3.116 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^9 (d+e x)} \, dx\)
Optimal. Leaf size=201 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4} \]
[Out]
-1/64*e^4*(-e^2*x^2+d^2)^(3/2)/d^3/x^4-1/8*(-e^2*x^2+d^2)^(5/2)/d/x^8+1/7*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^7-1/16*
e^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^6+2/35*e^3*(-e^2*x^2+d^2)^(5/2)/d^4/x^5-3/128*e^8*arctanh((-e^2*x^2+d^2)^(1/2)/
d)/d^4+3/128*e^6*(-e^2*x^2+d^2)^(1/2)/d^3/x^2
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Rubi [A] time = 0.19, antiderivative size = 201, normalized size of antiderivative = 1.00,
number of steps used = 10, 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} \[ \frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4} \]
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x]
[Out]
(3*e^6*Sqrt[d^2 - e^2*x^2])/(128*d^3*x^2) - (e^4*(d^2 - e^2*x^2)^(3/2))/(64*d^3*x^4) - (d^2 - e^2*x^2)^(5/2)/(
8*d*x^8) + (e*(d^2 - e^2*x^2)^(5/2))/(7*d^2*x^7) - (e^2*(d^2 - e^2*x^2)^(5/2))/(16*d^3*x^6) + (2*e^3*(d^2 - e^
2*x^2)^(5/2))/(35*d^4*x^5) - (3*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(128*d^4)
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^9 (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^9} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}-\frac {\int \frac {\left (8 d^2 e-3 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx}{8 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}+\frac {\int \frac {\left (21 d^3 e^2-16 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}-\frac {\int \frac {\left (96 d^4 e^3-21 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{336 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{16 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {e^4 \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{32 d^3}\\ &=-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}+\frac {\left (3 e^8\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{256 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {\left (3 e^6\right ) \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 )}{128 d^3}\\ &=\frac {3 e^6 \sqrt {d^2-e^2 x^2}}{128 d^3 x^2}-\frac {e^4 \left (d^2-e^2 x^2\right )^{3/2}}{64 d^3 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{8 d x^8}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{7 d^2 x^7}-\frac {e^2 \left (d^2-e^2 x^2\right )^{5/2}}{16 d^3 x^6}+\frac {2 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^4 x^5}-\frac {3 e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d^4}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 139, normalized size = 0.69 \[ \frac {-105 e^8 x^8 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-560 d^7+640 d^6 e x+840 d^5 e^2 x^2-1024 d^4 e^3 x^3-70 d^3 e^4 x^4+128 d^2 e^5 x^5-105 d e^6 x^6+256 e^7 x^7\right )+105 e^8 x^8 \log (x)}{4480 d^4 x^8} \]
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*(-560*d^7 + 640*d^6*e*x + 840*d^5*e^2*x^2 - 1024*d^4*e^3*x^3 - 70*d^3*e^4*x^4 + 128*d^2*e
^5*x^5 - 105*d*e^6*x^6 + 256*e^7*x^7) + 105*e^8*x^8*Log[x] - 105*e^8*x^8*Log[d + Sqrt[d^2 - e^2*x^2]])/(4480*d
^4*x^8)
________________________________________________________________________________________
fricas [A] time = 1.05, size = 130, normalized size = 0.65 \[ \frac {105 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (256 \, e^{7} x^{7} - 105 \, d e^{6} x^{6} + 128 \, d^{2} e^{5} x^{5} - 70 \, d^{3} e^{4} x^{4} - 1024 \, d^{4} e^{3} x^{3} + 840 \, d^{5} e^{2} x^{2} + 640 \, d^{6} e x - 560 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4480 \, d^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="fricas")
[Out]
1/4480*(105*e^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (256*e^7*x^7 - 105*d*e^6*x^6 + 128*d^2*e^5*x^5 - 70*d
^3*e^4*x^4 - 1024*d^4*e^3*x^3 + 840*d^5*e^2*x^2 + 640*d^6*e*x - 560*d^7)*sqrt(-e^2*x^2 + d^2))/(d^4*x^8)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 1/215040*((-1/2*(-2*d*exp(1)-2*sqrt(d^2-
x^2*exp(2))*exp(1))/x/exp(2))^7*(-107520*exp(1)^14*exp(2)^2+322560*exp(1)^12*exp(2)^3-403200*exp(1)^10*exp(2)^
4+349440*exp(1)^8*exp(2)^5-309120*exp(1)^6*exp(2)^6+201600*exp(1)^4*exp(2)^7-58800*exp(2)^9)+(-1/2*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*(26880*exp(1)^12*exp(2)^3-80640*exp(1)^10*exp(2)^4+107520*exp(1)^
