3.31 \(\int \frac{x^2 (d+e x)}{(d^2-e^2 x^2)^{11/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{16 x}{315 d^7 e^2 \sqrt{d^2-e^2 x^2}}-\frac{8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}} \]

[Out]

(x^2*(d + e*x))/(9*d*e*(d^2 - e^2*x^2)^(9/2)) - (2*(d - 3*e*x))/(63*d*e^3*(d^2 - e^2*x^2)^(7/2)) - (2*x)/(105*
d^3*e^2*(d^2 - e^2*x^2)^(5/2)) - (8*x)/(315*d^5*e^2*(d^2 - e^2*x^2)^(3/2)) - (16*x)/(315*d^7*e^2*Sqrt[d^2 - e^
2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0617767, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {796, 778, 192, 191} \[ \frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{16 x}{315 d^7 e^2 \sqrt{d^2-e^2 x^2}}-\frac{8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x]

[Out]

(x^2*(d + e*x))/(9*d*e*(d^2 - e^2*x^2)^(9/2)) - (2*(d - 3*e*x))/(63*d*e^3*(d^2 - e^2*x^2)^(7/2)) - (2*x)/(105*
d^3*e^2*(d^2 - e^2*x^2)^(5/2)) - (8*x)/(315*d^5*e^2*(d^2 - e^2*x^2)^(3/2)) - (16*x)/(315*d^7*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 )^{11/2}} \, dx &=\frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{\int \frac{x \left (2 d^2 e-6 d e^2 x\right )}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx}{9 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 \int \frac{1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{21 d e^2}\\ &=\frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 \int \frac{1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{105 d^3 e^2}\\ &=\frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{16 \int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{315 d^5 e^2}\\ &=\frac{x^2 (d+e x)}{9 d e \left (d^2-e^2 x^2\right )^{9/2}}-\frac{2 (d-3 e x)}{63 d e^3 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{2 x}{105 d^3 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{8 x}{315 d^5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{16 x}{315 d^7 e^2 \sqrt{d^2-e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0543511, size = 126, normalized size = 0.85 \[ \frac{35 d^6 e^2 x^2+70 d^5 e^3 x^3-70 d^4 e^4 x^4-56 d^3 e^5 x^5+56 d^2 e^6 x^6+10 d^7 e x-10 d^8+16 d e^7 x^7-16 e^8 x^8}{315 d^7 e^3 (d-e x)^4 (d+e x)^3 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x]

[Out]

(-10*d^8 + 10*d^7*e*x + 35*d^6*e^2*x^2 + 70*d^5*e^3*x^3 - 70*d^4*e^4*x^4 - 56*d^3*e^5*x^5 + 56*d^2*e^6*x^6 + 1
6*d*e^7*x^7 - 16*e^8*x^8)/(315*d^7*e^3*(d - e*x)^4*(d + e*x)^3*Sqrt[d^2 - e^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.052, size = 121, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( 16\,{e}^{8}{x}^{8}-16\,{e}^{7}{x}^{7}d-56\,{e}^{6}{x}^{6}{d}^{2}+56\,{e}^{5}{x}^{5}{d}^{3}+70\,{e}^{4}{x}^{4}{d}^{4}-70\,{x}^{3}{d}^{5}{e}^{3}-35\,{x}^{2}{d}^{6}{e}^{2}-10\,x{d}^{7}e+10\,{d}^{8} \right ) }{315\,{d}^{7}{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{11}{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)^(11/2),x)

[Out]

-1/315*(-e*x+d)*(e*x+d)^2*(16*e^8*x^8-16*d*e^7*x^7-56*d^2*e^6*x^6+56*d^3*e^5*x^5+70*d^4*e^4*x^4-70*d^5*e^3*x^3
-35*d^6*e^2*x^2-10*d^7*e*x+10*d^8)/d^7/e^3/(-e^2*x^2+d^2)^(11/2)

________________________________________________________________________________________

Maxima [A]  time = 1.01844, size = 213, normalized size = 1.44 \begin{align*} \frac{x^{2}}{7 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} e} + \frac{d x}{9 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} e^{2}} - \frac{2 \, d^{2}}{63 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{9}{2}} e^{3}} - \frac{x}{63 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d e^{2}} - \frac{2 \, x}{105 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} e^{2}} - \frac{8 \, x}{315 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} e^{2}} - \frac{16 \, x}{315 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} 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)^(11/2),x, algorithm="maxima")

[Out]

1/7*x^2/((-e^2*x^2 + d^2)^(9/2)*e) + 1/9*d*x/((-e^2*x^2 + d^2)^(9/2)*e^2) - 2/63*d^2/((-e^2*x^2 + d^2)^(9/2)*e
^3) - 1/63*x/((-e^2*x^2 + d^2)^(7/2)*d*e^2) - 2/105*x/((-e^2*x^2 + d^2)^(5/2)*d^3*e^2) - 8/315*x/((-e^2*x^2 +
d^2)^(3/2)*d^5*e^2) - 16/315*x/(sqrt(-e^2*x^2 + d^2)*d^7*e^2)

