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

Optimal. Leaf size=145 \[ \frac{2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{e (30 d+41 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(2*e*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e*(10*d + 13*e*x))/(15*d^4*(d^2
- e^2*x^2)^(3/2)) + (e*(30*d + 41*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2
- e^2*x^2]/(d^6*x) - (2*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^6

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Rubi [A]  time = 0.429258, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^6 x}+\frac{e (30 d+41 e x)}{15 d^6 \sqrt{d^2-e^2 x^2}}-\frac{2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e*(10*d + 13*e*x))/(15*d^4*(d^2
- e^2*x^2)^(3/2)) + (e*(30*d + 41*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2
- e^2*x^2]/(d^6*x) - (2*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^6

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Rubi in Sympy [A]  time = 52.5511, size = 143, normalized size = 0.99 \[ \frac{e}{5 d^{3} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{13 e}{15 d^{4} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} - \frac{1}{d^{4} x \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 e}{d^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{56 e^{2} x}{15 d^{6} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{2 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e/(5*d**3*(d - e*x)**2*sqrt(d**2 - e**2*x**2)) + 13*e/(15*d**4*(d - e*x)*sqrt(d*
*2 - e**2*x**2)) - 1/(d**4*x*sqrt(d**2 - e**2*x**2)) + 2*e/(d**5*sqrt(d**2 - e**
2*x**2)) + 56*e**2*x/(15*d**6*sqrt(d**2 - e**2*x**2)) - 2*e*atanh(sqrt(d**2 - e*
*2*x**2)/d)/d**6

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Mathematica [A]  time = 0.109448, size = 112, normalized size = 0.77 \[ \frac{-30 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^4-76 d^3 e x+32 d^2 e^2 x^2+82 d e^3 x^3-56 e^4 x^4\right )}{x (e x-d)^3 (d+e x)}+30 e \log (x)}{15 d^6} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(15*d^4 - 76*d^3*e*x + 32*d^2*e^2*x^2 + 82*d*e^3*x^3 - 56*
e^4*x^4))/(x*(-d + e*x)^3*(d + e*x)) + 30*e*Log[x] - 30*e*Log[d + Sqrt[d^2 - e^2
*x^2]])/(15*d^6)

