3.30 \(\int \frac{x^2 (d+e x)}{(d^2-e^2 x^2)^{9/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

(x^2*(d + e*x))/(7*d*e*(d^2 - e^2*x^2)^(7/2)) - (2*(d - 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*
d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*Sqrt[d^2 - e^2*x^2])

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Rubi [A]  time = 0.0525345, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {796, 778, 192, 191} \[ \frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(9/2),x]

[Out]

(x^2*(d + e*x))/(7*d*e*(d^2 - e^2*x^2)^(7/2)) - (2*(d - 2*e*x))/(35*d*e^3*(d^2 - e^2*x^2)^(5/2)) - (4*x)/(105*
d^3*e^2*(d^2 - e^2*x^2)^(3/2)) - (8*x)/(105*d^5*e^2*Sqrt[d^2 - e^2*x^2])

Rule 796

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(x^2*(a*g - c*f*x)*(a + c*x^2)^(p
+ 1))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[x*Simp[2*a*g - c*f*(2*p + 5)*x, x]*(a + c*x^2)^(p + 1
), x], x] /; FreeQ[{a, c, f, g}, x] && EqQ[a*g^2 + f^2*c, 0] && LtQ[p, -2]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx &=\frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{\int \frac{x \left (2 d^2 e-4 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d e^2}\\ &=\frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{105 d^3 e^2}\\ &=\frac{x^2 (d+e x)}{7 d e \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 (d-2 e x)}{35 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{4 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{8 x}{105 d^5 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0453019, size = 104, normalized size = 0.86 \[ \frac{15 d^4 e^2 x^2+20 d^3 e^3 x^3-20 d^2 e^4 x^4+6 d^5 e x-6 d^6-8 d e^5 x^5+8 e^6 x^6}{105 d^5 e^3 (d-e x)^3 (d+e x)^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(9/2),x]

[Out]

(-6*d^6 + 6*d^5*e*x + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 - 20*d^2*e^4*x^4 - 8*d*e^5*x^5 + 8*e^6*x^6)/(105*d^5*e^3
*(d - e*x)^3*(d + e*x)^2*Sqrt[d^2 - e^2*x^2])

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Maple [A]  time = 0.053, size = 99, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -8\,{e}^{6}{x}^{6}+8\,{e}^{5}{x}^{5}d+20\,{e}^{4}{x}^{4}{d}^{2}-20\,{x}^{3}{d}^{3}{e}^{3}-15\,{x}^{2}{d}^{4}{e}^{2}-6\,x{d}^{5}e+6\,{d}^{6} \right ) }{105\,{d}^{5}{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),x)

[Out]

-1/105*(-e*x+d)*(e*x+d)^2*(-8*e^6*x^6+8*d*e^5*x^5+20*d^2*e^4*x^4-20*d^3*e^3*x^3-15*d^4*e^2*x^2-6*d^5*e*x+6*d^6
)/d^5/e^3/(-e^2*x^2+d^2)^(9/2)

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Maxima [A]  time = 0.995122, size = 182, normalized size = 1.5 \begin{align*} \frac{x^{2}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e} + \frac{d x}{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e^{2}} - \frac{2 \, d^{2}}{35 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e^{3}} - \frac{x}{35 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d e^{2}} - \frac{4 \, x}{105 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} e^{2}} - \frac{8 \, x}{105 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{5} e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),x, algorithm="maxima")

[Out]

1/5*x^2/((-e^2*x^2 + d^2)^(7/2)*e) + 1/7*d*x/((-e^2*x^2 + d^2)^(7/2)*e^2) - 2/35*d^2/((-e^2*x^2 + d^2)^(7/2)*e
^3) - 1/35*x/((-e^2*x^2 + d^2)^(5/2)*d*e^2) - 4/105*x/((-e^2*x^2 + d^2)^(3/2)*d^3*e^2) - 8/105*x/(sqrt(-e^2*x^
2 + d^2)*d^5*e^2)

