\(\int \frac {1}{x (d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [216]

Optimal result

Integrand size = 27, antiderivative size = 234 \[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}} \]

[Out]

8/13*d*(-e*x+d)/(-e^2*x^2+d^2)^(13/2)-4/13*e*x/d/(-e^2*x^2+d^2)^(11/2)+1/117*(-40*e*x+13*d)/d^3/(-e^2*x^2+d^2)
^(9/2)+1/819*(-320*e*x+117*d)/d^5/(-e^2*x^2+d^2)^(7/2)+1/1365*(-640*e*x+273*d)/d^7/(-e^2*x^2+d^2)^(5/2)+1/819*
(-512*e*x+273*d)/d^9/(-e^2*x^2+d^2)^(3/2)-arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^11+1/819*(-1024*e*x+819*d)/d^11/(-
e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 837, 12, 272, 65, 214} \[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}} \]

[In]

Int[1/(x*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(8*d*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2)) - (4*e*x)/(13*d*(d^2 - e^2*x^2)^(11/2)) + (13*d - 40*e*x)/(117*d^3
*(d^2 - e^2*x^2)^(9/2)) + (117*d - 320*e*x)/(819*d^5*(d^2 - e^2*x^2)^(7/2)) + (273*d - 640*e*x)/(1365*d^7*(d^2
- e^2*x^2)^(5/2)) + (273*d - 512*e*x)/(819*d^9*(d^2 - e^2*x^2)^(3/2)) + (819*d - 1024*e*x)/(819*d^11*Sqrt[d^2
- e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]/d^11

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 866

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

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^4}{x \left (d^2-e^2 x^2\right )^{15/2}} \, dx \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {\int \frac {-13 d^4+44 d^3 e x+13 d^2 e^2 x^2}{x \left (d^2-e^2 x^2\right )^{13/2}} \, dx}{13 d^2} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {\int \frac {143 d^4-440 d^3 e x}{x \left (d^2-e^2 x^2\right )^{11/2}} \, dx}{143 d^4} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {\int \frac {1287 d^6 e^2-3520 d^5 e^3 x}{x \left (d^2-e^2 x^2\right )^{9/2}} \, dx}{1287 d^8 e^2} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {\int \frac {9009 d^8 e^4-21120 d^7 e^5 x}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{9009 d^{12} e^4} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {45045 d^{10} e^6-84480 d^9 e^7 x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{45045 d^{16} e^6} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {135135 d^{12} e^8-168960 d^{11} e^9 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{135135 d^{20} e^8} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {135135 d^{14} e^{10}}{x \sqrt {d^2-e^2 x^2}} \, dx}{135135 d^{24} e^{10}} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^{10}} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^{10}} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{d^{10} e^2} \\ & = \frac {8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac {13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {819 d-1024 e x}{819 d^{11} \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (9839 d^9+22976 d^8 e x-4466 d^7 e^2 x^2-56304 d^6 e^3 x^3-34156 d^5 e^4 x^4+40240 d^4 e^5 x^5+45735 d^3 e^6 x^6-1540 d^2 e^7 x^7-16385 d e^8 x^8-5120 e^9 x^9\right )}{(d-e x)^3 (d+e x)^7}+8190 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{4095 d^{11}} \]

[In]

Integrate[1/(x*(d + e*x)^4*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(9839*d^9 + 22976*d^8*e*x - 4466*d^7*e^2*x^2 - 56304*d^6*e^3*x^3 - 34156*d^5*e^4*x^4 + 4
0240*d^4*e^5*x^5 + 45735*d^3*e^6*x^6 - 1540*d^2*e^7*x^7 - 16385*d*e^8*x^8 - 5120*e^9*x^9))/((d - e*x)^3*(d + e
*x)^7) + 8190*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(4095*d^11)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1327\) vs. \(2(204)=408\).

Time = 0.44 (sec) , antiderivative size = 1328, normalized size of antiderivative = 5.68

method	result	size
default	\(\text {Expression too large to display}\)	\(1328\)

[In]

int(1/x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/d^4*(1/5/d^2/(-e^2*x^2+d^2)^(5/2)+1/d^2*(1/3/d^2/(-e^2*x^2+d^2)^(3/2)+1/d^2*(1/d^2/(-e^2*x^2+d^2)^(1/2)-1/d^
2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x))))-1/e^3/d*(-1/13/d/e/(x+d/e)^4/(-(x+d/e)^2*e^2
+2*d*e*(x+d/e))^(5/2)+9/13*e/d*(-1/11/d/e/(x+d/e)^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+8/11*e/d*(-1/9/d/e/(x
+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-1/7/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+6/
7*e/d*(-1/10*(-2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+4/5/d^2*(-1/6*(-2*(x+d/e)*e^2
+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2*(x+d/e)*e^2+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(1/2)))))))-1/e/d^3*(-1/9/d/e/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-1/7/d/e/(x+d/e
)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+6/7*e/d*(-1/10*(-2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(5/2)+4/5/d^2*(-1/6*(-2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2*
(x+d/e)*e^2+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))))-1/d^4*(-1/7/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(5/2)+6/7*e/d*(-1/10*(-2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+4/5/d^2*(-1/6*(-
2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2*(x+d/e)*e^2+2*d*e)/(-(x+d/e)
^2*e^2+2*d*e*(x+d/e))^(1/2))))-1/e^2/d^2*(-1/11/d/e/(x+d/e)^3/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+8/11*e/d*(-
1/9/d/e/(x+d/e)^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+7/9*e/d*(-1/7/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e)
)^(5/2)+6/7*e/d*(-1/10*(-2*(x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+4/5/d^2*(-1/6*(-2*(
x+d/e)*e^2+2*d*e)/d^2/e^2/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-1/3/e^2/d^4*(-2*(x+d/e)*e^2+2*d*e)/(-(x+d/e)^2*
e^2+2*d*e*(x+d/e))^(1/2))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (205) = 410\).

