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

Optimal. Leaf size=182 \[ \frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*e^2*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d + 31*e*x))/(15*d^4*
(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d + 107*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - S
qrt[d^2 - e^2*x^2]/(2*d^5*x^2) - (3*e*Sqrt[d^2 - e^2*x^2])/(d^6*x) - (13*e^2*Arc
Tanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^6)

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Rubi [A]  time = 0.563133, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (90 d+107 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{3 e \sqrt{d^2-e^2 x^2}}{d^6 x}-\frac{13 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^6}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^5 x^2}+\frac{e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*e^2*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d + 31*e*x))/(15*d^4*
(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d + 107*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - S
qrt[d^2 - e^2*x^2]/(2*d^5*x^2) - (3*e*Sqrt[d^2 - e^2*x^2])/(d^6*x) - (13*e^2*Arc
Tanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^6)

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Rubi in Sympy [A]  time = 65.8072, size = 158, normalized size = 0.87 \[ \frac{e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} \left (d - e x\right )^{3}} + \frac{17 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{5} \left (d - e x\right )^{2}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{2 d^{5} x^{2}} - \frac{13 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{6}} + \frac{107 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{6} \left (d - e x\right )} - \frac{3 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*sqrt(d**2 - e**2*x**2)/(5*d**4*(d - e*x)**3) + 17*e**2*sqrt(d**2 - e**2*x**
2)/(15*d**5*(d - e*x)**2) - sqrt(d**2 - e**2*x**2)/(2*d**5*x**2) - 13*e**2*atanh
(sqrt(d**2 - e**2*x**2)/d)/(2*d**6) + 107*e**2*sqrt(d**2 - e**2*x**2)/(15*d**6*(
d - e*x)) - 3*e*sqrt(d**2 - e**2*x**2)/(d**6*x)

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Mathematica [A]  time = 0.182114, size = 109, normalized size = 0.6 \[ \frac{-195 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^4+45 d^3 e x-479 d^2 e^2 x^2+717 d e^3 x^3-304 e^4 x^4\right )}{x^2 (e x-d)^3}+195 e^2 \log (x)}{30 d^6} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(15*d^4 + 45*d^3*e*x - 479*d^2*e^2*x^2 + 717*d*e^3*x^3 - 3
04*e^4*x^4))/(x^2*(-d + e*x)^3) + 195*e^2*Log[x] - 195*e^2*Log[d + Sqrt[d^2 - e^
2*x^2]])/(30*d^6)

