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

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

[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

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Rubi [A]
time = 0.43, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1819, 821, 272, 65, 214} \begin {gather*} -\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 {4 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{12}}-\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}} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.93, size = 193, normalized size = 0.71 \begin {gather*} \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 {8 e \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^{12}} \end {gather*}

Antiderivative was successfully verified.

[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) - (8*e*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^
2*x^2]/d])/d^12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1428\) vs. \(2(239)=478\).
time = 0.16, size = 1429, normalized size = 5.27

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*e/d^5*(-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*e^2*(x+d/e)+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*e^2*(x+d/e)+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*e^2*(x+d/e)+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))))+2/e/d^3*(-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*e^2*(x+d/e)+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*e^2*(x+d/e)+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*e^2*(x+d/e)+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))))))+1/e^2/d^2*(-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*e^2*(x+d/e)+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*e^2*(x+d/e)+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*e^2*(x+d/e)+2*d*e)/(-(x+d
/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))))))+1/d^4*(-1/d^2/x/(-e^2*x^2+d^2)^(5/2)+6*e^2/d^2*(1/5*x/d^2/(-e^2*x^2+d^2)^
(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))))+3/d^4*(-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*e^2*(x+d/e)+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*e^2*(x+d/e)+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*e^2*(x+d/e)+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/
2)))))-4/d^5*e*(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))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.67, size = 425, normalized size = 1.57 \begin {gather*} -\frac {366136 \, x^{11} e^{11} + 1464544 \, d x^{10} e^{10} + 1098408 \, d^{2} x^{9} e^{9} - 2929088 \, d^{3} x^{8} e^{8} - 5125904 \, d^{4} x^{7} e^{7} + 5125904 \, d^{6} x^{5} e^{5} + 2929088 \, d^{7} x^{4} e^{4} - 1098408 \, d^{8} x^{3} e^{3} - 1464544 \, d^{9} x^{2} e^{2} - 366136 \, d^{10} x e + 180180 \, {\left (x^{11} e^{11} + 4 \, d x^{10} e^{10} + 3 \, d^{2} x^{9} e^{9} - 8 \, d^{3} x^{8} e^{8} - 14 \, d^{4} x^{7} e^{7} + 14 \, d^{6} x^{5} e^{5} + 8 \, d^{7} x^{4} e^{4} - 3 \, d^{8} x^{3} e^{3} - 4 \, d^{9} x^{2} e^{2} - d^{10} x e\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (305920 \, x^{10} e^{10} + 1043500 \, d x^{9} e^{9} + 350000 \, d^{2} x^{8} e^{8} - 2496180 \, d^{3} x^{7} e^{7} - 2748320 \, d^{4} x^{6} e^{6} + 1301264 \, d^{5} x^{5} e^{5} + 3157776 \, d^{6} x^{4} e^{4} + 700504 \, d^{7} x^{3} e^{3} - 1014094 \, d^{8} x^{2} e^{2} - 546316 \, d^{9} x e - 45045 \, d^{10}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{45045 \, {\left (d^{12} x^{11} e^{10} + 4 \, d^{13} x^{10} e^{9} + 3 \, d^{14} x^{9} e^{8} - 8 \, d^{15} x^{8} e^{7} - 14 \, d^{16} x^{7} e^{6} + 14 \, d^{18} x^{5} e^{4} + 8 \, d^{19} x^{4} e^{3} - 3 \, d^{20} x^{3} e^{2} - 4 \, d^{21} x^{2} e - d^{22} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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