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

Optimal. Leaf size=117 \[ \frac{2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d+16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(2*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d + 8*e*x)/(15*d^3*(d^2 - e^2*x^2
)^(3/2)) + (15*d + 16*e*x)/(15*d^5*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2
*x^2]/d]/d^5

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Rubi [A]  time = 0.338625, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d+16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d + 8*e*x)/(15*d^3*(d^2 - e^2*x^2
)^(3/2)) + (15*d + 16*e*x)/(15*d^5*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2
*x^2]/d]/d^5

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Rubi in Sympy [A]  time = 41.5973, size = 114, normalized size = 0.97 \[ \frac{1}{5 d^{2} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{8}{15 d^{3} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{1}{d^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{16 e x}{15 d^{5} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(5*d**2*(d - e*x)**2*sqrt(d**2 - e**2*x**2)) + 8/(15*d**3*(d - e*x)*sqrt(d**2
- e**2*x**2)) + 1/(d**4*sqrt(d**2 - e**2*x**2)) + 16*e*x/(15*d**5*sqrt(d**2 - e*
*2*x**2)) - atanh(sqrt(d**2 - e**2*x**2)/d)/d**5

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Mathematica [A]  time = 0.0913727, size = 95, normalized size = 0.81 \[ \frac{-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (26 d^3-22 d^2 e x-17 d e^2 x^2+16 e^3 x^3\right )}{(d-e x)^3 (d+e x)}+15 \log (x)}{15 d^5} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(26*d^3 - 22*d^2*e*x - 17*d*e^2*x^2 + 16*e^3*x^3))/((d - e
*x)^3*(d + e*x)) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^5)

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Maple [A]  time = 0.013, size = 160, normalized size = 1.4 \[{\frac{2}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{3\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{4}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{2\,ex}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,ex}{15\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,ex}{15\,{d}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/(-e^2*x^2+d^2)^(5/2)+1/3/d^2/(-e^2*x^2+d^2)^(3/2)+1/d^4/(-e^2*x^2+d^2)^(1/2)
-1/d^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+2/5/d*e*x/(-
e^2*x^2+d^2)^(5/2)+8/15/d^3*e*x/(-e^2*x^2+d^2)^(3/2)+16/15/d^5*e*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*x + d)^2/((-e^2*x^2 + d^2)^(7/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279835, size = 540, normalized size = 4.62 \[ \frac{26 \, e^{6} x^{6} - 4 \, d e^{5} x^{5} - 155 \, d^{2} e^{4} x^{4} + 130 \, d^{3} e^{3} x^{3} + 120 \, d^{4} e^{2} x^{2} - 120 \, d^{5} e x + 15 \,{\left (e^{6} x^{6} - 2 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} + 10 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} - 8 \, d^{5} e x + 4 \, d^{6} +{\left (3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (16 \, e^{5} x^{5} - 95 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} + 120 \, d^{3} e^{2} x^{2} - 120 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{5} e^{6} x^{6} - 2 \, d^{6} e^{5} x^{5} - 4 \, d^{7} e^{4} x^{4} + 10 \, d^{8} e^{3} x^{3} - d^{9} e^{2} x^{2} - 8 \, d^{10} e x + 4 \, d^{11} +{\left (3 \, d^{6} e^{4} x^{4} - 6 \, d^{7} e^{3} x^{3} - d^{8} e^{2} x^{2} + 8 \, d^{9} e x - 4 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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),x, algorithm="fricas")

[Out]

1/15*(26*e^6*x^6 - 4*d*e^5*x^5 - 155*d^2*e^4*x^4 + 130*d^3*e^3*x^3 + 120*d^4*e^2
*x^2 - 120*d^5*e*x + 15*(e^6*x^6 - 2*d*e^5*x^5 - 4*d^2*e^4*x^4 + 10*d^3*e^3*x^3
- d^4*e^2*x^2 - 8*d^5*e*x + 4*d^6 + (3*d*e^4*x^4 - 6*d^2*e^3*x^3 - d^3*e^2*x^2 +
8*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (
16*e^5*x^5 - 95*d*e^4*x^4 + 70*d^2*e^3*x^3 + 120*d^3*e^2*x^2 - 120*d^4*e*x)*sqrt
(-e^2*x^2 + d^2))/(d^5*e^6*x^6 - 2*d^6*e^5*x^5 - 4*d^7*e^4*x^4 + 10*d^8*e^3*x^3
- d^9*e^2*x^2 - 8*d^10*e*x + 4*d^11 + (3*d^6*e^4*x^4 - 6*d^7*e^3*x^3 - d^8*e^2*x
^2 + 8*d^9*e*x - 4*d^10)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.297339, size = 159, normalized size = 1.36 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{16 \, x e^{5}}{d^{5}} + \frac{15 \, e^{4}}{d^{4}}\right )} - \frac{40 \, e^{3}}{d^{3}}\right )} x - \frac{35 \, e^{2}}{d^{2}}\right )} x + \frac{30 \, e}{d}\right )} x + 26\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{5}} \]

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),x, algorithm="giac")

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*((((x*(16*x*e^5/d^5 + 15*e^4/d^4) - 40*e^3/d^3)*x - 3
5*e^2/d^2)*x + 30*e/d)*x + 26)/(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^5