3.29 \(\int \frac{d+e x}{x^3 (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=184 \[ -\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]

[Out]

(d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + (35*d + 24*e*
x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^2 - e^2*x^2])/(2*d^7*x^2) - (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*
x) - (7*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

________________________________________________________________________________________

Rubi [A]  time = 0.15962, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {823, 835, 807, 266, 63, 208} \[ -\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + (35*d + 24*e*
x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^2 - e^2*x^2])/(2*d^7*x^2) - (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*
x) - (7*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

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 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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{d+e x}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{7 d^3 e^2+6 d^2 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^4 e^2}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{35 d^5 e^4+24 d^4 e^5 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^8 e^4}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{105 d^7 e^6+48 d^6 e^7 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{12} e^6}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}-\frac{\int \frac{-96 d^8 e^7-105 d^7 e^8 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^{14} e^6}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}+\frac{\left (7 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^7}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}+\frac{\left (7 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^7}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \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 )}{2 d^7}\\ &=\frac{d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{35 d+24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}-\frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8}\\ \end{align*}

Mathematica [A]  time = 0.147898, size = 183, normalized size = 0.99 \[ \frac{d \sqrt{1-\frac{e^2 x^2}{d^2}} \left (176 d^4 e^2 x^2+4 d^3 e^3 x^3-249 d^2 e^4 x^4-15 d^5 e x-15 d^6+9 d e^5 x^5+96 e^6 x^6\right )+105 e^2 x^2 (d+e x)^2 (e x-d)^3 \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )}{30 d^9 x^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(d*Sqrt[1 - (e^2*x^2)/d^2]*(-15*d^6 - 15*d^5*e*x + 176*d^4*e^2*x^2 + 4*d^3*e^3*x^3 - 249*d^2*e^4*x^4 + 9*d*e^5
*x^5 + 96*e^6*x^6) + 105*e^2*x^2*(-d + e*x)^3*(d + e*x)^2*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(30*d^9*x^2*(d - e
*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*Sqrt[1 - (e^2*x^2)/d^2])

________________________________________________________________________________________

Maple [A]  time = 0.061, size = 227, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,d{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{2}}{10\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{e}^{2}}{6\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{2\,{d}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{7\,{e}^{2}}{2\,{d}^{7}}\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{e}{{d}^{2}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{6\,{e}^{3}x}{5\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{e}^{3}x}{5\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{16\,{e}^{3}x}{5\,{d}^{8}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2/d/x^2/(-e^2*x^2+d^2)^(5/2)+7/10*e^2/d^3/(-e^2*x^2+d^2)^(5/2)+7/6*e^2/d^5/(-e^2*x^2+d^2)^(3/2)+7/2*e^2/d^7
/(-e^2*x^2+d^2)^(1/2)-7/2*e^2/d^7/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-e/d^2/x/(-e^2*x
^2+d^2)^(5/2)+6/5*e^3/d^4*x/(-e^2*x^2+d^2)^(5/2)+8/5*e^3/d^6*x/(-e^2*x^2+d^2)^(3/2)+16/5*e^3/d^8*x/(-e^2*x^2+d
^2)^(1/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.47298, size = 595, normalized size = 3.23 \begin{align*} \frac{116 \, e^{7} x^{7} - 116 \, d e^{6} x^{6} - 232 \, d^{2} e^{5} x^{5} + 232 \, d^{3} e^{4} x^{4} + 116 \, d^{4} e^{3} x^{3} - 116 \, d^{5} e^{2} x^{2} + 105 \,{\left (e^{7} x^{7} - d e^{6} x^{6} - 2 \, d^{2} e^{5} x^{5} + 2 \, d^{3} e^{4} x^{4} + d^{4} e^{3} x^{3} - d^{5} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (96 \, e^{6} x^{6} + 9 \, d e^{5} x^{5} - 249 \, d^{2} e^{4} x^{4} + 4 \, d^{3} e^{3} x^{3} + 176 \, d^{4} e^{2} x^{2} - 15 \, d^{5} e x - 15 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{8} e^{5} x^{7} - d^{9} e^{4} x^{6} - 2 \, d^{10} e^{3} x^{5} + 2 \, d^{11} e^{2} x^{4} + d^{12} e x^{3} - d^{13} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(116*e^7*x^7 - 116*d*e^6*x^6 - 232*d^2*e^5*x^5 + 232*d^3*e^4*x^4 + 116*d^4*e^3*x^3 - 116*d^5*e^2*x^2 + 10
5*(e^7*x^7 - d*e^6*x^6 - 2*d^2*e^5*x^5 + 2*d^3*e^4*x^4 + d^4*e^3*x^3 - d^5*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 +
d^2))/x) - (96*e^6*x^6 + 9*d*e^5*x^5 - 249*d^2*e^4*x^4 + 4*d^3*e^3*x^3 + 176*d^4*e^2*x^2 - 15*d^5*e*x - 15*d^6
)*sqrt(-e^2*x^2 + d^2))/(d^8*e^5*x^7 - d^9*e^4*x^6 - 2*d^10*e^3*x^5 + 2*d^11*e^2*x^4 + d^12*e*x^3 - d^13*x^2)

