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

Optimal. Leaf size=234 \[ -\frac{4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac{819 d-1024 e x}{819 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac{13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac{8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^{11}} \]

[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

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Rubi [A]  time = 0.384538, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {852, 1805, 823, 12, 266, 63, 208} \[ -\frac{4 e x}{13 d \left (d^2-e^2 x^2\right )^{11/2}}+\frac{819 d-1024 e x}{819 d^{11} \sqrt{d^2-e^2 x^2}}+\frac{273 d-512 e x}{819 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{273 d-640 e x}{1365 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{117 d-320 e x}{819 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac{13 d-40 e x}{117 d^3 \left (d^2-e^2 x^2\right )^{9/2}}+\frac{8 d (d-e x)}{13 \left (d^2-e^2 x^2\right )^{13/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^{11}} \]

Antiderivative was successfully verified.

[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 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 823

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] + Di
st[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 12

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

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 (d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\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{\operatorname{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{\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^{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]  time = 0.201152, size = 161, normalized size = 0.69 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-4466 d^7 e^2 x^2-56304 d^6 e^3 x^3-34156 d^5 e^4 x^4+40240 d^4 e^5 x^5+45735 d^3 e^6 x^6-1540 d^2 e^7 x^7+22976 d^8 e x+9839 d^9-16385 d e^8 x^8-5120 e^9 x^9\right )}{(d-e x)^3 (d+e x)^7}-4095 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+4095 \log (x)}{4095 d^{11}} \]

Antiderivative was successfully verified.

[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) + 4095*Log[x] - 4095*Log[d + Sqrt[d^2 - e^2*x^2]])/(4095*d^11)

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Maple [A]  time = 0.075, size = 385, normalized size = 1.7 \begin{align*}{\frac{320}{819\,{d}^{5}e} \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}}}}-{\frac{128\,ex}{273\,{d}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{512\,ex}{819\,{d}^{9}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{1024\,ex}{819\,{d}^{11}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}+{\frac{2}{13\,{e}^{3}{d}^{3}} \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{29}{117\,{d}^{4}{e}^{2}} \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{1}{13\,{e}^{4}{d}^{2}} \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{1}{{d}^{10}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{5\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{1}{3\,{d}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{{d}^{10}}{\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/(e*x+d)^4/(-e^2*x^2+d^2)^(7/2),x)

[Out]

320/819/d^5/e/(d/e+x)/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-128/273/d^7*e/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)*
x-512/819/d^9*e/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)*x-1024/819/d^11*e/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+
2/13/e^3/d^3/(d/e+x)^3/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)+29/117/e^2/d^4/(d/e+x)^2/(-(d/e+x)^2*e^2+2*d*e*(d/
e+x))^(5/2)+1/13/e^4/d^2/(d/e+x)^4/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(5/2)-1/d^10/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(
1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/5/d^6/(-e^2*x^2+d^2)^(5/2)+1/3/d^8/(-e^2*x^2+d^2)^(3/2)+1/d^10/(-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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 5.72544, size = 995, normalized size = 4.25 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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