Time = 0.76 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.85 \[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {9839 \, e^{10} x^{10} + 39356 \, d e^{9} x^{9} + 29517 \, d^{2} e^{8} x^{8} - 78712 \, d^{3} e^{7} x^{7} - 137746 \, d^{4} e^{6} x^{6} + 137746 \, d^{6} e^{4} x^{4} + 78712 \, d^{7} e^{3} x^{3} - 29517 \, d^{8} e^{2} x^{2} - 39356 \, d^{9} e x - 9839 \, d^{10} + 4095 \, {\left (e^{10} x^{10} + 4 \, d e^{9} x^{9} + 3 \, d^{2} e^{8} x^{8} - 8 \, d^{3} e^{7} x^{7} - 14 \, d^{4} e^{6} x^{6} + 14 \, d^{6} e^{4} x^{4} + 8 \, d^{7} e^{3} x^{3} - 3 \, d^{8} e^{2} x^{2} - 4 \, d^{9} e x - d^{10}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (5120 \, e^{9} x^{9} + 16385 \, d e^{8} x^{8} + 1540 \, d^{2} e^{7} x^{7} - 45735 \, d^{3} e^{6} x^{6} - 40240 \, d^{4} e^{5} x^{5} + 34156 \, d^{5} e^{4} x^{4} + 56304 \, d^{6} e^{3} x^{3} + 4466 \, d^{7} e^{2} x^{2} - 22976 \, d^{8} e x - 9839 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4095 \, {\left (d^{11} e^{10} x^{10} + 4 \, d^{12} e^{9} x^{9} + 3 \, d^{13} e^{8} x^{8} - 8 \, d^{14} e^{7} x^{7} - 14 \, d^{15} e^{6} x^{6} + 14 \, d^{17} e^{4} x^{4} + 8 \, d^{18} e^{3} x^{3} - 3 \, d^{19} e^{2} x^{2} - 4 \, d^{20} e x - d^{21}\right )}} \]

[In]

integrate(1/x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/4095*(9839*e^10*x^10 + 39356*d*e^9*x^9 + 29517*d^2*e^8*x^8 - 78712*d^3*e^7*x^7 - 137746*d^4*e^6*x^6 + 137746
*d^6*e^4*x^4 + 78712*d^7*e^3*x^3 - 29517*d^8*e^2*x^2 - 39356*d^9*e*x - 9839*d^10 + 4095*(e^10*x^10 + 4*d*e^9*x
^9 + 3*d^2*e^8*x^8 - 8*d^3*e^7*x^7 - 14*d^4*e^6*x^6 + 14*d^6*e^4*x^4 + 8*d^7*e^3*x^3 - 3*d^8*e^2*x^2 - 4*d^9*e
*x - d^10)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (5120*e^9*x^9 + 16385*d*e^8*x^8 + 1540*d^2*e^7*x^7 - 45735*d^3
*e^6*x^6 - 40240*d^4*e^5*x^5 + 34156*d^5*e^4*x^4 + 56304*d^6*e^3*x^3 + 4466*d^7*e^2*x^2 - 22976*d^8*e*x - 9839
*d^9)*sqrt(-e^2*x^2 + d^2))/(d^11*e^10*x^10 + 4*d^12*e^9*x^9 + 3*d^13*e^8*x^8 - 8*d^14*e^7*x^7 - 14*d^15*e^6*x
^6 + 14*d^17*e^4*x^4 + 8*d^18*e^3*x^3 - 3*d^19*e^2*x^2 - 4*d^20*e*x - d^21)

Sympy [F]

\[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]

[In]

integrate(1/x/(e*x+d)**4/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral(1/(x*(-(-d + e*x)*(d + e*x))**(7/2)*(d + e*x)**4), x)

Maxima [F]

\[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x), x)

Giac [F]

\[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(7/2)*(e*x + d)^4*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^4} \,d x \]

[In]

int(1/(x*(d^2 - e^2*x^2)^(7/2)*(d + e*x)^4),x)

[Out]

int(1/(x*(d^2 - e^2*x^2)^(7/2)*(d + e*x)^4), x)