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

Optimal. Leaf size=186 \[ \frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(7*d - 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^2*(d + e*x)*(d^2 -
e^2*x^2)^(3/2)) + (35*d - 24*e*x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^
2 - e^2*x^2])/(2*d^7*x^2) + (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*x) - (7*e^2*ArcTan
h[Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

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Rubi [A]  time = 0.524456, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(7*d - 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^2*(d + e*x)*(d^2 -
e^2*x^2)^(3/2)) + (35*d - 24*e*x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^
2 - e^2*x^2])/(2*d^7*x^2) + (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*x) - (7*e^2*ArcTan
h[Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

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Rubi in Sympy [A]  time = 65.1641, size = 162, normalized size = 0.87 \[ \frac{d - e x}{5 d^{2} x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{7 d - 6 e x}{15 d^{4} x^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{35 d - 24 e x}{15 d^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{7 \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{7} x^{2}} - \frac{7 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{8}} + \frac{16 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{8} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - e*x)/(5*d**2*x**2*(d**2 - e**2*x**2)**(5/2)) + (7*d - 6*e*x)/(15*d**4*x**2*
(d**2 - e**2*x**2)**(3/2)) + (35*d - 24*e*x)/(15*d**6*x**2*sqrt(d**2 - e**2*x**2
)) - 7*sqrt(d**2 - e**2*x**2)/(2*d**7*x**2) - 7*e**2*atanh(sqrt(d**2 - e**2*x**2
)/d)/(2*d**8) + 16*e*sqrt(d**2 - e**2*x**2)/(5*d**8*x)

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Mathematica [A]  time = 0.151049, size = 137, normalized size = 0.74 \[ \frac{-105 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-15 d^6+15 d^5 e x+176 d^4 e^2 x^2-4 d^3 e^3 x^3-249 d^2 e^4 x^4-9 d e^5 x^5+96 e^6 x^6\right )}{x^2 (d-e x)^2 (d+e x)^3}+105 e^2 \log (x)}{30 d^8} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-15*d^6 + 15*d^5*e*x + 176*d^4*e^2*x^2 - 4*d^3*e^3*x^3 -
249*d^2*e^4*x^4 - 9*d*e^5*x^5 + 96*e^6*x^6))/(x^2*(d - e*x)^2*(d + e*x)^3) + 105
*e^2*Log[x] - 105*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^8)

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Maple [A]  time = 0.023, size = 298, normalized size = 1.6 \[ -{\frac{1}{2\,{d}^{3}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{6\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{2\,{d}^{7}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{7\,{e}^{2}}{2\,{d}^{7}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{e}{5\,{d}^{4}} \left ( x+{\frac{d}{e}} \right ) ^{-1} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{e}^{3}x}{15\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{e}^{3}x}{15\,{d}^{8}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}+{\frac{e}{{d}^{4}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{e}^{3}x}{3\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{e}^{3}x}{3\,{d}^{8}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/d^3/x^2/(-e^2*x^2+d^2)^(3/2)+7/6*e^2/d^5/(-e^2*x^2+d^2)^(3/2)+7/2*e^2/d^7/(
-e^2*x^2+d^2)^(1/2)-7/2*e^2/d^7/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^
2)^(1/2))/x)+1/5/d^4*e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-4/15/d^6*e^3
/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-8/15/d^8*e^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/
e))^(1/2)*x+e/d^4/x/(-e^2*x^2+d^2)^(3/2)-4/3*e^3/d^6*x/(-e^2*x^2+d^2)^(3/2)-8/3*
e^3/d^8*x/(-e^2*x^2+d^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^3), x)

