3.2.15 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)} \, dx\)
Optimal. Leaf size=172 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
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Rubi [A] time = 0.15, antiderivative size = 172, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used =
{850, 835, 807, 266, 47, 63, 208} \begin {gather*} -\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
-(e^5*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) + (e^3*(d^2 - e^2*x^2)^(3/2))/(24*d^2*x^4) - (d^2 - e^2*x^2)^(5/2)/(7*
d*x^7) + (e*(d^2 - e^2*x^2)^(5/2))/(6*d^2*x^6) - (2*e^2*(d^2 - e^2*x^2)^(5/2))/(35*d^3*x^5) + (e^7*ArcTanh[Sqr
t[d^2 - e^2*x^2]/d])/(16*d^3)
Rule 47
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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]
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]
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 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 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 850
Int[((x_)^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*x)/e)*(a + c*x
^2)^(p - 1), x] /; FreeQ[{a, c, d, e, n, p}, x] && EqQ[c*d^2 + a*e^2, 0] && !IntegerQ[p] && ( !IntegerQ[n] ||
!IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2]))
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8 (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {\int \frac {\left (7 d^2 e-2 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{7 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}+\frac {\int \frac {\left (12 d^3 e^2-7 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{42 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^3 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^3 \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d^2}\\ &=\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^5 \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=-\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^5 \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 )}{16 d^2}\\ &=-\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 128, normalized size = 0.74 \begin {gather*} \frac {105 e^7 x^7 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-240 d^6+280 d^5 e x+384 d^4 e^2 x^2-490 d^3 e^3 x^3-48 d^2 e^4 x^4+105 d e^5 x^5-96 e^6 x^6\right )-105 e^7 x^7 \log (x)}{1680 d^3 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(-240*d^6 + 280*d^5*e*x + 384*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 48*d^2*e^4*x^4 + 105*d*e^5*
x^5 - 96*e^6*x^6) - 105*e^7*x^7*Log[x] + 105*e^7*x^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(1680*d^3*x^7)
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IntegrateAlgebraic [A] time = 0.68, size = 137, normalized size = 0.80 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-240 d^6+280 d^5 e x+384 d^4 e^2 x^2-490 d^3 e^3 x^3-48 d^2 e^4 x^4+105 d e^5 x^5-96 e^6 x^6\right )}{1680 d^3 x^7}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(-240*d^6 + 280*d^5*e*x + 384*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 48*d^2*e^4*x^4 + 105*d*e^5*
x^5 - 96*e^6*x^6))/(1680*d^3*x^7) - (e^7*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d])/(8*d^3)
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fricas [A] time = 0.43, size = 119, normalized size = 0.69 \begin {gather*} -\frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (96 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} + 490 \, d^{3} e^{3} x^{3} - 384 \, d^{4} e^{2} x^{2} - 280 \, d^{5} e x + 240 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),x, algorithm="fricas")
[Out]
-1/1680*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (96*e^6*x^6 - 105*d*e^5*x^5 + 48*d^2*e^4*x^4 + 490*d
^3*e^3*x^3 - 384*d^4*e^2*x^2 - 280*d^5*e*x + 240*d^6)*sqrt(-e^2*x^2 + d^2))/(d^3*x^7)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 1/13440*((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x
^2*exp(2))*exp(1))/x/exp(2))^6*(6720*exp(1)^12*exp(2)^2-20160*exp(1)^10*exp(2)^3+25200*exp(1)^8*exp(2)^4-21840
*exp(1)^6*exp(2)^5+19320*exp(1)^4*exp(2)^6-8925*exp(2)^8)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/
exp(2))^5*(-1680*exp(1)^10*exp(2)^3+5040*exp(1)^8*exp(2)^4-6720*exp(1)^6*exp(2)^5+5040*exp(1)^4*exp(2)^6-1575*
exp(2)^8)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*(560*exp(1)^8*exp(2)^4-1680*exp(1)^6*e
xp(2)^5+2380*exp(1)^4*exp(2)^6-1365*exp(2)^8)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*(-
210*exp(1)^6*exp(2)^5+630*exp(1)^4*exp(2)^6-315*exp(2)^8)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/
exp(2))^2*(84*exp(1)^4*exp(2)^6-105*exp(2)^8)+15*exp(2)^8+35/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp
(2)^8/x/exp(2))/d^3/(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7/exp(1)^9+1/562949953421312*(
70368744177664*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^64*exp(2)^10-70368744
177664/3*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^62*exp(2)^11+8796093022208*
d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^60*exp(2)^12-17592186044416/5*d^18*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^58*exp(2)^13+4398046511104/3*d^18*(-1/2*(-
2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^56*exp(2)^14-4398046511104/7*d^18*(-1/2*(-2*d*exp
(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^7*exp(1)^54*exp(2)^15-211106232532992*d^18*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^62*exp(2)^11+70368744177664*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^60*exp(2)^12-26388279066624*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^4*exp(1)^58*exp(2)^13+52776558133248/5*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))/x/exp(2))^5*exp(1)^56*exp(2)^14+281474976710656*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x
/exp(2))^2*exp(1)^60*exp(2)^12-299067162755072/3*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(
