3.45 \(\int \frac{x^4 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{17 d^2 (d+e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (15 d+13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}+\frac{d^3 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(d^3*(d + e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) - (17*d^2*(d + e*x))/(15*e^5*(d^
2 - e^2*x^2)^(3/2)) + (2*(15*d + 13*e*x))/(15*e^5*Sqrt[d^2 - e^2*x^2]) - ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]]/e^5

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Rubi [A]  time = 0.316319, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{17 d^2 (d+e x)}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 (15 d+13 e x)}{15 e^5 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}+\frac{d^3 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(d^3*(d + e*x)^2)/(5*e^5*(d^2 - e^2*x^2)^(5/2)) - (17*d^2*(d + e*x))/(15*e^5*(d^
2 - e^2*x^2)^(3/2)) + (2*(15*d + 13*e*x))/(15*e^5*Sqrt[d^2 - e^2*x^2]) - ArcTan[
(e*x)/Sqrt[d^2 - e^2*x^2]]/e^5

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Rubi in Sympy [A]  time = 58.0747, size = 121, normalized size = 1. \[ \frac{d^{3}}{5 e^{5} \left (d - e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{17 d^{2}}{15 e^{5} \left (d - e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{2 d}{e^{5} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{26 x}{15 e^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d**3/(5*e**5*(d - e*x)**2*sqrt(d**2 - e**2*x**2)) - 17*d**2/(15*e**5*(d - e*x)*s
qrt(d**2 - e**2*x**2)) + 2*d/(e**5*sqrt(d**2 - e**2*x**2)) + 26*x/(15*e**4*sqrt(
d**2 - e**2*x**2)) - atan(e*x/sqrt(d**2 - e**2*x**2))/e**5

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Mathematica [A]  time = 0.118366, size = 106, normalized size = 0.88 \[ \frac{15 (d+e x) (d-e x)^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-16 d^3+17 d^2 e x+22 d e^2 x^2-26 e^3 x^3\right )}{15 e^5 (e x-d)^3 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d + e*x)^2)/(d^2 - e^2*x^2)^(7/2),x]

[Out]

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

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Maple [B]  time = 0.023, size = 236, normalized size = 2. \[{\frac{{d}^{2}{x}^{3}}{2\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{3\,{d}^{4}x}{10\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{{d}^{2}x}{10\,{e}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{6\,x}{5\,{e}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}+{\frac{{x}^{5}}{5} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{3}}{3\,{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+2\,{\frac{d{x}^{4}}{e \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}}-{\frac{8\,{d}^{3}{x}^{2}}{3\,{e}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{16\,{d}^{5}}{15\,{e}^{5}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)^2/(-e^2*x^2+d^2)^(7/2),x)

[Out]

1/2*d^2*x^3/e^2/(-e^2*x^2+d^2)^(5/2)-3/10*d^4/e^4*x/(-e^2*x^2+d^2)^(5/2)+1/10*d^
2/e^4*x/(-e^2*x^2+d^2)^(3/2)+6/5/e^4*x/(-e^2*x^2+d^2)^(1/2)+1/5*x^5/(-e^2*x^2+d^
2)^(5/2)-1/3*x^3/e^2/(-e^2*x^2+d^2)^(3/2)-1/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x
/(-e^2*x^2+d^2)^(1/2))+2*d/e*x^4/(-e^2*x^2+d^2)^(5/2)-8/3*d^3/e^3*x^2/(-e^2*x^2+
d^2)^(5/2)+16/15*d^5/e^5/(-e^2*x^2+d^2)^(5/2)

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Maxima [A]  time = 0.80027, size = 420, normalized size = 3.47 \[ \frac{1}{15} \, e^{2} x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{1}{3} \, x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )} + \frac{2 \, d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{2} x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{8 \, d^{3} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{3 \, d^{4} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{16 \, d^{5}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{5}} + \frac{11 \, d^{2} x}{30 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}} - \frac{4 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{4}} - \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*e^2*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(
5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 1/3*x*(3*x^2/((-e^2*x^2 + d^2)
^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 2*d*x^4/((-e^2*x^2 + d^2)^(5
/2)*e) + 1/2*d^2*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 8/3*d^3*x^2/((-e^2*x^2 + d^2
)^(5/2)*e^3) - 3/10*d^4*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 16/15*d^5/((-e^2*x^2 +
d^2)^(5/2)*e^5) + 11/30*d^2*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 4/15*x/(sqrt(-e^2*x
^2 + d^2)*e^4) - arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^4)

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Fricas [A]  time = 0.280343, size = 548, normalized size = 4.53 \[ \frac{16 \, e^{6} x^{6} + 46 \, d e^{5} x^{5} - 130 \, d^{2} e^{4} x^{4} + 5 \, d^{3} e^{3} x^{3} + 120 \, d^{4} e^{2} x^{2} - 60 \, d^{5} e x + 30 \,{\left (e^{6} x^{6} - 2 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} + 10 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} - 8 \, d^{5} e x + 4 \, d^{6} +{\left (3 \, d e^{4} x^{4} - 6 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (26 \, e^{5} x^{5} - 70 \, d e^{4} x^{4} - 25 \, d^{2} e^{3} x^{3} + 120 \, d^{3} e^{2} x^{2} - 60 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{11} x^{6} - 2 \, d e^{10} x^{5} - 4 \, d^{2} e^{9} x^{4} + 10 \, d^{3} e^{8} x^{3} - d^{4} e^{7} x^{2} - 8 \, d^{5} e^{6} x + 4 \, d^{6} e^{5} +{\left (3 \, d e^{9} x^{4} - 6 \, d^{2} e^{8} x^{3} - d^{3} e^{7} x^{2} + 8 \, d^{4} e^{6} x - 4 \, d^{5} e^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(16*e^6*x^6 + 46*d*e^5*x^5 - 130*d^2*e^4*x^4 + 5*d^3*e^3*x^3 + 120*d^4*e^2*
x^2 - 60*d^5*e*x + 30*(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))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)
) - (26*e^5*x^5 - 70*d*e^4*x^4 - 25*d^2*e^3*x^3 + 120*d^3*e^2*x^2 - 60*d^4*e*x)*
sqrt(-e^2*x^2 + d^2))/(e^11*x^6 - 2*d*e^10*x^5 - 4*d^2*e^9*x^4 + 10*d^3*e^8*x^3
- d^4*e^7*x^2 - 8*d^5*e^6*x + 4*d^6*e^5 + (3*d*e^9*x^4 - 6*d^2*e^8*x^3 - d^3*e^7
*x^2 + 8*d^4*e^6*x - 4*d^5*e^5)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)**2/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(x**4*(d + e*x)**2/(-(-d + e*x)*(d + e*x))**(7/2), x)

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GIAC/XCAS [A]  time = 0.292406, size = 128, normalized size = 1.06 \[ -\arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )}{\rm sign}\left (d\right ) - \frac{{\left (16 \, d^{5} e^{\left (-5\right )} +{\left (15 \, d^{4} e^{\left (-4\right )} -{\left (40 \, d^{3} e^{\left (-3\right )} +{\left (35 \, d^{2} e^{\left (-2\right )} - 2 \,{\left (15 \, d e^{\left (-1\right )} + 13 \, x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*x^4/(-e^2*x^2 + d^2)^(7/2),x, algorithm="giac")

[Out]

-arcsin(x*e/d)*e^(-5)*sign(d) - 1/15*(16*d^5*e^(-5) + (15*d^4*e^(-4) - (40*d^3*e
^(-3) + (35*d^2*e^(-2) - 2*(15*d*e^(-1) + 13*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)
/(x^2*e^2 - d^2)^3