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

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

[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

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Rubi [A]  time = 0.679868, antiderivative size = 271, normalized size of antiderivative = 1., 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 = {852, 1805, 807, 266, 63, 208} \[ -\frac{e (36036 d-52175 e x)}{9009 d^{12} \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^{12} x}-\frac{e (12012 d-21583 e x)}{9009 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}-\frac{e (12012 d-23225 e x)}{15015 d^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5148 d-10111 e x)}{9009 d^6 \left (d^2-e^2 x^2\right )^{7/2}}-\frac{e (572 d-1103 e x)}{1287 d^4 \left (d^2-e^2 x^2\right )^{9/2}}-\frac{4 e (13 d-24 e x)}{143 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac{8 e (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}+\frac{4 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^{12}} \]

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 852

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 1805

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]

Rule 807

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 266

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 63

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 208

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

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) \operatorname{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 \operatorname{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]  time = 0.230546, size = 183, normalized size = 0.68 \[ \frac{\sqrt{d^2-e^2 x^2} \left (1014094 d^8 e^2 x^2-700504 d^7 e^3 x^3-3157776 d^6 e^4 x^4-1301264 d^5 e^5 x^5+2748320 d^4 e^6 x^6+2496180 d^3 e^7 x^7-350000 d^2 e^8 x^8+546316 d^9 e x+45045 d^{10}-1043500 d e^9 x^9-305920 e^{10} x^{10}\right )}{45045 d^{12} x (e x-d)^3 (d+e x)^7}+\frac{4 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )}{d^{12}}-\frac{4 e \log (x)}{d^{12}} \]

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) - (4*e*Log[x])/d^12 + (4*e*Log[d + Sqrt[d^2 - e
^2*x^2]])/d^12

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Maple [B]  time = 0.071, size = 484, normalized size = 1.8 \begin{align*}{\frac{20222\,{e}^{2}x}{15015\,{d}^{8}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{80888\,{e}^{2}x}{45045\,{d}^{10}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}+{\frac{161776\,{e}^{2}x}{45045\,{d}^{12}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{{d}^{6}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{10111}{9009\,{d}^{6}} \left ({\frac{d}{e}}+x \right ) ^{-1} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}+4\,{\frac{e}{{d}^{11}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}{x}} \right ) }-{\frac{4\,e}{3\,{d}^{9}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-4\,{\frac{e}{{d}^{11}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}-{\frac{4\,e}{5\,{d}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{1}{13\,{d}^{3}{e}^{3}} \left ({\frac{d}{e}}+x \right ) ^{-4} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{35}{143\,{d}^{4}{e}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-3} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{709}{1287\,{d}^{5}e} \left ({\frac{d}{e}}+x \right ) ^{-2} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}+{\frac{6\,{e}^{2}x}{5\,{d}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{e}^{2}x}{5\,{d}^{10}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{e}^{2}x}{5\,{d}^{12}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \end{align*}

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)

[Out]

20222/15015/d^8*e^2/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*x+80888/45045/d^10*e^2/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))
^(3/2)*x+161776/45045/d^12*e^2/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-1/d^6/x/(-e^2*x^2+d^2)^(5/2)-10111/9009/
d^6/(d/e+x)/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+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)-4/3/d^9*e/(-e^2*x^2+d^2)^(3/2)-4/d^11*e/(-e^2*x^2+d^2)^(1/2)-4/5/d^7*e/(-e^2*x^2+d^2)^(5/2)-1/13/d^3/
e^3/(d/e+x)^4/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-35/143/d^4/e^2/(d/e+x)^3/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/
2)-709/1287/d^5/e/(d/e+x)^2/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+6/5/d^8*e^2*x/(-e^2*x^2+d^2)^(5/2)+8/5/d^10*e
^2*x/(-e^2*x^2+d^2)^(3/2)+16/5/d^12*e^2*x/(-e^2*x^2+d^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{4} x^{2}}\,{d x} \end{align*}

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

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Fricas [A]  time = 8.26641, size = 1116, normalized size = 4.12 \begin{align*} -\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 )}} \end{align*}

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*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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

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]

[undef, undef, undef, undef, undef, undef, undef, 1]