}^{6}{x}^{6}+48\,{e}^{5}{x}^{5}d+24\,{e}^{4}{x}^{4}{d}^{2}-56\,{e}^{3}{x}^{3}{d}^{3}-66\,{e}^{2}{x}^{2}{d}^{4}-6\,x{d}^{5}e+19\,{d}^{6} \right ) }{63\,e{d}^{7} \left ( ex+d \right ) ^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/63*(-e*x+d)*(16*e^6*x^6+48*d*e^5*x^5+24*d^2*e^4*x^4-56*d^3*e^3*x^3-66*d^4*e^2
*x^2-6*d^5*e*x+19*d^6)/(e*x+d)^2/d^7/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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281532, size = 676, normalized size = 4.57 \[ -\frac{16 \, e^{11} x^{12} + 162 \, d e^{10} x^{11} + 78 \, d^{2} e^{9} x^{10} - 1414 \, d^{3} e^{8} x^{9} - 2124 \, d^{4} e^{7} x^{8} + 2736 \, d^{5} e^{6} x^{7} + 6825 \, d^{6} e^{5} x^{6} - 126 \, d^{7} e^{4} x^{5} - 7812 \, d^{8} e^{3} x^{4} - 3360 \, d^{9} e^{2} x^{3} + 3024 \, d^{10} e x^{2} + 2016 \, d^{11} x -{\left (19 \, e^{10} x^{11} - 39 \, d e^{9} x^{10} - 592 \, d^{2} e^{8} x^{9} - 696 \, d^{3} e^{7} x^{8} + 2043 \, d^{4} e^{6} x^{7} + 4053 \, d^{5} e^{5} x^{6} - 1050 \, d^{6} e^{4} x^{5} - 6300 \, d^{7} e^{3} x^{4} - 2352 \, d^{8} e^{2} x^{3} + 3024 \, d^{9} e x^{2} + 2016 \, d^{10} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{63 \,{\left (6 \, d^{8} e^{11} x^{11} + 18 \, d^{9} e^{10} x^{10} - 26 \, d^{10} e^{9} x^{9} - 126 \, d^{11} e^{8} x^{8} - 30 \, d^{12} e^{7} x^{7} + 262 \, d^{13} e^{6} x^{6} + 210 \, d^{14} e^{5} x^{5} - 186 \, d^{15} e^{4} x^{4} - 256 \, d^{16} e^{3} x^{3} + 96 \, d^{18} e x + 32 \, d^{19} -{\left (d^{7} e^{11} x^{11} + 3 \, d^{8} e^{10} x^{10} - 16 \, d^{9} e^{9} x^{9} - 56 \, d^{10} e^{8} x^{8} + 9 \, d^{11} e^{7} x^{7} + 179 \, d^{12} e^{6} x^{6} + 118 \, d^{13} e^{5} x^{5} - 174 \, d^{14} e^{4} x^{4} - 208 \, d^{15} e^{3} x^{3} + 16 \, d^{16} e^{2} x^{2} + 96 \, d^{17} e x + 32 \, d^{18}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)^3),x, algorithm="fricas")

[Out]

-1/63*(16*e^11*x^12 + 162*d*e^10*x^11 + 78*d^2*e^9*x^10 - 1414*d^3*e^8*x^9 - 212
4*d^4*e^7*x^8 + 2736*d^5*e^6*x^7 + 6825*d^6*e^5*x^6 - 126*d^7*e^4*x^5 - 7812*d^8
*e^3*x^4 - 3360*d^9*e^2*x^3 + 3024*d^10*e*x^2 + 2016*d^11*x - (19*e^10*x^11 - 39
*d*e^9*x^10 - 592*d^2*e^8*x^9 - 696*d^3*e^7*x^8 + 2043*d^4*e^6*x^7 + 4053*d^5*e^
5*x^6 - 1050*d^6*e^4*x^5 - 6300*d^7*e^3*x^4 - 2352*d^8*e^2*x^3 + 3024*d^9*e*x^2
+ 2016*d^10*x)*sqrt(-e^2*x^2 + d^2))/(6*d^8*e^11*x^11 + 18*d^9*e^10*x^10 - 26*d^
10*e^9*x^9 - 126*d^11*e^8*x^8 - 30*d^12*e^7*x^7 + 262*d^13*e^6*x^6 + 210*d^14*e^
5*x^5 - 186*d^15*e^4*x^4 - 256*d^16*e^3*x^3 + 96*d^18*e*x + 32*d^19 - (d^7*e^11*
x^11 + 3*d^8*e^10*x^10 - 16*d^9*e^9*x^9 - 56*d^10*e^8*x^8 + 9*d^11*e^7*x^7 + 179
*d^12*e^6*x^6 + 118*d^13*e^5*x^5 - 174*d^14*e^4*x^4 - 208*d^15*e^3*x^3 + 16*d^16
*e^2*x^2 + 96*d^17*e*x + 32*d^18)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)**3), 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)^3),x, algorithm="giac")

[Out]

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