________________________________________________________________________________________

Sympy [C]  time = 22.2567, size = 2696, normalized size = 14.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((30*I*d**8*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 +
60*d**9*e**6*x**8) - 322*I*d**6*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 1
80*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 105*d**6*e**2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e
**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*d**6*e**2*x**2*log(e*x/d)/(-60*d**15*x**2 + 180*d**1
3*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*I*d**6*e**2*x**2*asin(d/(e*x))/(-60*d**15*x**2 +
180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 490*I*d**4*e**4*x**4*sqrt(-1 + e**2*x**2/d**2
)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e*
*2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 630*d**4*e**4
*x**4*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 630*I*d**4
*e**4*x**4*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 21
0*I*d**2*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*
d**9*e**6*x**8) - 315*d**2*e**6*x**6*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**
4*x**6 + 60*d**9*e**6*x**8) + 630*d**2*e**6*x**6*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*
e**4*x**6 + 60*d**9*e**6*x**8) - 630*I*d**2*e**6*x**6*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 18
0*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 105*e**8*x**8*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x*
*4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*e**8*x**8*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*I*e**8*x**8*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x
**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8), Abs(e**2*x**2)/Abs(d**2) > 1), (30*d**8*sqrt(1 - e**2*x**2/d**
2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 322*d**6*e**2*x**2*sqrt(
1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 105*d**
6*e**2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**
8) + 210*d**6*e**2*x**2*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e*
*4*x**6 + 60*d**9*e**6*x**8) - 105*I*pi*d**6*e**2*x**2/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*
x**6 + 60*d**9*e**6*x**8) + 490*d**4*e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**1
3*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 630*d**4*e**4*x**4*log(sqrt(1 - e**2*x**2/d**2) + 1)/
(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*I*pi*d**4*e**4*x**4/(-6
0*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*d**2*e**6*x**6*sqrt(1 - e*
*2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 315*d**2*e**6
*x**6*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 6
30*d**2*e**6*x**6*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**
6 + 60*d**9*e**6*x**8) - 315*I*pi*d**2*e**6*x**6/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 +
60*d**9*e**6*x**8) + 105*e**8*x**8*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4
*x**6 + 60*d**9*e**6*x**8) - 210*e**8*x**8*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*
x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 105*I*pi*e**8*x**8/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 1
80*d**11*e**4*x**6 + 60*d**9*e**6*x**8), True)) + e*Piecewise((5*d**6*e*sqrt(d**2/(e**2*x**2) - 1)/(-5*d**14 +
15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) - 30*d**4*e**3*x**2*sqrt(d**2/(e**2*x**2) - 1)/(-
5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) + 40*d**2*e**5*x**4*sqrt(d**2/(e**2*x**2
) - 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) - 16*e**7*x**6*sqrt(d**2/(e**2*
x**2) - 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6), Abs(d**2)/(Abs(e**2)*Abs(x
**2)) > 1), (5*I*d**6*e*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d*
*8*e**6*x**6) - 30*I*d**4*e**3*x**2*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4
*x**4 + 5*d**8*e**6*x**6) + 40*I*d**2*e**5*x**4*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 1
5*d**10*e**4*x**4 + 5*d**8*e**6*x**6) - 16*I*e**7*x**6*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x
**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6), True))

________________________________________________________________________________________

Giac [A]  time = 1.18364, size = 351, normalized size = 1.91 \begin{align*} -\frac{\sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left ({\left (3 \,{\left (x{\left (\frac{11 \, x e^{7}}{d^{8}} + \frac{15 \, e^{6}}{d^{7}}\right )} - \frac{25 \, e^{5}}{d^{6}}\right )} x - \frac{100 \, e^{4}}{d^{5}}\right )} x + \frac{45 \, e^{3}}{d^{4}}\right )} x + \frac{58 \, e^{2}}{d^{3}}\right )}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac{7 \, e^{2} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{2 \, d^{8}} + \frac{x^{2}{\left (\frac{4 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} + e^{6}\right )}}{8 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8}} - \frac{{\left (\frac{4 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{8} e^{8}}{x} + \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{8} e^{6}}{x^{2}}\right )} e^{\left (-8\right )}}{8 \, d^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-1/15*sqrt(-x^2*e^2 + d^2)*(((3*(x*(11*x*e^7/d^8 + 15*e^6/d^7) - 25*e^5/d^6)*x - 100*e^4/d^5)*x + 45*e^3/d^4)*
x + 58*e^2/d^3)/(x^2*e^2 - d^2)^3 - 7/2*e^2*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^8
+ 1/8*x^2*(4*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^4/x + e^6)/((d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^8) - 1/8*(4*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)*d^8*e^8/x + (d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^8*e^6/x^2)*e^(-8)/d^16