3.27 \(\int \frac{d+e x}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)
Optimal. Leaf size=117 \[ \frac{d+e x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d+8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{5 d+4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
(d + e*x)/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (5*d + 4*e*x)/(15*d^4*(d^2 - e^2*x^2)^
(3/2)) + (15*d + 8*e*x)/(15*d^6*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*x^
2]/d]/d^6
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Rubi [A] time = 0.314073, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2
\[ \frac{d+e x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d+8 e x}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{5 d+4 e x}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(x*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
(d + e*x)/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (5*d + 4*e*x)/(15*d^4*(d^2 - e^2*x^2)^
(3/2)) + (15*d + 8*e*x)/(15*d^6*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*x^
2]/d]/d^6
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Rubi in Sympy [A] time = 44.7554, size = 97, normalized size = 0.83 \[ \frac{d + e x}{5 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{5 d + 4 e x}{15 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{15 d + 8 e x}{15 d^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/x/(-e**2*x**2+d**2)**(7/2),x)
[Out]
(d + e*x)/(5*d**2*(d**2 - e**2*x**2)**(5/2)) + (5*d + 4*e*x)/(15*d**4*(d**2 - e*
*2*x**2)**(3/2)) + (15*d + 8*e*x)/(15*d**6*sqrt(d**2 - e**2*x**2)) - atanh(sqrt(
d**2 - e**2*x**2)/d)/d**6
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Mathematica [A] time = 0.101851, size = 106, normalized size = 0.91 \[ \frac{-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (23 d^4-8 d^3 e x-27 d^2 e^2 x^2+7 d e^3 x^3+8 e^4 x^4\right )}{(d-e x)^3 (d+e x)^2}+15 \log (x)}{15 d^6} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(x*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
((Sqrt[d^2 - e^2*x^2]*(23*d^4 - 8*d^3*e*x - 27*d^2*e^2*x^2 + 7*d*e^3*x^3 + 8*e^4
*x^4))/((d - e*x)^3*(d + e*x)^2) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/
(15*d^6)
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Maple [A] time = 0.018, size = 163, normalized size = 1.4 \[{\frac{ex}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{4\,ex}{15\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,ex}{15\,{d}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{1}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{3\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{5}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/x/(-e^2*x^2+d^2)^(7/2),x)
[Out]
1/5*e*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/15*e/d^4*x/(-e^2*x^2+d^2)^(3/2)+8/15*e/d^6*x/
(-e^2*x^2+d^2)^(1/2)+1/5/d/(-e^2*x^2+d^2)^(5/2)+1/3/d^3/(-e^2*x^2+d^2)^(3/2)+1/d
^5/(-e^2*x^2+d^2)^(1/2)-1/d^5/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)
^(1/2))/x)