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Fricas [B]  time = 2.31488, size = 486, normalized size = 4.02 \begin{align*} -\frac{6 \, e^{7} x^{7} - 6 \, d e^{6} x^{6} - 18 \, d^{2} e^{5} x^{5} + 18 \, d^{3} e^{4} x^{4} + 18 \, d^{4} e^{3} x^{3} - 18 \, d^{5} e^{2} x^{2} - 6 \, d^{6} e x + 6 \, d^{7} -{\left (8 \, e^{6} x^{6} - 8 \, d e^{5} x^{5} - 20 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x - 6 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{105 \,{\left (d^{5} e^{10} x^{7} - d^{6} e^{9} x^{6} - 3 \, d^{7} e^{8} x^{5} + 3 \, d^{8} e^{7} x^{4} + 3 \, d^{9} e^{6} x^{3} - 3 \, d^{10} e^{5} x^{2} - d^{11} e^{4} x + d^{12} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),x, algorithm="fricas")

[Out]

-1/105*(6*e^7*x^7 - 6*d*e^6*x^6 - 18*d^2*e^5*x^5 + 18*d^3*e^4*x^4 + 18*d^4*e^3*x^3 - 18*d^5*e^2*x^2 - 6*d^6*e*
x + 6*d^7 - (8*e^6*x^6 - 8*d*e^5*x^5 - 20*d^2*e^4*x^4 + 20*d^3*e^3*x^3 + 15*d^4*e^2*x^2 + 6*d^5*e*x - 6*d^6)*s
qrt(-e^2*x^2 + d^2))/(d^5*e^10*x^7 - d^6*e^9*x^6 - 3*d^7*e^8*x^5 + 3*d^8*e^7*x^4 + 3*d^9*e^6*x^3 - 3*d^10*e^5*
x^2 - d^11*e^4*x + d^12*e^3)

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Sympy [C]  time = 17.9308, size = 904, normalized size = 7.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x+d)/(-e**2*x**2+d**2)**(9/2),x)

[Out]

d*Piecewise((35*I*d**4*x**3/(-105*d**13*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d*
*2) - 315*d**9*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)) - 28*I*d**2
*e**2*x**5/(-105*d**13*sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) - 315*d**9*e*
*4*x**4*sqrt(-1 + e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)) + 8*I*e**4*x**7/(-105*d**13*
sqrt(-1 + e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(-1 + e**2*
x**2/d**2) + 105*d**7*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (-35*d**4*x**3/(-10
5*d**13*sqrt(1 - e**2*x**2/d**2) + 315*d**11*e**2*x**2*sqrt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 -
e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(1 - e**2*x**2/d**2)) + 28*d**2*e**2*x**5/(-105*d**13*sqrt(1 - e**2*x
**2/d**2) + 315*d**11*e**2*x**2*sqrt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 - e**2*x**2/d**2) + 105*d
**7*e**6*x**6*sqrt(1 - e**2*x**2/d**2)) - 8*e**4*x**7/(-105*d**13*sqrt(1 - e**2*x**2/d**2) + 315*d**11*e**2*x*
*2*sqrt(1 - e**2*x**2/d**2) - 315*d**9*e**4*x**4*sqrt(1 - e**2*x**2/d**2) + 105*d**7*e**6*x**6*sqrt(1 - e**2*x
**2/d**2)), True)) + e*Piecewise((2*d**2/(-35*d**6*e**4*sqrt(d**2 - e**2*x**2) + 105*d**4*e**6*x**2*sqrt(d**2
- e**2*x**2) - 105*d**2*e**8*x**4*sqrt(d**2 - e**2*x**2) + 35*e**10*x**6*sqrt(d**2 - e**2*x**2)) - 7*e**2*x**2
/(-35*d**6*e**4*sqrt(d**2 - e**2*x**2) + 105*d**4*e**6*x**2*sqrt(d**2 - e**2*x**2) - 105*d**2*e**8*x**4*sqrt(d
**2 - e**2*x**2) + 35*e**10*x**6*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(9/2)), True))

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Giac [A]  time = 1.19429, size = 104, normalized size = 0.86 \begin{align*} \frac{{\left ({\left ({\left (4 \, x^{2}{\left (\frac{2 \, x^{2} e^{4}}{d^{5}} - \frac{7 \, e^{2}}{d^{3}}\right )} + \frac{35}{d}\right )} x + 21 \, e^{\left (-1\right )}\right )} x^{2} - 6 \, d^{2} e^{\left (-3\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{105 \,{\left (x^{2} e^{2} - d^{2}\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(9/2),x, algorithm="giac")

[Out]

1/105*(((4*x^2*(2*x^2*e^4/d^5 - 7*e^2/d^3) + 35/d)*x + 21*e^(-1))*x^2 - 6*d^2*e^(-3))*sqrt(-x^2*e^2 + d^2)/(x^
2*e^2 - d^2)^4