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Maple [A]  time = 0.016, size = 193, normalized size = 1.3 \[{\frac{7\,{e}^{2}x}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{28\,{e}^{2}x}{15\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{56\,{e}^{2}x}{15\,{d}^{6}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,e}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{2\,e}{3\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{e}{{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-2\,{\frac{e}{{d}^{5}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/5*e^2*x/d^2/(-e^2*x^2+d^2)^(5/2)+28/15*e^2/d^4*x/(-e^2*x^2+d^2)^(3/2)+56/15*e^
2/d^6*x/(-e^2*x^2+d^2)^(1/2)-1/x/(-e^2*x^2+d^2)^(5/2)+2/5/d*e/(-e^2*x^2+d^2)^(5/
2)+2/3/d^3*e/(-e^2*x^2+d^2)^(3/2)+2/d^5*e/(-e^2*x^2+d^2)^(1/2)-2/d^5*e/(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)^2/((-e^2*x^2 + d^2)^(7/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283614, size = 670, normalized size = 4.62 \[ -\frac{56 \, e^{8} x^{8} - 266 \, d e^{7} x^{7} - 112 \, d^{2} e^{6} x^{6} + 1100 \, d^{3} e^{5} x^{5} - 415 \, d^{4} e^{4} x^{4} - 1080 \, d^{5} e^{3} x^{3} + 600 \, d^{6} e^{2} x^{2} + 240 \, d^{7} e x - 120 \, d^{8} - 30 \,{\left (4 \, d e^{7} x^{7} - 8 \, d^{2} e^{6} x^{6} - 8 \, d^{3} e^{5} x^{5} + 24 \, d^{4} e^{4} x^{4} - 4 \, d^{5} e^{3} x^{3} - 16 \, d^{6} e^{2} x^{2} + 8 \, d^{7} e x -{\left (e^{7} x^{7} - 2 \, d e^{6} x^{6} - 7 \, d^{2} e^{5} x^{5} + 16 \, d^{3} e^{4} x^{4} - 16 \, d^{5} e^{2} x^{2} + 8 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + 2 \,{\left (23 \, e^{7} x^{7} + 66 \, d e^{6} x^{6} - 325 \, d^{2} e^{5} x^{5} + 80 \, d^{3} e^{4} x^{4} + 480 \, d^{4} e^{3} x^{3} - 270 \, d^{5} e^{2} x^{2} - 120 \, d^{6} e x + 60 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (4 \, d^{7} e^{6} x^{7} - 8 \, d^{8} e^{5} x^{6} - 8 \, d^{9} e^{4} x^{5} + 24 \, d^{10} e^{3} x^{4} - 4 \, d^{11} e^{2} x^{3} - 16 \, d^{12} e x^{2} + 8 \, d^{13} x -{\left (d^{6} e^{6} x^{7} - 2 \, d^{7} e^{5} x^{6} - 7 \, d^{8} e^{4} x^{5} + 16 \, d^{9} e^{3} x^{4} - 16 \, d^{11} e x^{2} + 8 \, d^{12} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(56*e^8*x^8 - 266*d*e^7*x^7 - 112*d^2*e^6*x^6 + 1100*d^3*e^5*x^5 - 415*d^4
*e^4*x^4 - 1080*d^5*e^3*x^3 + 600*d^6*e^2*x^2 + 240*d^7*e*x - 120*d^8 - 30*(4*d*
e^7*x^7 - 8*d^2*e^6*x^6 - 8*d^3*e^5*x^5 + 24*d^4*e^4*x^4 - 4*d^5*e^3*x^3 - 16*d^
6*e^2*x^2 + 8*d^7*e*x - (e^7*x^7 - 2*d*e^6*x^6 - 7*d^2*e^5*x^5 + 16*d^3*e^4*x^4
- 16*d^5*e^2*x^2 + 8*d^6*e*x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^
2))/x) + 2*(23*e^7*x^7 + 66*d*e^6*x^6 - 325*d^2*e^5*x^5 + 80*d^3*e^4*x^4 + 480*d
^4*e^3*x^3 - 270*d^5*e^2*x^2 - 120*d^6*e*x + 60*d^7)*sqrt(-e^2*x^2 + d^2))/(4*d^
7*e^6*x^7 - 8*d^8*e^5*x^6 - 8*d^9*e^4*x^5 + 24*d^10*e^3*x^4 - 4*d^11*e^2*x^3 - 1
6*d^12*e*x^2 + 8*d^13*x - (d^6*e^6*x^7 - 2*d^7*e^5*x^6 - 7*d^8*e^4*x^5 + 16*d^9*
e^3*x^4 - 16*d^11*e*x^2 + 8*d^12*x)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.295941, size = 254, normalized size = 1.75 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{41 \, x e^{6}}{d^{6}} + \frac{30 \, e^{5}}{d^{5}}\right )} - \frac{95 \, e^{4}}{d^{4}}\right )} x - \frac{70 \, e^{3}}{d^{3}}\right )} x + \frac{60 \, e^{2}}{d^{2}}\right )} x + \frac{46 \, e}{d}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{2 \, e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{6}} + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{6}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((((x*(41*x*e^6/d^6 + 30*e^5/d^5) - 95*e^4/d^4)*x - 7
0*e^3/d^3)*x + 60*e^2/d^2)*x + 46*e/d)/(x^2*e^2 - d^2)^3 - 2*e*ln(1/2*abs(-2*d*e
- 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^6 + 1/2*x*e^3/((d*e + sqrt(-x^2*e^
2 + d^2)*e)*d^6) - 1/2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-1)/(d^6*x)