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Fricas [A]  time = 0.330299, size = 1010, normalized size = 5.43 \[ \frac{96 \, e^{12} x^{12} + 687 \, d e^{11} x^{11} - 1281 \, d^{2} e^{10} x^{10} - 4946 \, d^{3} e^{9} x^{9} + 4162 \, d^{4} e^{8} x^{8} + 11487 \, d^{5} e^{7} x^{7} - 6375 \, d^{6} e^{6} x^{6} - 11310 \, d^{7} e^{5} x^{5} + 5550 \, d^{8} e^{4} x^{4} + 4560 \, d^{9} e^{3} x^{3} - 2640 \, d^{10} e^{2} x^{2} - 480 \, d^{11} e x + 480 \, d^{12} + 105 \,{\left (6 \, d e^{11} x^{11} + 6 \, d^{2} e^{10} x^{10} - 44 \, d^{3} e^{9} x^{9} - 44 \, d^{4} e^{8} x^{8} + 102 \, d^{5} e^{7} x^{7} + 102 \, d^{6} e^{6} x^{6} - 96 \, d^{7} e^{5} x^{5} - 96 \, d^{8} e^{4} x^{4} + 32 \, d^{9} e^{3} x^{3} + 32 \, d^{10} e^{2} x^{2} -{\left (e^{11} x^{11} + d e^{10} x^{10} - 19 \, d^{2} e^{9} x^{9} - 19 \, d^{3} e^{8} x^{8} + 66 \, d^{4} e^{7} x^{7} + 66 \, d^{5} e^{6} x^{6} - 80 \, d^{6} e^{5} x^{5} - 80 \, d^{7} e^{4} x^{4} + 32 \, d^{8} e^{3} x^{3} + 32 \, d^{9} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 2 \,{\left (58 \, e^{11} x^{11} - 230 \, d e^{10} x^{10} - 1075 \, d^{2} e^{9} x^{9} + 1181 \, d^{3} e^{8} x^{8} + 3696 \, d^{4} e^{7} x^{7} - 2220 \, d^{5} e^{6} x^{6} - 4605 \, d^{6} e^{5} x^{5} + 2205 \, d^{7} e^{4} x^{4} + 2160 \, d^{8} e^{3} x^{3} - 1200 \, d^{9} e^{2} x^{2} - 240 \, d^{10} e x + 240 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (6 \, d^{9} e^{9} x^{11} + 6 \, d^{10} e^{8} x^{10} - 44 \, d^{11} e^{7} x^{9} - 44 \, d^{12} e^{6} x^{8} + 102 \, d^{13} e^{5} x^{7} + 102 \, d^{14} e^{4} x^{6} - 96 \, d^{15} e^{3} x^{5} - 96 \, d^{16} e^{2} x^{4} + 32 \, d^{17} e x^{3} + 32 \, d^{18} x^{2} -{\left (d^{8} e^{9} x^{11} + d^{9} e^{8} x^{10} - 19 \, d^{10} e^{7} x^{9} - 19 \, d^{11} e^{6} x^{8} + 66 \, d^{12} e^{5} x^{7} + 66 \, d^{13} e^{4} x^{6} - 80 \, d^{14} e^{3} x^{5} - 80 \, d^{15} e^{2} x^{4} + 32 \, d^{16} e x^{3} + 32 \, d^{17} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(96*e^12*x^12 + 687*d*e^11*x^11 - 1281*d^2*e^10*x^10 - 4946*d^3*e^9*x^9 + 4
162*d^4*e^8*x^8 + 11487*d^5*e^7*x^7 - 6375*d^6*e^6*x^6 - 11310*d^7*e^5*x^5 + 555
0*d^8*e^4*x^4 + 4560*d^9*e^3*x^3 - 2640*d^10*e^2*x^2 - 480*d^11*e*x + 480*d^12 +
105*(6*d*e^11*x^11 + 6*d^2*e^10*x^10 - 44*d^3*e^9*x^9 - 44*d^4*e^8*x^8 + 102*d^
5*e^7*x^7 + 102*d^6*e^6*x^6 - 96*d^7*e^5*x^5 - 96*d^8*e^4*x^4 + 32*d^9*e^3*x^3 +
32*d^10*e^2*x^2 - (e^11*x^11 + d*e^10*x^10 - 19*d^2*e^9*x^9 - 19*d^3*e^8*x^8 +
66*d^4*e^7*x^7 + 66*d^5*e^6*x^6 - 80*d^6*e^5*x^5 - 80*d^7*e^4*x^4 + 32*d^8*e^3*x
^3 + 32*d^9*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) -
2*(58*e^11*x^11 - 230*d*e^10*x^10 - 1075*d^2*e^9*x^9 + 1181*d^3*e^8*x^8 + 3696*d
^4*e^7*x^7 - 2220*d^5*e^6*x^6 - 4605*d^6*e^5*x^5 + 2205*d^7*e^4*x^4 + 2160*d^8*e
^3*x^3 - 1200*d^9*e^2*x^2 - 240*d^10*e*x + 240*d^11)*sqrt(-e^2*x^2 + d^2))/(6*d^
9*e^9*x^11 + 6*d^10*e^8*x^10 - 44*d^11*e^7*x^9 - 44*d^12*e^6*x^8 + 102*d^13*e^5*
x^7 + 102*d^14*e^4*x^6 - 96*d^15*e^3*x^5 - 96*d^16*e^2*x^4 + 32*d^17*e*x^3 + 32*
d^18*x^2 - (d^8*e^9*x^11 + d^9*e^8*x^10 - 19*d^10*e^7*x^9 - 19*d^11*e^6*x^8 + 66
*d^12*e^5*x^7 + 66*d^13*e^4*x^6 - 80*d^14*e^3*x^5 - 80*d^15*e^2*x^4 + 32*d^16*e*
x^3 + 32*d^17*x^2)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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