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

Optimal. Leaf size=145 \[ \frac{4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{e (15 d+19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(4*e*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (e*(5*d + 7*e*x))/(5*d^3*(d^2 - e^
2*x^2)^(3/2)) + (e*(15*d + 19*e*x))/(5*d^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2
*x^2]/(d^5*x) - (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

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Rubi [A]  time = 0.448311, 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 \[ \frac{4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{e (15 d+19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(4*e*(d + e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (e*(5*d + 7*e*x))/(5*d^3*(d^2 - e^
2*x^2)^(3/2)) + (e*(15*d + 19*e*x))/(5*d^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2
*x^2]/(d^5*x) - (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

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Rubi in Sympy [A]  time = 55.9283, size = 124, normalized size = 0.86 \[ \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{3} \left (d - e x\right )^{3}} + \frac{4 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} \left (d - e x\right )^{2}} - \frac{3 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{5}} + \frac{19 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{5} \left (d - e x\right )} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*sqrt(d**2 - e**2*x**2)/(5*d**3*(d - e*x)**3) + 4*e*sqrt(d**2 - e**2*x**2)/(5*d
**4*(d - e*x)**2) - 3*e*atanh(sqrt(d**2 - e**2*x**2)/d)/d**5 + 19*e*sqrt(d**2 -
e**2*x**2)/(5*d**5*(d - e*x)) - sqrt(d**2 - e**2*x**2)/(d**5*x)

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Mathematica [A]  time = 0.151774, size = 94, normalized size = 0.65 \[ \frac{-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (5 d^3-39 d^2 e x+57 d e^2 x^2-24 e^3 x^3\right )}{x (e x-d)^3}+15 e \log (x)}{5 d^5} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(5*d^3 - 39*d^2*e*x + 57*d*e^2*x^2 - 24*e^3*x^3))/(x*(-d +
e*x)^3) + 15*e*Log[x] - 15*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(5*d^5)

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Maple [A]  time = 0.015, size = 190, normalized size = 1.3 \[ -{\frac{d}{x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{9\,{e}^{2}x}{5\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{12\,{e}^{2}x}{5\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{24\,{e}^{2}x}{5\,{d}^{5}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{4\,e}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{e}{{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+3\,{\frac{e}{{d}^{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}-3\,{\frac{e}{{d}^{4}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/x/(-e^2*x^2+d^2)^(5/2)+9/5/d*e^2*x/(-e^2*x^2+d^2)^(5/2)+12/5/d^3*e^2*x/(-e^2*
x^2+d^2)^(3/2)+24/5/d^5*e^2*x/(-e^2*x^2+d^2)^(1/2)+4/5*e/(-e^2*x^2+d^2)^(5/2)+e/
d^2/(-e^2*x^2+d^2)^(3/2)+3*e/d^4/(-e^2*x^2+d^2)^(1/2)-3*e/d^4/(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)^3/((-e^2*x^2 + d^2)^(7/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295272, size = 620, normalized size = 4.28 \[ \frac{153 \, d e^{6} x^{6} - 330 \, d^{2} e^{5} x^{5} - 70 \, d^{3} e^{4} x^{4} + 525 \, d^{4} e^{3} x^{3} - 220 \, d^{5} e^{2} x^{2} - 100 \, d^{6} e x + 40 \, d^{7} + 15 \,{\left (e^{7} x^{7} + d e^{6} x^{6} - 13 \, d^{2} e^{5} x^{5} + 15 \, d^{3} e^{4} x^{4} + 8 \, d^{4} e^{3} x^{3} - 20 \, d^{5} e^{2} x^{2} + 8 \, d^{6} e x -{\left (e^{6} x^{6} - 6 \, d e^{5} x^{5} + 5 \, d^{2} e^{4} x^{4} + 12 \, d^{3} e^{3} x^{3} - 20 \, d^{4} e^{2} x^{2} + 8 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (48 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} - 165 \, d^{2} e^{4} x^{4} + 475 \, d^{3} e^{3} x^{3} - 200 \, d^{4} e^{2} x^{2} - 100 \, d^{5} e x + 40 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{5} e^{6} x^{7} + d^{6} e^{5} x^{6} - 13 \, d^{7} e^{4} x^{5} + 15 \, d^{8} e^{3} x^{4} + 8 \, d^{9} e^{2} x^{3} - 20 \, d^{10} e x^{2} + 8 \, d^{11} x -{\left (d^{5} e^{5} x^{6} - 6 \, d^{6} e^{4} x^{5} + 5 \, d^{7} e^{3} x^{4} + 12 \, d^{8} e^{2} x^{3} - 20 \, d^{9} e x^{2} + 8 \, d^{10} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x^2),x, algorithm="fricas")

[Out]

1/5*(153*d*e^6*x^6 - 330*d^2*e^5*x^5 - 70*d^3*e^4*x^4 + 525*d^4*e^3*x^3 - 220*d^
5*e^2*x^2 - 100*d^6*e*x + 40*d^7 + 15*(e^7*x^7 + d*e^6*x^6 - 13*d^2*e^5*x^5 + 15
*d^3*e^4*x^4 + 8*d^4*e^3*x^3 - 20*d^5*e^2*x^2 + 8*d^6*e*x - (e^6*x^6 - 6*d*e^5*x
^5 + 5*d^2*e^4*x^4 + 12*d^3*e^3*x^3 - 20*d^4*e^2*x^2 + 8*d^5*e*x)*sqrt(-e^2*x^2
+ d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (48*e^6*x^6 - 105*d*e^5*x^5 - 165*d
^2*e^4*x^4 + 475*d^3*e^3*x^3 - 200*d^4*e^2*x^2 - 100*d^5*e*x + 40*d^6)*sqrt(-e^2
*x^2 + d^2))/(d^5*e^6*x^7 + d^6*e^5*x^6 - 13*d^7*e^4*x^5 + 15*d^8*e^3*x^4 + 8*d^
9*e^2*x^3 - 20*d^10*e*x^2 + 8*d^11*x - (d^5*e^5*x^6 - 6*d^6*e^4*x^5 + 5*d^7*e^3*
x^4 + 12*d^8*e^2*x^3 - 20*d^9*e*x^2 + 8*d^10*x)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3/(x**2*(-(-d + e*x)*(d + e*x))**(7/2)), x)

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GIAC/XCAS [A]  time = 0.30003, size = 250, normalized size = 1.72 \[ -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left ({\left (x{\left (\frac{19 \, x e^{6}}{d^{5}} + \frac{15 \, e^{5}}{d^{4}}\right )} - \frac{45 \, e^{4}}{d^{3}}\right )} x - \frac{35 \, e^{3}}{d^{2}}\right )} x + \frac{30 \, e^{2}}{d}\right )} x + 24 \, e\right )}}{5 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{3 \, e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d^{5}} + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{5}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*x^2),x, algorithm="giac")

[Out]

-1/5*sqrt(-x^2*e^2 + d^2)*((((x*(19*x*e^6/d^5 + 15*e^5/d^4) - 45*e^4/d^3)*x - 35
*e^3/d^2)*x + 30*e^2/d)*x + 24*e)/(x^2*e^2 - d^2)^3 - 3*e*ln(1/2*abs(-2*d*e - 2*
sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^5 + 1/2*x*e^3/((d*e + sqrt(-x^2*e^2 + d
^2)*e)*d^5) - 1/2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-1)/(d^5*x)