\(\int \frac {1}{x^2 (d+e x)^4 (d^2-e^2 x^2)^{7/2}} \, dx\) [217]

Optimal result

Integrand size = 27, antiderivative size = 271 \[ \int \frac {1}{x^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}+\frac {4 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{12}} \]

[Out]

-8/13*e*(-e*x+d)/(-e^2*x^2+d^2)^(13/2)-4/143*e*(-24*e*x+13*d)/d^2/(-e^2*x^2+d^2)^(11/2)-1/1287*e*(-1103*e*x+57
2*d)/d^4/(-e^2*x^2+d^2)^(9/2)-1/9009*e*(-10111*e*x+5148*d)/d^6/(-e^2*x^2+d^2)^(7/2)-1/15015*e*(-23225*e*x+1201
2*d)/d^8/(-e^2*x^2+d^2)^(5/2)-1/9009*e*(-21583*e*x+12012*d)/d^10/(-e^2*x^2+d^2)^(3/2)+4*e*arctanh((-e^2*x^2+d^
2)^(1/2)/d)/d^12-1/9009*e*(-52175*e*x+36036*d)/d^12/(-e^2*x^2+d^2)^(1/2)-(-e^2*x^2+d^2)^(1/2)/d^12/x

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1819, 821, 272, 65, 214} \[ \int \frac {1}{x^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{12}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}} \]

[In]

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

[Out]

(-8*e*(d - e*x))/(13*(d^2 - e^2*x^2)^(13/2)) - (4*e*(13*d - 24*e*x))/(143*d^2*(d^2 - e^2*x^2)^(11/2)) - (e*(57
2*d - 1103*e*x))/(1287*d^4*(d^2 - e^2*x^2)^(9/2)) - (e*(5148*d - 10111*e*x))/(9009*d^6*(d^2 - e^2*x^2)^(7/2))
- (e*(12012*d - 23225*e*x))/(15015*d^8*(d^2 - e^2*x^2)^(5/2)) - (e*(12012*d - 21583*e*x))/(9009*d^10*(d^2 - e^
2*x^2)^(3/2)) - (e*(36036*d - 52175*e*x))/(9009*d^12*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(d^12*x) + (4*
e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^12

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 821

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

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^2 \left (d^2-e^2 x^2\right )^{15/2}} \, dx \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {\int \frac {-13 d^4+52 d^3 e x-83 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{13/2}} \, dx}{13 d^2} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}+\frac {\int \frac {143 d^4-572 d^3 e x+960 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx}{143 d^4} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\int \frac {-1287 d^4+5148 d^3 e x-8824 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{9/2}} \, dx}{1287 d^6} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {\int \frac {9009 d^4-36036 d^3 e x+60666 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx}{9009 d^8} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-45045 d^4+180180 d^3 e x-278700 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{45045 d^{10}} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {135135 d^4-540540 d^3 e x+647490 d^2 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{135135 d^{12}} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-135135 d^4+540540 d^3 e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{135135 d^{14}} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}-\frac {(4 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^{11}} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}-\frac {(2 e) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^{11}} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}+\frac {4 \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^{11} e} \\ & = -\frac {8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac {4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e (36036 d-52175 e x)}{9009 d^{12} \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^{12} x}+\frac {4 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{12}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (45045 d^{10}+546316 d^9 e x+1014094 d^8 e^2 x^2-700504 d^7 e^3 x^3-3157776 d^6 e^4 x^4-1301264 d^5 e^5 x^5+2748320 d^4 e^6 x^6+2496180 d^3 e^7 x^7-350000 d^2 e^8 x^8-1043500 d e^9 x^9-305920 e^{10} x^{10}\right )}{45045 d^{12} x (-d+e x)^3 (d+e x)^7}+\frac {4 \sqrt {d^2} e \log (x)}{d^{13}}-\frac {4 \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{d^{13}} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(45045*d^10 + 546316*d^9*e*x + 1014094*d^8*e^2*x^2 - 700504*d^7*e^3*x^3 - 3157776*d^6*e^4
*x^4 - 1301264*d^5*e^5*x^5 + 2748320*d^4*e^6*x^6 + 2496180*d^3*e^7*x^7 - 350000*d^2*e^8*x^8 - 1043500*d*e^9*x^
9 - 305920*e^10*x^10))/(45045*d^12*x*(-d + e*x)^3*(d + e*x)^7) + (4*Sqrt[d^2]*e*Log[x])/d^13 - (4*Sqrt[d^2]*e*
Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/d^13

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(519\) vs. \(2(239)=478\).

