\(\int \frac {x^2}{(d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [213]

Optimal result

Integrand size = 27, antiderivative size = 209 \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}} \]

[Out]

14/2145*x/d^4/e^2/(-e^2*x^2+d^2)^(5/2)-1/13*d/e^3/(e*x+d)^4/(-e^2*x^2+d^2)^(5/2)+17/143/e^3/(e*x+d)^3/(-e^2*x^
2+d^2)^(5/2)-7/1287/d/e^3/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2)-7/1287/d^2/e^3/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+56/6435*x
/d^6/e^2/(-e^2*x^2+d^2)^(3/2)+112/6435*x/d^8/e^2/(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1653, 807, 673, 198, 197} \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}} \]

[In]

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

[Out]

(14*x)/(2145*d^4*e^2*(d^2 - e^2*x^2)^(5/2)) - d/(13*e^3*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + 17/(143*e^3*(d +
e*x)^3*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d*e^3*(d + e*x)^2*(d^2 - e^2*x^2)^(5/2)) - 7/(1287*d^2*e^3*(d + e*x)*(
d^2 - e^2*x^2)^(5/2)) + (56*x)/(6435*d^6*e^2*(d^2 - e^2*x^2)^(3/2)) + (112*x)/(6435*d^8*e^2*Sqrt[d^2 - e^2*x^2
])

Rule 197

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 198

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 673

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a + c*x^2)^(p +
1)/(2*c*d*(m + p + 1))), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^
p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p +
2], 0]

Rule 807

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d
+ e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

Rule 1653

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {3 d^2 e^2-5 d e^3 x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{8 e^4} \\ & = -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{8 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(7 d) \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{104 e^2} \\ & = -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{143 e^2} \\ & = -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {49 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{1287 d e^2} \\ & = -\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {14 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{429 d^2 e^2} \\ & = \frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{2145 d^4 e^2} \\ & = \frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{6435 d^6 e^2} \\ & = \frac {14 x}{2145 d^4 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {d}{13 e^3 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {17}{143 e^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d e^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {7}{1287 d^2 e^3 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {56 x}{6435 d^6 e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {112 x}{6435 d^8 e^2 \sqrt {d^2-e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (200 d^9+800 d^8 e x+700 d^7 e^2 x^2+945 d^6 e^3 x^3-280 d^5 e^4 x^4-1358 d^4 e^5 x^5-672 d^3 e^6 x^6+392 d^2 e^7 x^7+448 d e^8 x^8+112 e^9 x^9\right )}{6435 d^8 e^3 (d-e x)^3 (d+e x)^7} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(200*d^9 + 800*d^8*e*x + 700*d^7*e^2*x^2 + 945*d^6*e^3*x^3 - 280*d^5*e^4*x^4 - 1358*d^4*e
^5*x^5 - 672*d^3*e^6*x^6 + 392*d^2*e^7*x^7 + 448*d*e^8*x^8 + 112*e^9*x^9))/(6435*d^8*e^3*(d - e*x)^3*(d + e*x)
^7)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.63

method	result	size
gosper	\(\frac {\left (-e x +d \right ) \left (112 e^{9} x^{9}+448 d \,e^{8} x^{8}+392 d^{2} e^{7} x^{7}-672 d^{3} e^{6} x^{6}-1358 d^{4} e^{5} x^{5}-280 d^{5} e^{4} x^{4}+945 d^{6} e^{3} x^{3}+700 x^{2} d^{7} e^{2}+800 x \,d^{8} e +200 d^{9}\right )}{6435 \left (e x +d \right )^{3} d^{8} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(132\)
trager	\(\frac {\left (112 e^{9} x^{9}+448 d \,e^{8} x^{8}+392 d^{2} e^{7} x^{7}-672 d^{3} e^{6} x^{6}-1358 d^{4} e^{5} x^{5}-280 d^{5} e^{4} x^{4}+945 d^{6} e^{3} x^{3}+700 x^{2} d^{7} e^{2}+800 x \,d^{8} e +200 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6435 d^{8} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e^{3}}\)	\(134\)
default	\(\text {Expression too large to display}\)	\(985\)

[In]

int(x^2/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/6435*(-e*x+d)*(112*e^9*x^9+448*d*e^8*x^8+392*d^2*e^7*x^7-672*d^3*e^6*x^6-1358*d^4*e^5*x^5-280*d^5*e^4*x^4+94
5*d^6*e^3*x^3+700*d^7*e^2*x^2+800*d^8*e*x+200*d^9)/(e*x+d)^3/d^8/e^3/(-e^2*x^2+d^2)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.61 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.52 \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {200 \, e^{10} x^{10} + 800 \, d e^{9} x^{9} + 600 \, d^{2} e^{8} x^{8} - 1600 \, d^{3} e^{7} x^{7} - 2800 \, d^{4} e^{6} x^{6} + 2800 \, d^{6} e^{4} x^{4} + 1600 \, d^{7} e^{3} x^{3} - 600 \, d^{8} e^{2} x^{2} - 800 \, d^{9} e x - 200 \, d^{10} - {\left (112 \, e^{9} x^{9} + 448 \, d e^{8} x^{8} + 392 \, d^{2} e^{7} x^{7} - 672 \, d^{3} e^{6} x^{6} - 1358 \, d^{4} e^{5} x^{5} - 280 \, d^{5} e^{4} x^{4} + 945 \, d^{6} e^{3} x^{3} + 700 \, d^{7} e^{2} x^{2} + 800 \, d^{8} e x + 200 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{8} e^{13} x^{10} + 4 \, d^{9} e^{12} x^{9} + 3 \, d^{10} e^{11} x^{8} - 8 \, d^{11} e^{10} x^{7} - 14 \, d^{12} e^{9} x^{6} + 14 \, d^{14} e^{7} x^{4} + 8 \, d^{15} e^{6} x^{3} - 3 \, d^{16} e^{5} x^{2} - 4 \, d^{17} e^{4} x - d^{18} e^{3}\right )}} \]