8*exp(2)^5-107520*exp(1)^6*exp(2)^6+105840*exp(1)^4*exp(2)^7-52080*exp(2)^9)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^5*(-8960*exp(1)^10*exp(2)^4+26880*exp(1)^8*exp(2)^5-38080*exp(1)^6*exp(2)^6+33600*e
xp(1)^4*exp(2)^7-11760*exp(2)^9)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*(3360*exp(1)^8*
exp(2)^5-10080*exp(1)^6*exp(2)^6+15120*exp(1)^4*exp(2)^7-9240*exp(2)^9)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^3*(-1344*exp(1)^6*exp(2)^6+4032*exp(1)^4*exp(2)^7-2352*exp(2)^9)+(-1/2*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*(560*exp(1)^4*exp(2)^7-560*exp(2)^9)+105*exp(2)^9+120*(-2*d*exp(1)-2*sq
rt(d^2-x^2*exp(2))*exp(1))*exp(2)^9/x/exp(2))/d^4/(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^
8/exp(1)^10+1/18446744073709551616*(-2305843009213693952*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x/exp(2))^2*exp(1)^82*exp(2)^11+2305843009213693952/3*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^3*exp(1)^80*exp(2)^12-288230376151711744*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/e
xp(2))^4*exp(1)^78*exp(2)^13+576460752303423488/5*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp
(2))^5*exp(1)^76*exp(2)^14-144115188075855872/3*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2
))^6*exp(1)^74*exp(2)^15+144115188075855872/7*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))
^7*exp(1)^72*exp(2)^16-9007199254740992*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^8*exp
(1)^70*exp(2)^17+6917529027641081856*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)
^80*exp(2)^12-2305843009213693952*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^78
*exp(2)^13+864691128455135232*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^76*exp
(2)^14-1729382256910270464/5*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^74*exp(
2)^15+144115188075855872*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^72*exp(2)^1
6-9223372036854775808*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^78*exp(2)^13+9
799832789158199296/3*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^76*exp(2)^14-12
97036692682702848*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^74*exp(2)^15+10088
06316530991104/5*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^72*exp(2)^16-288230
376151711744/3*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^70*exp(2)^17+92233720
36854775808*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^76*exp(2)^14-28823037615
17117440*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^74*exp(2)^15+12970366926827
02848*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^72*exp(2)^16-90792568487789199
36*d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^74*exp(2)^15+1008806316530991104*
d^28*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^72*exp(2)^16-504403158265495552*d^28
*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^70*exp(2)^17+6485183463413514240*d^28*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^72*exp(2)^16-2017612633061982208*d^28*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^70*exp(2)^17-2522015791327477760*d^28*(-2*d*ex
p(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^72*exp(2)^16/x/exp(2)+8646911284551352320*d^28*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))*exp(1)^74*exp(2)^15/x/exp(2)-13258597302978740224*d^28*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp
(2))*exp(1))*exp(1)^76*exp(2)^14/x/exp(2)+14987979559889010688*d^28*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)*exp(1)^78*exp(2)^13/x/exp(2)-17293822569102704640*d^28*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^80
*exp(2)^12/x/exp(2)+13835058055282163712*d^28*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^82*exp(2)^11/
x/exp(2)-4611686018427387904*d^28*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^84*exp(2)^10/x/exp(2))/d^
32/exp(1)^80/exp(2)^8+1/2*(-12*exp(1)^7*exp(2)^2+12*exp(1)^5*exp(2)^3-4*exp(1)^3*exp(2)^4+4*exp(1)^9*exp(2))*a
tan((-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^4/sqrt(-exp(1)^4+e
xp(2)^2)/exp(1)+1/128*(384*exp(1)^14*exp(2)^2-448*exp(1)^12*exp(2)^3+320*exp(1)^10*exp(2)^4-240*exp(1)^8*exp(2
)^5+208*exp(1)^6*exp(2)^6-184*exp(1)^4*exp(2)^7+85*exp(2)^9-128*exp(1)^16*exp(2))*ln(1/2*abs(-2*d*exp(1)-2*sqr
t(d^2-x^2*exp(2))*exp(1))/abs(x)/exp(2))/d^4/exp(1)^9/exp(1)
