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

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

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Rubi [A]  time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {796, 778, 192, 191} \begin {gather*} \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}} \end {gather*}

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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.05, size = 126, normalized size = 0.85 \begin {gather*} \frac {-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+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}} \end {gather*}

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])

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IntegrateAlgebraic [A]  time = 0.62, size = 126, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-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+16 d e^7 x^7-16 e^8 x^8\right )}{315 d^7 e^3 (d-e x)^5 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-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 + 16*d*e^7*x^7 - 16*e^8*x^8))/(315*d^7*e^3*(d - e*x)^5*(d + e*x)^4)

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fricas [B]  time = 0.76, size = 305, normalized size = 2.06 \begin {gather*} -\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 {gather*}

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)

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giac [A]  time = 0.30, size = 90, normalized size = 0.61 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 121, normalized size = 0.82 \begin {gather*} -\frac {\left (-e x +d \right ) \left (e x +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 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}} d^{7} e^{3}} \end {gather*}

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)

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maxima [A]  time = 0.45, size = 158, normalized size = 1.07 \begin {gather*} \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 {gather*}

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)

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mupad [B]  time = 2.74, size = 202, normalized size = 1.36 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}}{144\,d^3\,e^3\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{252\,e^3}-\frac {17\,x}{252\,d\,e^2}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {5}{144\,d^2\,e^3}+\frac {131\,x}{5040\,d^3\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e^2\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{315\,d^7\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(d + e*x))/(d^2 - e^2*x^2)^(11/2),x)

[Out]

(d^2 - e^2*x^2)^(1/2)/(144*d^3*e^3*(d - e*x)^5) - ((d^2 - e^2*x^2)^(1/2)*(1/(252*e^3) - (17*x)/(252*d*e^2)))/(
(d + e*x)^4*(d - e*x)^4) - ((d^2 - e^2*x^2)^(1/2)*(5/(144*d^2*e^3) + (131*x)/(5040*d^3*e^2)))/((d + e*x)^3*(d
- e*x)^3) - (8*x*(d^2 - e^2*x^2)^(1/2))/(315*d^5*e^2*(d + e*x)^2*(d - e*x)^2) - (16*x*(d^2 - e^2*x^2)^(1/2))/(
315*d^7*e^2*(d + e*x)*(d - e*x))

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sympy [C]  time = 48.46, size = 1401, normalized size = 9.47

result too large to display

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/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**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*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)) + 7
2*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*sq
rt(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)), T
rue))

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