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

Optimal. Leaf size=89 \[ \frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.122402, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(d+e x)^2}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{4 x}{15 d^3 e \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2917, size = 76, normalized size = 0.85 \[ \frac{\left (d + e x\right )^{2}}{5 e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{2 \left (d + e x\right )}{15 d e^{2} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{4 x}{15 d^{3} e \sqrt{d^{2} - e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**2/(5*e**2*(d**2 - e**2*x**2)**(5/2)) - 2*(d + e*x)/(15*d*e**2*(d**2 -
e**2*x**2)**(3/2)) - 4*x/(15*d**3*e*sqrt(d**2 - e**2*x**2))

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Mathematica [A]  time = 0.0504812, size = 69, normalized size = 0.78 \[ \frac{\sqrt{d^2-e^2 x^2} \left (d^3-2 d^2 e x+8 d e^2 x^2-4 e^3 x^3\right )}{15 d^3 e^2 (d-e x)^3 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(d^3 - 2*d^2*e*x + 8*d*e^2*x^2 - 4*e^3*x^3))/(15*d^3*e^2*(d
- e*x)^3*(d + e*x))

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Maple [A]  time = 0.013, size = 64, normalized size = 0.7 \[{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3} \left ( -4\,{e}^{3}{x}^{3}+8\,d{e}^{2}{x}^{2}-2\,{d}^{2}ex+{d}^{3} \right ) }{15\,{d}^{3}{e}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-e*x+d)*(e*x+d)^3*(-4*e^3*x^3+8*d*e^2*x^2-2*d^2*e*x+d^3)/d^3/e^2/(-e^2*x^2
+d^2)^(7/2)

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Maxima [A]  time = 0.709691, size = 147, normalized size = 1.65 \[ \frac{x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}} + \frac{2 \, d x}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d^{2}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{2 \, x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d e} - \frac{4 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} e} \]

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.276486, size = 306, normalized size = 3.44 \[ \frac{e^{4} x^{6} - 14 \, d e^{3} x^{5} + 20 \, d^{2} e^{2} x^{4} + 20 \, d^{3} e x^{3} - 30 \, d^{4} x^{2} +{\left (4 \, e^{3} x^{5} - 5 \, d e^{2} x^{4} - 20 \, d^{2} e x^{3} + 30 \, d^{3} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{3} e^{6} x^{6} - 2 \, d^{4} e^{5} x^{5} - 4 \, d^{5} e^{4} x^{4} + 10 \, d^{6} e^{3} x^{3} - d^{7} e^{2} x^{2} - 8 \, d^{8} e x + 4 \, d^{9} +{\left (3 \, d^{4} e^{4} x^{4} - 6 \, d^{5} e^{3} x^{3} - d^{6} e^{2} x^{2} + 8 \, d^{7} e x - 4 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(e^4*x^6 - 14*d*e^3*x^5 + 20*d^2*e^2*x^4 + 20*d^3*e*x^3 - 30*d^4*x^2 + (4*e
^3*x^5 - 5*d*e^2*x^4 - 20*d^2*e*x^3 + 30*d^3*x^2)*sqrt(-e^2*x^2 + d^2))/(d^3*e^6
*x^6 - 2*d^4*e^5*x^5 - 4*d^5*e^4*x^4 + 10*d^6*e^3*x^3 - d^7*e^2*x^2 - 8*d^8*e*x
+ 4*d^9 + (3*d^4*e^4*x^4 - 6*d^5*e^3*x^3 - d^6*e^2*x^2 + 8*d^7*e*x - 4*d^8)*sqrt
(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.293586, size = 86, normalized size = 0.97 \[ \frac{{\left ({\left (2 \, x{\left (\frac{2 \, x^{2} e^{3}}{d^{3}} - \frac{5 \, e}{d}\right )} - 5\right )} x^{2} - d^{2} e^{\left (-2\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*((2*x*(2*x^2*e^3/d^3 - 5*e/d) - 5)*x^2 - d^2*e^(-2))*sqrt(-x^2*e^2 + d^2)/(
x^2*e^2 - d^2)^3