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Maple [A]  time = 0.018, size = 222, normalized size = 1.2 \[{\frac{19\,{e}^{3}x}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{76\,{e}^{3}x}{15\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{152\,{e}^{3}x}{15\,{d}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{d}{2\,{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{13\,{e}^{2}}{10\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{13\,{e}^{2}}{6\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{13\,{e}^{2}}{2\,{d}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{13\,{e}^{2}}{2\,{d}^{5}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-3\,{\frac{e}{x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19/5*e^3*x/d^2/(-e^2*x^2+d^2)^(5/2)+76/15*e^3/d^4*x/(-e^2*x^2+d^2)^(3/2)+152/15*
e^3/d^6*x/(-e^2*x^2+d^2)^(1/2)-1/2*d/x^2/(-e^2*x^2+d^2)^(5/2)+13/10/d*e^2/(-e^2*
x^2+d^2)^(5/2)+13/6/d^3*e^2/(-e^2*x^2+d^2)^(3/2)+13/2/d^5*e^2/(-e^2*x^2+d^2)^(1/
2)-13/2/d^5*e^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-3*e
/x/(-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((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.301746, size = 765, normalized size = 4.2 \[ \frac{558 \, e^{9} x^{9} - 975 \, d e^{8} x^{8} - 4776 \, d^{2} e^{7} x^{7} + 9880 \, d^{3} e^{6} x^{6} + 2540 \, d^{4} e^{5} x^{5} - 13215 \, d^{5} e^{4} x^{4} + 3540 \, d^{6} e^{3} x^{3} + 3540 \, d^{7} e^{2} x^{2} - 840 \, d^{8} e x - 240 \, d^{9} + 195 \,{\left (e^{9} x^{9} - 7 \, d e^{8} x^{8} + 3 \, d^{2} e^{7} x^{7} + 31 \, d^{3} e^{6} x^{6} - 40 \, d^{4} e^{5} x^{5} - 12 \, d^{5} e^{4} x^{4} + 40 \, d^{6} e^{3} x^{3} - 16 \, d^{7} e^{2} x^{2} +{\left (e^{8} x^{8} + 2 \, d e^{7} x^{7} - 19 \, d^{2} e^{6} x^{6} + 20 \, d^{3} e^{5} x^{5} + 20 \, d^{4} e^{4} x^{4} - 40 \, d^{5} e^{3} x^{3} + 16 \, d^{6} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (50 \, e^{8} x^{8} - 2441 \, d e^{7} x^{7} + 4525 \, d^{2} e^{6} x^{6} + 3995 \, d^{3} e^{5} x^{5} - 11535 \, d^{4} e^{4} x^{4} + 3120 \, d^{5} e^{3} x^{3} + 3420 \, d^{6} e^{2} x^{2} - 840 \, d^{7} e x - 240 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{6} e^{7} x^{9} - 7 \, d^{7} e^{6} x^{8} + 3 \, d^{8} e^{5} x^{7} + 31 \, d^{9} e^{4} x^{6} - 40 \, d^{10} e^{3} x^{5} - 12 \, d^{11} e^{2} x^{4} + 40 \, d^{12} e x^{3} - 16 \, d^{13} x^{2} +{\left (d^{6} e^{6} x^{8} + 2 \, d^{7} e^{5} x^{7} - 19 \, d^{8} e^{4} x^{6} + 20 \, d^{9} e^{3} x^{5} + 20 \, d^{10} e^{2} x^{4} - 40 \, d^{11} e x^{3} + 16 \, d^{12} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(558*e^9*x^9 - 975*d*e^8*x^8 - 4776*d^2*e^7*x^7 + 9880*d^3*e^6*x^6 + 2540*d
^4*e^5*x^5 - 13215*d^5*e^4*x^4 + 3540*d^6*e^3*x^3 + 3540*d^7*e^2*x^2 - 840*d^8*e
*x - 240*d^9 + 195*(e^9*x^9 - 7*d*e^8*x^8 + 3*d^2*e^7*x^7 + 31*d^3*e^6*x^6 - 40*
d^4*e^5*x^5 - 12*d^5*e^4*x^4 + 40*d^6*e^3*x^3 - 16*d^7*e^2*x^2 + (e^8*x^8 + 2*d*
e^7*x^7 - 19*d^2*e^6*x^6 + 20*d^3*e^5*x^5 + 20*d^4*e^4*x^4 - 40*d^5*e^3*x^3 + 16
*d^6*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (50*e^8
*x^8 - 2441*d*e^7*x^7 + 4525*d^2*e^6*x^6 + 3995*d^3*e^5*x^5 - 11535*d^4*e^4*x^4
+ 3120*d^5*e^3*x^3 + 3420*d^6*e^2*x^2 - 840*d^7*e*x - 240*d^8)*sqrt(-e^2*x^2 + d
^2))/(d^6*e^7*x^9 - 7*d^7*e^6*x^8 + 3*d^8*e^5*x^7 + 31*d^9*e^4*x^6 - 40*d^10*e^3
*x^5 - 12*d^11*e^2*x^4 + 40*d^12*e*x^3 - 16*d^13*x^2 + (d^6*e^6*x^8 + 2*d^7*e^5*
x^7 - 19*d^8*e^4*x^6 + 20*d^9*e^3*x^5 + 20*d^10*e^2*x^4 - 40*d^11*e*x^3 + 16*d^1
2*x^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3/(x**3*(-(-d + e*x)*(d + e*x))**(7/2)), x)

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GIAC/XCAS [A]  time = 0.310203, size = 350, normalized size = 1.92 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{107 \, x e^{7}}{d^{6}} + \frac{90 \, e^{6}}{d^{5}}\right )} - \frac{245 \, e^{5}}{d^{4}}\right )} x - \frac{205 \, e^{4}}{d^{3}}\right )} x + \frac{150 \, e^{3}}{d^{2}}\right )} x + \frac{127 \, e^{2}}{d}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{13 \, e^{2}{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{2 \, d^{6}} + \frac{x^{2}{\left (\frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6}} - \frac{{\left (\frac{12 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{6} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{6} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((((x*(107*x*e^7/d^6 + 90*e^6/d^5) - 245*e^5/d^4)*x -
205*e^4/d^3)*x + 150*e^3/d^2)*x + 127*e^2/d)/(x^2*e^2 - d^2)^3 - 13/2*e^2*ln(1/
2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^6 + 1/8*x^2*(12*(d*e +
sqrt(-x^2*e^2 + d^2)*e)*e^4/x + e^6)/((d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^6) - 1
/8*(12*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^6*e^8/x + (d*e + sqrt(-x^2*e^2 + d^2)*e)
^2*d^6*e^6/x^2)*e^(-8)/d^12