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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)/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.294215, size = 718, normalized size = 6.14 \[ -\frac{8 \, e^{8} x^{8} - 85 \, d e^{7} x^{7} + d^{2} e^{6} x^{6} + 304 \, d^{3} e^{5} x^{5} - 65 \, d^{4} e^{4} x^{4} - 340 \, d^{5} e^{3} x^{3} + 60 \, d^{6} e^{2} x^{2} + 120 \, d^{7} e x - 15 \,{\left (4 \, d e^{7} x^{7} - 4 \, d^{2} e^{6} x^{6} - 16 \, d^{3} e^{5} x^{5} + 16 \, d^{4} e^{4} x^{4} + 20 \, d^{5} e^{3} x^{3} - 20 \, d^{6} e^{2} x^{2} - 8 \, d^{7} e x + 8 \, d^{8} -{\left (e^{7} x^{7} - d e^{6} x^{6} - 9 \, d^{2} e^{5} x^{5} + 9 \, d^{3} e^{4} x^{4} + 16 \, d^{4} e^{3} x^{3} - 16 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x + 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (23 \, e^{7} x^{7} + 9 \, d e^{6} x^{6} - 179 \, d^{2} e^{5} x^{5} + 35 \, d^{3} e^{4} x^{4} + 280 \, d^{4} e^{3} x^{3} - 60 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{7} e^{7} x^{7} - 4 \, d^{8} e^{6} x^{6} - 16 \, d^{9} e^{5} x^{5} + 16 \, d^{10} e^{4} x^{4} + 20 \, d^{11} e^{3} x^{3} - 20 \, d^{12} e^{2} x^{2} - 8 \, d^{13} e x + 8 \, d^{14} -{\left (d^{6} e^{7} x^{7} - d^{7} e^{6} x^{6} - 9 \, d^{8} e^{5} x^{5} + 9 \, d^{9} e^{4} x^{4} + 16 \, d^{10} e^{3} x^{3} - 16 \, d^{11} e^{2} x^{2} - 8 \, d^{12} e x + 8 \, d^{13}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="fricas")
[Out]
-1/15*(8*e^8*x^8 - 85*d*e^7*x^7 + d^2*e^6*x^6 + 304*d^3*e^5*x^5 - 65*d^4*e^4*x^4
- 340*d^5*e^3*x^3 + 60*d^6*e^2*x^2 + 120*d^7*e*x - 15*(4*d*e^7*x^7 - 4*d^2*e^6*
x^6 - 16*d^3*e^5*x^5 + 16*d^4*e^4*x^4 + 20*d^5*e^3*x^3 - 20*d^6*e^2*x^2 - 8*d^7*
e*x + 8*d^8 - (e^7*x^7 - d*e^6*x^6 - 9*d^2*e^5*x^5 + 9*d^3*e^4*x^4 + 16*d^4*e^3*
x^3 - 16*d^5*e^2*x^2 - 8*d^6*e*x + 8*d^7)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-
e^2*x^2 + d^2))/x) + (23*e^7*x^7 + 9*d*e^6*x^6 - 179*d^2*e^5*x^5 + 35*d^3*e^4*x^
4 + 280*d^4*e^3*x^3 - 60*d^5*e^2*x^2 - 120*d^6*e*x)*sqrt(-e^2*x^2 + d^2))/(4*d^7
*e^7*x^7 - 4*d^8*e^6*x^6 - 16*d^9*e^5*x^5 + 16*d^10*e^4*x^4 + 20*d^11*e^3*x^3 -
20*d^12*e^2*x^2 - 8*d^13*e*x + 8*d^14 - (d^6*e^7*x^7 - d^7*e^6*x^6 - 9*d^8*e^5*x
^5 + 9*d^9*e^4*x^4 + 16*d^10*e^3*x^3 - 16*d^11*e^2*x^2 - 8*d^12*e*x + 8*d^13)*sq
rt(-e^2*x^2 + d^2))
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Sympy [A] time = 36.5228, size = 2378, normalized size = 20.32 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/x/(-e**2*x**2+d**2)**(7/2),x)
[Out]
d*Piecewise((-46*I*d**6*sqrt(-1 + e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**
2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 15*d**6*log(e**2*x**2/d**2)/(-30*d*
*13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 30*d**6*log(
e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6)
- 30*I*d**6*asin(d/(e*x))/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 +
30*d**7*e**6*x**6) + 70*I*d**4*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-30*d**13 +
90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 45*d**4*e**2*x**2*
log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**
7*e**6*x**6) - 90*d**4*e**2*x**2*log(e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90
*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 90*I*d**4*e**2*x**2*asin(d/(e*x))/(-30*d*
*13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*I*d**2*e*
*4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4
*x**4 + 30*d**7*e**6*x**6) - 45*d**2*e**4*x**4*log(e**2*x**2/d**2)/(-30*d**13 +
90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 90*d**2*e**4*x**4*
log(e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x*
*6) - 90*I*d**2*e**4*x**4*asin(d/(e*x))/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**
9*e**4*x**4 + 30*d**7*e**6*x**6) + 15*e**6*x**6*log(e**2*x**2/d**2)/(-30*d**13 +