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maple [B] time = 0.02, size = 571, normalized size = 2.84 \[ -\frac {3 e^{8} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}\, d^{3}}-\frac {3 e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}+\frac {3 e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{9} x}{8 d^{6}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{9} x}{8 d^{6}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8}}{128 d^{5}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{9} x}{4 d^{8}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{9} x}{4 d^{8}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8}}{128 d^{7}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{9} x}{5 d^{10}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{8}}{5 d^{9}}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8}}{640 d^{9}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{7}}{5 d^{10} x}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{128 d^{9} x^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{5 d^{8} x^{3}}-\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{64 d^{7} x^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{5 d^{6} x^{5}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{16 d^{5} x^{6}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{7 d^{4} x^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{3} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x)
[Out]
-1/4/d^8*e^9*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x-3/8/d^6*e^9*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x-3/8/d^4*e
^9/(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^6*e^3/x^5*(-e^2*x^2+d^2)^(7/2)-
3/16/d^5*e^2/x^6*(-e^2*x^2+d^2)^(7/2)+1/5/d^8*e^5/x^3*(-e^2*x^2+d^2)^(7/2)+1/5/d^10*e^7/x*(-e^2*x^2+d^2)^(7/2)
+1/5/d^10*e^9*x*(-e^2*x^2+d^2)^(5/2)+1/4/d^8*e^9*x*(-e^2*x^2+d^2)^(3/2)+3/8/d^6*e^9*x*(-e^2*x^2+d^2)^(1/2)+3/8
/d^4*e^9/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)-13/64/d^7*e^4/x^4*(-e^2*x^2+d^2)^(7/2)-25/128/
d^9*e^6/x^2*(-e^2*x^2+d^2)^(7/2)+1/7*e/d^4/x^7*(-e^2*x^2+d^2)^(7/2)-3/128/(d^2)^(1/2)/d^3*e^8*ln((2*d^2+2*(d^2
)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/5/d^9*e^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)-1/8/d^3/x^8*(-e^2*x^2+d^2)^(7
/2)+3/640/d^9*e^8*(-e^2*x^2+d^2)^(5/2)+1/128*(-e^2*x^2+d^2)^(3/2)/d^7*e^8+3/128*(-e^2*x^2+d^2)^(1/2)/d^5*e^8
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maxima [A] time = 1.00, size = 228, normalized size = 1.13 \[ -\frac {3 \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d^{4}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d^{5}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{128 \, d^{5} x^{2}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{35 \, d^{4} x^{3}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{64 \, d^{3} x^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{35 \, d^{2} x^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{16 \, d x^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{7 \, x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{8 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^9/(e*x+d),x, algorithm="maxima")
[Out]
-3/128*e^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^4 + 3/128*sqrt(-e^2*x^2 + d^2)*e^8/d^5 + 3/12
8*(-e^2*x^2 + d^2)^(3/2)*e^6/(d^5*x^2) - 2/35*(-e^2*x^2 + d^2)^(3/2)*e^5/(d^4*x^3) + 3/64*(-e^2*x^2 + d^2)^(3/
2)*e^4/(d^3*x^4) - 3/35*(-e^2*x^2 + d^2)^(3/2)*e^3/(d^2*x^5) + 1/16*(-e^2*x^2 + d^2)^(3/2)*e^2/(d*x^6) + 1/7*(
-e^2*x^2 + d^2)^(3/2)*e/x^7 - 1/8*(-e^2*x^2 + d^2)^(3/2)*d/x^8
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^9\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)),x)
[Out]
int((d^2 - e^2*x^2)^(5/2)/(x^9*(d + e*x)), x)
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sympy [C] time = 27.71, size = 1159, normalized size = 5.77 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e**2*x**2+d**2)**(5/2)/x**9/(e*x+d),x)
[Out]
d**3*Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d
**6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(8*e
*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*x**5*sqrt(
-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-d*
*2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) - d**2*e*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*e**2*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**3*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), Ab
s(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 + 15*d*e**2*x**7), True))
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