2))^3*exp(1)^58*exp(2)^13+13194139533312*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*ex
p(1)^56*exp(2)^14-30786325577728/5*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^5
4*exp(2)^15-211106232532992*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^58*exp(2
)^13+87960930222080*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^56*exp(2)^14+659
70697666560*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^56*exp(2)^14-30786325577
728*d^18*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^54*exp(2)^15+76965813944320*d^18
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^54*exp(2)^15/x/exp(2)-263882790666240*d^18*(-2*d*exp(1)-2*
sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^56*exp(2)^14/x/exp(2)+404620279021568*d^18*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))*exp(1)^58*exp(2)^13/x/exp(2)-457396837154816*d^18*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp
(1)^60*exp(2)^12/x/exp(2)+527765581332480*d^18*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^62*exp(2)^11
/x/exp(2)-422212465065984*d^18*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^64*exp(2)^10/x/exp(2)+140737
488355328*d^18*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^66*exp(2)^9/x/exp(2))/d^21/exp(1)^63/exp(2)^
7+1/2*(12*exp(1)^6*exp(2)^2-12*exp(1)^4*exp(2)^3+4*exp(2)^5-4*exp(1)^8*exp(2))*atan((-1/2*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/d^3/sqrt(-exp(1)^4+exp(2)^2)/exp(1)+1/16*(-48*exp(
1)^10*exp(2)^2+56*exp(1)^8*exp(2)^3-40*exp(1)^6*exp(2)^4+30*exp(1)^4*exp(2)^5-13*exp(2)^7+16*exp(1)^12*exp(2))
*ln(1/2*abs(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/abs(x)/exp(2))/d^3/exp(1)^6/exp(1)
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maple [B] time = 0.02, size = 546, normalized size = 3.17 \begin {gather*} \frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}\, d^{2}}+\frac {3 e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{3}}-\frac {3 e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{3}}+\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{8} x}{8 d^{5}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8} x}{8 d^{5}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}{16 d^{4}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{8} x}{4 d^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8} x}{4 d^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}{48 d^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8} x}{5 d^{9}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{7}}{5 d^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7}}{80 d^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{5 d^{9} x}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{16 d^{8} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{5 d^{7} x^{3}}+\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{24 d^{6} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d^{5} x^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{6 d^{4} x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),x)
[Out]
1/4/d^7*e^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x+3/8/d^5*e^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x+3/8/d^3*e^
8/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)-1/5/d^5*e^2/x^5*(-e^2*x^2+d^2)^(7/2)+1
/6*e/d^4/x^6*(-e^2*x^2+d^2)^(7/2)-1/5/d^7*e^4/x^3*(-e^2*x^2+d^2)^(7/2)-1/5/d^9*e^6/x*(-e^2*x^2+d^2)^(7/2)-1/5/
d^9*e^8*x*(-e^2*x^2+d^2)^(5/2)-1/4/d^7*e^8*x*(-e^2*x^2+d^2)^(3/2)-3/8/d^5*e^8*x*(-e^2*x^2+d^2)^(1/2)-3/8/d^3*e
^8/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+5/24/d^6*e^3/x^4*(-e^2*x^2+d^2)^(7/2)+3/16/d^8*e^5/x
^2*(-e^2*x^2+d^2)^(7/2)+1/16/(d^2)^(1/2)/d^2*e^7*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/5/d^8*e^7*
(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)-1/80/d^8*e^7*(-e^2*x^2+d^2)^(5/2)-1/7/d^3/x^7*(-e^2*x^2+d^2)^(7/2)-1/48*(-
e^2*x^2+d^2)^(3/2)/d^6*e^7-1/16*(-e^2*x^2+d^2)^(1/2)/d^4*e^7
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maxima [A] time = 0.99, size = 203, normalized size = 1.18 \begin {gather*} \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 )}{16 \, d^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{16 \, d^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{16 \, d^{4} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{35 \, d^{3} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{8 \, d^{2} x^{4}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d x^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{6 \, x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d),x, algorithm="maxima")
[Out]
1/16*e^7*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 1/16*sqrt(-e^2*x^2 + d^2)*e^7/d^4 - 1/16*(-
e^2*x^2 + d^2)^(3/2)*e^5/(d^4*x^2) + 2/35*(-e^2*x^2 + d^2)^(3/2)*e^4/(d^3*x^3) - 1/8*(-e^2*x^2 + d^2)^(3/2)*e^
3/(d^2*x^4) + 3/35*(-e^2*x^2 + d^2)^(3/2)*e^2/(d*x^5) + 1/6*(-e^2*x^2 + d^2)^(3/2)*e/x^6 - 1/7*(-e^2*x^2 + d^2
)^(3/2)*d/x^7
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^8\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)),x)
[Out]
int((d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)), x)
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sympy [C] time = 18.79, size = 1037, normalized size = 6.03
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e**2*x**2+d**2)**(5/2)/x**8/(e*x+d),x)
[Out]
d**3*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))
- d**2*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*sqr
t(-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)) - d*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*sqr
t(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
+ 15*d*e**2*x**7), True)) + e**3*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**
2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**
2*x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I
*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True))
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