3.114 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=143 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2}+\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2} \]

[Out]

(e^4*Sqrt[d^2 - e^2*x^2])/(16*d*x^2) - (e^2*(d^2 - e^2*x^2)^(3/2))/(24*d*x^4) -
(d^2 - e^2*x^2)^(5/2)/(6*d*x^6) + (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5) - (e^6*A
rcTanh[Sqrt[d^2 - e^2*x^2]/d])/(16*d^2)

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Rubi [A]  time = 0.348569, antiderivative size = 143, 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 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2}+\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(e^4*Sqrt[d^2 - e^2*x^2])/(16*d*x^2) - (e^2*(d^2 - e^2*x^2)^(3/2))/(24*d*x^4) -
(d^2 - e^2*x^2)^(5/2)/(6*d*x^6) + (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5) - (e^6*A
rcTanh[Sqrt[d^2 - e^2*x^2]/d])/(16*d^2)

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Rubi in Sympy [A]  time = 37.8971, size = 116, normalized size = 0.81 \[ \frac{e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{16 d x^{2}} - \frac{e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{24 d x^{4}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{6 d x^{6}} - \frac{e^{6} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16 d^{2}} + \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}}{5 d^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**7/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.118481, size = 117, normalized size = 0.82 \[ \frac{-15 e^6 x^6 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (-40 d^5+48 d^4 e x+70 d^3 e^2 x^2-96 d^2 e^3 x^3-15 d e^4 x^4+48 e^5 x^5\right )+15 e^6 x^6 \log (x)}{240 d^2 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-40*d^5 + 48*d^4*e*x + 70*d^3*e^2*x^2 - 96*d^2*e^3*x^3 - 1
5*d*e^4*x^4 + 48*e^5*x^5) + 15*e^6*x^6*Log[x] - 15*e^6*x^6*Log[d + Sqrt[d^2 - e^
2*x^2]])/(240*d^2*x^6)

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Maple [B]  time = 0.021, size = 521, normalized size = 3.6 \[ -{\frac{1}{6\,{d}^{3}{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{5\,{e}^{2}}{24\,{d}^{5}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{e}^{4}}{16\,{d}^{7}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{6}}{80\,{d}^{7}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{48\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{6}}{16\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{6}}{16\,d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{6}}{5\,{d}^{7}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{7}x}{4\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{7}x}{8\,{d}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{3\,{e}^{7}}{8\,{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{5\,{d}^{4}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{3}}{5\,{d}^{6}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{5}}{5\,{d}^{8}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{7}x}{5\,{d}^{8}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{7}x}{4\,{d}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{e}^{7}x}{8\,{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{3\,{e}^{7}}{8\,{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x)

[Out]

-1/6/d^3/x^6*(-e^2*x^2+d^2)^(7/2)-5/24/d^5*e^2/x^4*(-e^2*x^2+d^2)^(7/2)-3/16/d^7
*e^4/x^2*(-e^2*x^2+d^2)^(7/2)+1/80/d^7*e^6*(-e^2*x^2+d^2)^(5/2)+1/48*e^6/d^5*(-e
^2*x^2+d^2)^(3/2)+1/16*e^6/d^3*(-e^2*x^2+d^2)^(1/2)-1/16*e^6/d/(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^7*e^6*(-(x+d/e)^2*e^2+2*d*e*(x
+d/e))^(5/2)-1/4/d^6*e^7*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-3/8/d^4*e^7*(-(x
+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-3/8/d^2*e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/
(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))+1/5*e/d^4/x^5*(-e^2*x^2+d^2)^(7/2)+1/5*e^3
/d^6/x^3*(-e^2*x^2+d^2)^(7/2)+1/5*e^5/d^8/x*(-e^2*x^2+d^2)^(7/2)+1/5*e^7/d^8*x*(
-e^2*x^2+d^2)^(5/2)+1/4*e^7/d^6*x*(-e^2*x^2+d^2)^(3/2)+3/8*e^7/d^4*x*(-e^2*x^2+d
^2)^(1/2)+3/8*e^7/d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

