\(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)^2} \, dx\) [170]

Optimal result

Integrand size = 27, antiderivative size = 198 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

[Out]

-1/7*(-e^2*x^2+d^2)^(3/2)/x^7+1/3*e*(-e^2*x^2+d^2)^(3/2)/d/x^6-11/35*e^2*(-e^2*x^2+d^2)^(3/2)/d^2/x^5+1/4*e^3*
(-e^2*x^2+d^2)^(3/2)/d^3/x^4-22/105*e^4*(-e^2*x^2+d^2)^(3/2)/d^4/x^3-1/8*e^7*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d
^4+1/8*e^5*(-e^2*x^2+d^2)^(1/2)/d^3/x^2

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {866, 1821, 849, 821, 272, 43, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=-\frac {e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4} \]

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(8*d^3*x^2) - (d^2 - e^2*x^2)^(3/2)/(7*x^7) + (e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^6) -
(11*e^2*(d^2 - e^2*x^2)^(3/2))/(35*d^2*x^5) + (e^3*(d^2 - e^2*x^2)^(3/2))/(4*d^3*x^4) - (22*e^4*(d^2 - e^2*x^2
)^(3/2))/(105*d^4*x^3) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 849

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 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 1821

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^8} \, dx \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac {\int \frac {\left (14 d^3 e-11 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^7} \, dx}{7 d^2} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}+\frac {\int \frac {\left (66 d^4 e^2-42 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^6} \, dx}{42 d^4} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {\int \frac {\left (210 d^5 e^3-132 d^4 e^4 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{210 d^6} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}+\frac {\int \frac {\left (528 d^6 e^4-210 d^5 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{840 d^8} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{4 d^3} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{8 d^3} \\ & = \frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^7 \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3} \\ & = \frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \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 )}{8 d^3} \\ & = \frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.73 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (-120 d^6+280 d^5 e x-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4-105 d e^5 x^5+176 e^6 x^6\right )}{x^7}-105 \sqrt {d^2} e^7 \log (x)+105 \sqrt {d^2} e^7 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{840 d^5} \]

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

((d*Sqrt[d^2 - e^2*x^2]*(-120*d^6 + 280*d^5*e*x - 144*d^4*e^2*x^2 - 70*d^3*e^3*x^3 + 88*d^2*e^4*x^4 - 105*d*e^
5*x^5 + 176*e^6*x^6))/x^7 - 105*Sqrt[d^2]*e^7*Log[x] + 105*Sqrt[d^2]*e^7*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])
/(840*d^5)

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-176 e^{6} x^{6}+105 d \,e^{5} x^{5}-88 d^{2} e^{4} x^{4}+70 d^{3} x^{3} e^{3}+144 d^{4} e^{2} x^{2}-280 d^{5} e x +120 d^{6}\right )}{840 x^{7} d^{4}}-\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{3} \sqrt {d^{2}}}\)	\(132\)
default	\(\text {Expression too large to display}\)	\(1575\)

[In]

int((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

-1/840*(-e^2*x^2+d^2)^(1/2)*(-176*e^6*x^6+105*d*e^5*x^5-88*d^2*e^4*x^4+70*d^3*e^3*x^3+144*d^4*e^2*x^2-280*d^5*
e*x+120*d^6)/x^7/d^4-1/8/d^3*e^7/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=\frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (176 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 88 \, d^{2} e^{4} x^{4} - 70 \, d^{3} e^{3} x^{3} - 144 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x - 120 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{840 \, d^{4} x^{7}} \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/840*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (176*e^6*x^6 - 105*d*e^5*x^5 + 88*d^2*e^4*x^4 - 70*d^3
*e^3*x^3 - 144*d^4*e^2*x^2 + 280*d^5*e*x - 120*d^6)*sqrt(-e^2*x^2 + d^2))/(d^4*x^7)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.26 (sec) , antiderivative size = 835, normalized size of antiderivative = 4.22 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{7 x^{6}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{35 d^{2} x^{4}} + \frac {4 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{4} x^{2}} + \frac {8 e^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{7 x^{6}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{35 d^{2} x^{4}} + \frac {4 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{4} x^{2}} + \frac {8 i e^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{6}} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((-e**2*x**2+d**2)**(5/2)/x**8/(e*x+d)**2,x)

[Out]

d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e*
*5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x
**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4
*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True))
- 2*d*e*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e*
*3/(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x
))/(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt
(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e
**2*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) + e**2*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/
(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*
I*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d*
*2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**
5 + 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1
- e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 1
5*d*e**2*x**7), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=-\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{4}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{8 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{8 \, d^{5} x^{2}} - \frac {22 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{105 \, d^{4} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d^{2} x^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{3 \, d x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{7 \, x^{7}} \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="maxima")

[Out]

-1/8*e^7*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^4 + 1/8*sqrt(-e^2*x^2 + d^2)*e^7/d^5 + 1/8*(-e^
2*x^2 + d^2)^(3/2)*e^5/(d^5*x^2) - 22/105*(-e^2*x^2 + d^2)^(3/2)*e^4/(d^4*x^3) + 1/4*(-e^2*x^2 + d^2)^(3/2)*e^
3/(d^3*x^4) - 11/35*(-e^2*x^2 + d^2)^(3/2)*e^2/(d^2*x^5) + 1/3*(-e^2*x^2 + d^2)^(3/2)*e/(d*x^6) - 1/7*(-e^2*x^
2 + d^2)^(3/2)/x^7

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.68 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=-\frac {1}{53760} \, {\left (\frac {6720 \, e^{6} \log \left (\sqrt {\frac {2 \, d}{e x + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4}} - \frac {6720 \, e^{6} \log \left ({\left | \sqrt {\frac {2 \, d}{e x + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4}} + \frac {32 \, {\left (105 \, e^{6} \log \left (2\right ) - 210 \, e^{6} \log \left (i + 1\right ) + 352 i \, e^{6}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4}} - \frac {105 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 3780 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 189 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 4992 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1981 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 700 \, e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 105 \, e^{6} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4} {\left (\frac {d}{e x + d} - 1\right )}^{7}}\right )} {\left | e \right |} \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="giac")

[Out]

-1/53760*(6720*e^6*log(sqrt(2*d/(e*x + d) - 1) + 1)*sgn(1/(e*x + d))*sgn(e)/d^4 - 6720*e^6*log(abs(sqrt(2*d/(e
*x + d) - 1) - 1))*sgn(1/(e*x + d))*sgn(e)/d^4 + 32*(105*e^6*log(2) - 210*e^6*log(I + 1) + 352*I*e^6)*sgn(1/(e
*x + d))*sgn(e)/d^4 - (105*e^6*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))*sgn(e) + 3780*e^6*(2*d/(e*x + d) -
1)^(11/2)*sgn(1/(e*x + d))*sgn(e) + 189*e^6*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e) + 4992*e^6*(2*d/
(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 1981*e^6*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e) + 70
0*e^6*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e) - 105*e^6*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn
(e))/(d^4*(d/(e*x + d) - 1)^7))*abs(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^8\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

int((d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x)

[Out]

int((d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2), x)