90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*e**6*x**6*log(
e*x/d)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6)
+ 30*I*e**6*x**6*asin(d/(e*x))/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x*
*4 + 30*d**7*e**6*x**6), Abs(e**2*x**2/d**2) > 1), (-46*d**6*sqrt(1 - e**2*x**2/
d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) -
15*d**6*log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4
+ 30*d**7*e**6*x**6) + 30*d**6*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-30*d**13 + 9
0*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 15*I*pi*d**6/(-30*d
**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) + 70*d**4*e**
2*x**2*sqrt(1 - e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x
**4 + 30*d**7*e**6*x**6) + 45*d**4*e**2*x**2*log(e**2*x**2/d**2)/(-30*d**13 + 90
*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 90*d**4*e**2*x**2*lo
g(sqrt(1 - e**2*x**2/d**2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x
**4 + 30*d**7*e**6*x**6) + 45*I*pi*d**4*e**2*x**2/(-30*d**13 + 90*d**11*e**2*x**
2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*d**2*e**4*x**4*sqrt(1 - e**2*x**
2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6)
- 45*d**2*e**4*x**4*log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d*
*9*e**4*x**4 + 30*d**7*e**6*x**6) + 90*d**2*e**4*x**4*log(sqrt(1 - e**2*x**2/d**
2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6)
- 45*I*pi*d**2*e**4*x**4/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 +
30*d**7*e**6*x**6) + 15*e**6*x**6*log(e**2*x**2/d**2)/(-30*d**13 + 90*d**11*e**2
*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e**6*x**6) - 30*e**6*x**6*log(sqrt(1 - e**2*
x**2/d**2) + 1)/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4 + 30*d**7*e*
*6*x**6) + 15*I*pi*e**6*x**6/(-30*d**13 + 90*d**11*e**2*x**2 - 90*d**9*e**4*x**4
+ 30*d**7*e**6*x**6), True)) + e*Piecewise((-15*I*d**4*x/(15*d**11*sqrt(-1 + e*
*2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*
sqrt(-1 + e**2*x**2/d**2)) + 20*I*d**2*e**2*x**3/(15*d**11*sqrt(-1 + e**2*x**2/d
**2) - 30*d**9*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 +
e**2*x**2/d**2)) - 8*I*e**4*x**5/(15*d**11*sqrt(-1 + e**2*x**2/d**2) - 30*d**9*
e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(-1 + e**2*x**2/d**2
)), Abs(e**2*x**2/d**2) > 1), (15*d**4*x/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30
*d**9*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/
d**2)) - 20*d**2*e**2*x**3/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**
2*sqrt(1 - e**2*x**2/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)) + 8*e**
4*x**5/(15*d**11*sqrt(1 - e**2*x**2/d**2) - 30*d**9*e**2*x**2*sqrt(1 - e**2*x**2
/d**2) + 15*d**7*e**4*x**4*sqrt(1 - e**2*x**2/d**2)), True))
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GIAC/XCAS [A] time = 0.292394, size = 165, normalized size = 1.41 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{8 \, x e^{5}}{d^{6}} + \frac{15 \, e^{4}}{d^{5}}\right )} - \frac{20 \, e^{3}}{d^{4}}\right )} x - \frac{35 \, e^{2}}{d^{3}}\right )} x + \frac{15 \, e}{d^{2}}\right )} x + \frac{23}{d}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="giac")
[Out]
-1/15*sqrt(-x^2*e^2 + d^2)*((((x*(8*x*e^5/d^6 + 15*e^4/d^5) - 20*e^3/d^4)*x - 35
*e^2/d^3)*x + 15*e/d^2)*x + 23/d)/(x^2*e^2 - d^2)^3 - ln(1/2*abs(-2*d*e - 2*sqrt
(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^6