Time = 0.43 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.92

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{12} x}-\frac {1257577 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{1921920 d^{10} e^{2} \left (x +\frac {d}{e}\right )^{3}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{104 d^{6} e^{6} \left (x +\frac {d}{e}\right )^{7}}-\frac {103 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{2288 d^{7} e^{5} \left (x +\frac {d}{e}\right )^{6}}-\frac {665 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5148 d^{8} e^{4} \left (x +\frac {d}{e}\right )^{5}}-\frac {86917 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{288288 d^{9} e^{3} \left (x +\frac {d}{e}\right )^{4}}-\frac {17417683 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{11531520 d^{11} e \left (x +\frac {d}{e}\right )^{2}}+\frac {59 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3840 d^{11} e \left (x -\frac {d}{e}\right )^{2}}-\frac {569 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3840 d^{12} \left (x -\frac {d}{e}\right )}-\frac {65075293 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{11531520 d^{12} \left (x +\frac {d}{e}\right )}+\frac {4 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{11} \sqrt {d^{2}}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{640 d^{10} e^{2} \left (x -\frac {d}{e}\right )^{3}}\)	\(520\)
default	\(\text {Expression too large to display}\)	\(1429\)

[In]

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

[Out]

-(-e^2*x^2+d^2)^(1/2)/d^12/x-1257577/1921920/d^10/e^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-1/104/d^6
/e^6/(x+d/e)^7*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-103/2288/d^7/e^5/(x+d/e)^6*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^
(1/2)-665/5148/d^8/e^4/(x+d/e)^5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-86917/288288/d^9/e^3/(x+d/e)^4*(-(x+d/e)
^2*e^2+2*d*e*(x+d/e))^(1/2)-17417683/11531520/d^11/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+59/3840/d^
11/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-569/3840/d^12/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
-65075293/11531520/d^12/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+4/d^11*e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1
/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/640/d^10/e^2/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.12 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {366136 \, e^{11} x^{11} + 1464544 \, d e^{10} x^{10} + 1098408 \, d^{2} e^{9} x^{9} - 2929088 \, d^{3} e^{8} x^{8} - 5125904 \, d^{4} e^{7} x^{7} + 5125904 \, d^{6} e^{5} x^{5} + 2929088 \, d^{7} e^{4} x^{4} - 1098408 \, d^{8} e^{3} x^{3} - 1464544 \, d^{9} e^{2} x^{2} - 366136 \, d^{10} e x + 180180 \, {\left (e^{11} x^{11} + 4 \, d e^{10} x^{10} + 3 \, d^{2} e^{9} x^{9} - 8 \, d^{3} e^{8} x^{8} - 14 \, d^{4} e^{7} x^{7} + 14 \, d^{6} e^{5} x^{5} + 8 \, d^{7} e^{4} x^{4} - 3 \, d^{8} e^{3} x^{3} - 4 \, d^{9} e^{2} x^{2} - d^{10} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (305920 \, e^{10} x^{10} + 1043500 \, d e^{9} x^{9} + 350000 \, d^{2} e^{8} x^{8} - 2496180 \, d^{3} e^{7} x^{7} - 2748320 \, d^{4} e^{6} x^{6} + 1301264 \, d^{5} e^{5} x^{5} + 3157776 \, d^{6} e^{4} x^{4} + 700504 \, d^{7} e^{3} x^{3} - 1014094 \, d^{8} e^{2} x^{2} - 546316 \, d^{9} e x - 45045 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{45045 \, {\left (d^{12} e^{10} x^{11} + 4 \, d^{13} e^{9} x^{10} + 3 \, d^{14} e^{8} x^{9} - 8 \, d^{15} e^{7} x^{8} - 14 \, d^{16} e^{6} x^{7} + 14 \, d^{18} e^{4} x^{5} + 8 \, d^{19} e^{3} x^{4} - 3 \, d^{20} e^{2} x^{3} - 4 \, d^{21} e x^{2} - d^{22} x\right )}} \]

[In]

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

[Out]

-1/45045*(366136*e^11*x^11 + 1464544*d*e^10*x^10 + 1098408*d^2*e^9*x^9 - 2929088*d^3*e^8*x^8 - 5125904*d^4*e^7
*x^7 + 5125904*d^6*e^5*x^5 + 2929088*d^7*e^4*x^4 - 1098408*d^8*e^3*x^3 - 1464544*d^9*e^2*x^2 - 366136*d^10*e*x
+ 180180*(e^11*x^11 + 4*d*e^10*x^10 + 3*d^2*e^9*x^9 - 8*d^3*e^8*x^8 - 14*d^4*e^7*x^7 + 14*d^6*e^5*x^5 + 8*d^7
*e^4*x^4 - 3*d^8*e^3*x^3 - 4*d^9*e^2*x^2 - d^10*e*x)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (305920*e^10*x^10 +
1043500*d*e^9*x^9 + 350000*d^2*e^8*x^8 - 2496180*d^3*e^7*x^7 - 2748320*d^4*e^6*x^6 + 1301264*d^5*e^5*x^5 + 315
7776*d^6*e^4*x^4 + 700504*d^7*e^3*x^3 - 1014094*d^8*e^2*x^2 - 546316*d^9*e*x - 45045*d^10)*sqrt(-e^2*x^2 + d^2
))/(d^12*e^10*x^11 + 4*d^13*e^9*x^10 + 3*d^14*e^8*x^9 - 8*d^15*e^7*x^8 - 14*d^16*e^6*x^7 + 14*d^18*e^4*x^5 + 8
*d^19*e^3*x^4 - 3*d^20*e^2*x^3 - 4*d^21*e*x^2 - d^22*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(1/x^2/(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^2), x)

Giac [F]

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

[In]

integrate(1/x^2/(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^2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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