________________________________________________________________________________________

Fricas [B]  time = 4.28955, size = 640, normalized size = 4.32 \begin{align*} -\frac{10 \, e^{9} x^{9} - 10 \, d e^{8} x^{8} - 40 \, d^{2} e^{7} x^{7} + 40 \, d^{3} e^{6} x^{6} + 60 \, d^{4} e^{5} x^{5} - 60 \, d^{5} e^{4} x^{4} - 40 \, d^{6} e^{3} x^{3} + 40 \, d^{7} e^{2} x^{2} + 10 \, d^{8} e x - 10 \, d^{9} -{\left (16 \, e^{8} x^{8} - 16 \, d e^{7} x^{7} - 56 \, d^{2} e^{6} x^{6} + 56 \, d^{3} e^{5} x^{5} + 70 \, d^{4} e^{4} x^{4} - 70 \, d^{5} e^{3} x^{3} - 35 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x + 10 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{315 \,{\left (d^{7} e^{12} x^{9} - d^{8} e^{11} x^{8} - 4 \, d^{9} e^{10} x^{7} + 4 \, d^{10} e^{9} x^{6} + 6 \, d^{11} e^{8} x^{5} - 6 \, d^{12} e^{7} x^{4} - 4 \, d^{13} e^{6} x^{3} + 4 \, d^{14} e^{5} x^{2} + d^{15} e^{4} x - d^{16} 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)^(11/2),x, algorithm="fricas")

[Out]

-1/315*(10*e^9*x^9 - 10*d*e^8*x^8 - 40*d^2*e^7*x^7 + 40*d^3*e^6*x^6 + 60*d^4*e^5*x^5 - 60*d^5*e^4*x^4 - 40*d^6
*e^3*x^3 + 40*d^7*e^2*x^2 + 10*d^8*e*x - 10*d^9 - (16*e^8*x^8 - 16*d*e^7*x^7 - 56*d^2*e^6*x^6 + 56*d^3*e^5*x^5
+ 70*d^4*e^4*x^4 - 70*d^5*e^3*x^3 - 35*d^6*e^2*x^2 - 10*d^7*e*x + 10*d^8)*sqrt(-e^2*x^2 + d^2))/(d^7*e^12*x^9
- d^8*e^11*x^8 - 4*d^9*e^10*x^7 + 4*d^10*e^9*x^6 + 6*d^11*e^8*x^5 - 6*d^12*e^7*x^4 - 4*d^13*e^6*x^3 + 4*d^14*
e^5*x^2 + d^15*e^4*x - d^16*e^3)

________________________________________________________________________________________

Sympy [C]  time = 38.6534, size = 1402, normalized size = 9.47 \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)**(11/2),x)

[Out]

d*Piecewise((-105*I*d**6*x**3/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/
d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*
d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) + 126*I*d**4*e**2*x**5/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d
**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x*
*6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2)) - 72*I*d**2*e**4*x**7/(315*d**17*
sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(-1 + e*
*2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(-1 + e**2*x**2/d**2))
+ 16*I*e**6*x**9/(315*d**17*sqrt(-1 + e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(-1 + e**2*x**2/d**2) + 1890
*d**13*e**4*x**4*sqrt(-1 + e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(-1 + e**2*x**2/d**2) + 315*d**9*e**8*x*
*8*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (105*d**6*x**3/(315*d**17*sqrt(1 - e**2*x**2/d**
2) - 1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**1
1*e**6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)) - 126*d**4*e**2*x**5/(315*
d**17*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 -
e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)
) + 72*d**2*e**4*x**7/(315*d**17*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1 - e**2*x**2/d**2) + 18
90*d**13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d**2) + 315*d**9*e**8*x*
*8*sqrt(1 - e**2*x**2/d**2)) - 16*e**6*x**9/(315*d**17*sqrt(1 - e**2*x**2/d**2) - 1260*d**15*e**2*x**2*sqrt(1
- e**2*x**2/d**2) + 1890*d**13*e**4*x**4*sqrt(1 - e**2*x**2/d**2) - 1260*d**11*e**6*x**6*sqrt(1 - e**2*x**2/d*
*2) + 315*d**9*e**8*x**8*sqrt(1 - e**2*x**2/d**2)), True)) + e*Piecewise((-2*d**2/(63*d**8*e**4*sqrt(d**2 - e*
*2*x**2) - 252*d**6*e**6*x**2*sqrt(d**2 - e**2*x**2) + 378*d**4*e**8*x**4*sqrt(d**2 - e**2*x**2) - 252*d**2*e*
*10*x**6*sqrt(d**2 - e**2*x**2) + 63*e**12*x**8*sqrt(d**2 - e**2*x**2)) + 9*e**2*x**2/(63*d**8*e**4*sqrt(d**2
- e**2*x**2) - 252*d**6*e**6*x**2*sqrt(d**2 - e**2*x**2) + 378*d**4*e**8*x**4*sqrt(d**2 - e**2*x**2) - 252*d**
2*e**10*x**6*sqrt(d**2 - e**2*x**2) + 63*e**12*x**8*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**4/(4*(d**2)**(11/2
)), True))

________________________________________________________________________________________

Giac [A]  time = 1.19468, size = 122, normalized size = 0.82 \begin{align*} \frac{{\left ({\left ({\left (2 \,{\left (4 \, x^{2}{\left (\frac{2 \, x^{2} e^{6}}{d^{7}} - \frac{9 \, e^{4}}{d^{5}}\right )} + \frac{63 \, e^{2}}{d^{3}}\right )} x^{2} - \frac{105}{d}\right )} x - 45 \, e^{\left (-1\right )}\right )} x^{2} + 10 \, d^{2} e^{\left (-3\right )}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{315 \,{\left (x^{2} e^{2} - d^{2}\right )}^{5}} \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)^(11/2),x, algorithm="giac")

[Out]

1/315*(((2*(4*x^2*(2*x^2*e^6/d^7 - 9*e^4/d^5) + 63*e^2/d^3)*x^2 - 105/d)*x - 45*e^(-1))*x^2 + 10*d^2*e^(-3))*s
qrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^5