3.844 \(\int \frac{1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 x}{35 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(6*x)/(35*d^3*(d^2 - e^2*x^2)^(5/2)) - 1/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2))
+ (8*x)/(35*d^5*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(35*d^7*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0861359, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac{16 x}{35 d^7 \sqrt{d^2-e^2 x^2}}+\frac{8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(6*x)/(35*d^3*(d^2 - e^2*x^2)^(5/2)) - 1/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2))
+ (8*x)/(35*d^5*(d^2 - e^2*x^2)^(3/2)) + (16*x)/(35*d^7*Sqrt[d^2 - e^2*x^2])

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Rubi in Sympy [A]  time = 10.3103, size = 90, normalized size = 0.85 \[ - \frac{1}{7 d e \left (d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{6 x}{35 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{8 x}{35 d^{5} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{16 x}{35 d^{7} \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

-1/(7*d*e*(d + e*x)*(d**2 - e**2*x**2)**(5/2)) + 6*x/(35*d**3*(d**2 - e**2*x**2)
**(5/2)) + 8*x/(35*d**5*(d**2 - e**2*x**2)**(3/2)) + 16*x/(35*d**7*sqrt(d**2 - e
**2*x**2))

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Mathematica [A]  time = 0.0862658, size = 104, normalized size = 0.98 \[ \frac{\sqrt{d^2-e^2 x^2} \left (-5 d^6+30 d^5 e x+30 d^4 e^2 x^2-40 d^3 e^3 x^3-40 d^2 e^4 x^4+16 d e^5 x^5+16 e^6 x^6\right )}{35 d^7 e (d-e x)^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5*d^6 + 30*d^5*e*x + 30*d^4*e^2*x^2 - 40*d^3*e^3*x^3 - 40
*d^2*e^4*x^4 + 16*d*e^5*x^5 + 16*e^6*x^6))/(35*d^7*e*(d - e*x)^3*(d + e*x)^4)

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Maple [A]  time = 0.013, size = 92, normalized size = 0.9 \[ -{\frac{ \left ( -ex+d \right ) \left ( -16\,{e}^{6}{x}^{6}-16\,{e}^{5}{x}^{5}d+40\,{e}^{4}{x}^{4}{d}^{2}+40\,{e}^{3}{x}^{3}{d}^{3}-30\,{e}^{2}{x}^{2}{d}^{4}-30\,x{d}^{5}e+5\,{d}^{6} \right ) }{35\,{d}^{7}e} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/35*(-e*x+d)*(-16*e^6*x^6-16*d*e^5*x^5+40*d^2*e^4*x^4+40*d^3*e^3*x^3-30*d^4*e^
2*x^2-30*d^5*e*x+5*d^6)/d^7/e/(-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(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284305, size = 690, normalized size = 6.51 \[ -\frac{16 \, e^{11} x^{12} + 46 \, d e^{10} x^{11} - 298 \, d^{2} e^{9} x^{10} - 578 \, d^{3} e^{8} x^{9} + 1268 \, d^{4} e^{7} x^{8} + 2248 \, d^{5} e^{6} x^{7} - 2247 \, d^{6} e^{5} x^{6} - 3962 \, d^{7} e^{4} x^{5} + 1820 \, d^{8} e^{3} x^{4} + 3360 \, d^{9} e^{2} x^{3} - 560 \, d^{10} e x^{2} - 1120 \, d^{11} x -{\left (5 \, e^{10} x^{11} - 91 \, d e^{9} x^{10} - 196 \, d^{2} e^{8} x^{9} + 652 \, d^{3} e^{7} x^{8} + 1177 \, d^{4} e^{6} x^{7} - 1547 \, d^{5} e^{5} x^{6} - 2702 \, d^{6} e^{4} x^{5} + 1540 \, d^{7} e^{3} x^{4} + 2800 \, d^{8} e^{2} x^{3} - 560 \, d^{9} e x^{2} - 1120 \, d^{10} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{35 \,{\left (6 \, d^{8} e^{11} x^{11} + 6 \, d^{9} e^{10} x^{10} - 50 \, d^{10} e^{9} x^{9} - 50 \, d^{11} e^{8} x^{8} + 146 \, d^{12} e^{7} x^{7} + 146 \, d^{13} e^{6} x^{6} - 198 \, d^{14} e^{5} x^{5} - 198 \, d^{15} e^{4} x^{4} + 128 \, d^{16} e^{3} x^{3} + 128 \, d^{17} e^{2} x^{2} - 32 \, d^{18} e x - 32 \, d^{19} -{\left (d^{7} e^{11} x^{11} + d^{8} e^{10} x^{10} - 20 \, d^{9} e^{9} x^{9} - 20 \, d^{10} e^{8} x^{8} + 85 \, d^{11} e^{7} x^{7} + 85 \, d^{12} e^{6} x^{6} - 146 \, d^{13} e^{5} x^{5} - 146 \, d^{14} e^{4} x^{4} + 112 \, d^{15} e^{3} x^{3} + 112 \, d^{16} e^{2} x^{2} - 32 \, d^{17} e x - 32 \, d^{18}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)),x, algorithm="fricas")

[Out]

-1/35*(16*e^11*x^12 + 46*d*e^10*x^11 - 298*d^2*e^9*x^10 - 578*d^3*e^8*x^9 + 1268
*d^4*e^7*x^8 + 2248*d^5*e^6*x^7 - 2247*d^6*e^5*x^6 - 3962*d^7*e^4*x^5 + 1820*d^8
*e^3*x^4 + 3360*d^9*e^2*x^3 - 560*d^10*e*x^2 - 1120*d^11*x - (5*e^10*x^11 - 91*d
*e^9*x^10 - 196*d^2*e^8*x^9 + 652*d^3*e^7*x^8 + 1177*d^4*e^6*x^7 - 1547*d^5*e^5*
x^6 - 2702*d^6*e^4*x^5 + 1540*d^7*e^3*x^4 + 2800*d^8*e^2*x^3 - 560*d^9*e*x^2 - 1
120*d^10*x)*sqrt(-e^2*x^2 + d^2))/(6*d^8*e^11*x^11 + 6*d^9*e^10*x^10 - 50*d^10*e
^9*x^9 - 50*d^11*e^8*x^8 + 146*d^12*e^7*x^7 + 146*d^13*e^6*x^6 - 198*d^14*e^5*x^
5 - 198*d^15*e^4*x^4 + 128*d^16*e^3*x^3 + 128*d^17*e^2*x^2 - 32*d^18*e*x - 32*d^
19 - (d^7*e^11*x^11 + d^8*e^10*x^10 - 20*d^9*e^9*x^9 - 20*d^10*e^8*x^8 + 85*d^11
*e^7*x^7 + 85*d^12*e^6*x^6 - 146*d^13*e^5*x^5 - 146*d^14*e^4*x^4 + 112*d^15*e^3*
x^3 + 112*d^16*e^2*x^2 - 32*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. \[ \int \frac{1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/((-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)),x, algorithm="giac")

[Out]

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