[In]

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

[Out]

1/6435*(200*e^10*x^10 + 800*d*e^9*x^9 + 600*d^2*e^8*x^8 - 1600*d^3*e^7*x^7 - 2800*d^4*e^6*x^6 + 2800*d^6*e^4*x
^4 + 1600*d^7*e^3*x^3 - 600*d^8*e^2*x^2 - 800*d^9*e*x - 200*d^10 - (112*e^9*x^9 + 448*d*e^8*x^8 + 392*d^2*e^7*
x^7 - 672*d^3*e^6*x^6 - 1358*d^4*e^5*x^5 - 280*d^5*e^4*x^4 + 945*d^6*e^3*x^3 + 700*d^7*e^2*x^2 + 800*d^8*e*x +
200*d^9)*sqrt(-e^2*x^2 + d^2))/(d^8*e^13*x^10 + 4*d^9*e^12*x^9 + 3*d^10*e^11*x^8 - 8*d^11*e^10*x^7 - 14*d^12*
e^9*x^6 + 14*d^14*e^7*x^4 + 8*d^15*e^6*x^3 - 3*d^16*e^5*x^2 - 4*d^17*e^4*x - d^18*e^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (181) = 362\).

Time = 0.20 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.92 \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {d}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{6} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{5} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{3}\right )}} + \frac {17}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} - \frac {7}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} - \frac {7}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3}\right )}} + \frac {14 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}} + \frac {56 \, x}{6435 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} e^{2}} + \frac {112 \, x}{6435 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} e^{2}} \]

[In]

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

[Out]

-1/13*d/((-e^2*x^2 + d^2)^(5/2)*e^7*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d*e^6*x^3 + 6*(-e^2*x^2 + d^2)^(5/2)*d^2*e^
5*x^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^3*e^4*x + (-e^2*x^2 + d^2)^(5/2)*d^4*e^3) + 17/143/((-e^2*x^2 + d^2)^(5/2)*
e^6*x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d*e^5*x^2 + 3*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x + (-e^2*x^2 + d^2)^(5/2)*d^3
*e^3) - 7/1287/((-e^2*x^2 + d^2)^(5/2)*d*e^5*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x + (-e^2*x^2 + d^2)^(5/2)
*d^3*e^3) - 7/1287/((-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x + (-e^2*x^2 + d^2)^(5/2)*d^3*e^3) + 14/2145*x/((-e^2*x^2
+ d^2)^(5/2)*d^4*e^2) + 56/6435*x/((-e^2*x^2 + d^2)^(3/2)*d^6*e^2) + 112/6435*x/(sqrt(-e^2*x^2 + d^2)*d^8*e^2)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.99 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {227}{6864\,d^3\,e^3}-\frac {353\,x}{17160\,d^4\,e^2}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {353}{41184\,d^5\,e^3}-\frac {56\,x}{6435\,d^6\,e^2}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^2\,e^3\,{\left (d+e\,x\right )}^7}+\frac {\sqrt {d^2-e^2\,x^2}}{2288\,d^3\,e^3\,{\left (d+e\,x\right )}^6}+\frac {37\,\sqrt {d^2-e^2\,x^2}}{5148\,d^4\,e^3\,{\left (d+e\,x\right )}^5}+\frac {353\,\sqrt {d^2-e^2\,x^2}}{41184\,d^5\,e^3\,{\left (d+e\,x\right )}^4}+\frac {112\,x\,\sqrt {d^2-e^2\,x^2}}{6435\,d^8\,e^2\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*(227/(6864*d^3*e^3) - (353*x)/(17160*d^4*e^2)))/((d + e*x)^3*(d - e*x)^3) - ((d^2 - e^2
*x^2)^(1/2)*(353/(41184*d^5*e^3) - (56*x)/(6435*d^6*e^2)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(
104*d^2*e^3*(d + e*x)^7) + (d^2 - e^2*x^2)^(1/2)/(2288*d^3*e^3*(d + e*x)^6) + (37*(d^2 - e^2*x^2)^(1/2))/(5148
*d^4*e^3*(d + e*x)^5) + (353*(d^2 - e^2*x^2)^(1/2))/(41184*d^5*e^3*(d + e*x)^4) + (112*x*(d^2 - e^2*x^2)^(1/2)
)/(6435*d^8*e^2*(d + e*x)*(d - e*x))