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

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 = 0.256613, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac{e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac{e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac{e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^2}+\frac{e^4 \sqrt{d^2-e^2 x^2}}{16 d x^2} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^7,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 = 31.9964, 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*x+d)*(-e**2*x**2+d**2)**(3/2)/x**7,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 = 0.21337, 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 + e*x)*(d^2 - e^2*x^2)^(3/2))/x^7,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 [A]  time = 0.037, size = 186, normalized size = 1.3 \[ -{\frac{1}{6\,d{x}^{6}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{24\,{d}^{3}{x}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{4}}{48\,{d}^{5}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{6}}{48\,{d}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{6}}{16\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{6}}{16\,d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{e}{5\,{d}^{2}{x}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-e^2*x^2+d^2)^(5/2)/d/x^6-1/24*e^2/d^3/x^4*(-e^2*x^2+d^2)^(5/2)+1/48*e^4/d
^5/x^2*(-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*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^5

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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)^(3/2)*(e*x + d)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.348645, 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)^(3/2)*(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 + 614
4*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 = 40.6252, 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*x+d)*(-e**2*x**2+d**2)**(3/2)/x**7,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 = 0.29495, 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)^(3/2)*(e*x + d)/x^7,x, algorithm="giac")

[Out]

Done