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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^2*x^2 + d^2)^(5/2)/((e*x + d)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.352663, size = 624, normalized size = 4.36 \[ -\frac{288 \, d e^{11} x^{11} - 90 \, d^{2} e^{10} x^{10} - 2400 \, d^{3} e^{9} x^{9} + 990 \, d^{4} e^{8} x^{8} + 7008 \, d^{5} e^{7} x^{7} - 3860 \, d^{6} e^{6} x^{6} - 9504 \, d^{7} e^{5} x^{5} + 6480 \, d^{8} e^{4} x^{4} + 6144 \, d^{9} e^{3} x^{3} - 4800 \, d^{10} e^{2} x^{2} - 1536 \, d^{11} e x + 1280 \, d^{12} - 15 \,{\left (e^{12} x^{12} - 18 \, d^{2} e^{10} x^{10} + 48 \, d^{4} e^{8} x^{8} - 32 \, d^{6} e^{6} x^{6} + 2 \,{\left (3 \, d e^{10} x^{10} - 16 \, d^{3} e^{8} x^{8} + 16 \, d^{5} e^{6} x^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (48 \, e^{11} x^{11} - 15 \, d e^{10} x^{10} - 960 \, d^{2} e^{9} x^{9} + 340 \, d^{3} e^{8} x^{8} + 4080 \, d^{4} e^{7} x^{7} - 2020 \, d^{5} e^{6} x^{6} - 7008 \, d^{6} e^{5} x^{5} + 4560 \, d^{7} e^{4} x^{4} + 5376 \, d^{8} e^{3} x^{3} - 4160 \, d^{9} e^{2} x^{2} - 1536 \, d^{10} e x + 1280 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \,{\left (d^{2} e^{6} x^{12} - 18 \, d^{4} e^{4} x^{10} + 48 \, d^{6} e^{2} x^{8} - 32 \, d^{8} x^{6} + 2 \,{\left (3 \, d^{3} e^{4} x^{10} - 16 \, d^{5} e^{2} x^{8} + 16 \, d^{7} x^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^7),x, algorithm="fricas")

[Out]

-1/240*(288*d*e^11*x^11 - 90*d^2*e^10*x^10 - 2400*d^3*e^9*x^9 + 990*d^4*e^8*x^8
+ 7008*d^5*e^7*x^7 - 3860*d^6*e^6*x^6 - 9504*d^7*e^5*x^5 + 6480*d^8*e^4*x^4 + 61
44*d^9*e^3*x^3 - 4800*d^10*e^2*x^2 - 1536*d^11*e*x + 1280*d^12 - 15*(e^12*x^12 -
18*d^2*e^10*x^10 + 48*d^4*e^8*x^8 - 32*d^6*e^6*x^6 + 2*(3*d*e^10*x^10 - 16*d^3*
e^8*x^8 + 16*d^5*e^6*x^6)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/
x) - (48*e^11*x^11 - 15*d*e^10*x^10 - 960*d^2*e^9*x^9 + 340*d^3*e^8*x^8 + 4080*d
^4*e^7*x^7 - 2020*d^5*e^6*x^6 - 7008*d^6*e^5*x^5 + 4560*d^7*e^4*x^4 + 5376*d^8*e
^3*x^3 - 4160*d^9*e^2*x^2 - 1536*d^10*e*x + 1280*d^11)*sqrt(-e^2*x^2 + d^2))/(d^
2*e^6*x^12 - 18*d^4*e^4*x^10 + 48*d^6*e^2*x^8 - 32*d^8*x^6 + 2*(3*d^3*e^4*x^10 -
16*d^5*e^2*x^8 + 16*d^7*x^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 79.6741, size = 918, normalized size = 6.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/x**7/(e*x+d),x)

[Out]

d**3*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(
d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(
16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d**2/
(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x*
*5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) +
1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*
d**5), True)) - d**2*e*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x
**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 +
15*e**2*x**7) + 2*I*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**
3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*
x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5
+ 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e*
*2*x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x*
*7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), True)
) - d*e**2*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*
sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*
acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**
2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**3/(8*d**
2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + e**3*
Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) -
1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x
**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True))

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GIAC/XCAS [A]  time = 0.315372, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^7),x, algorithm="giac")

[Out]

Done