3.209 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^6 (d+e x)^4} \, dx\)
Optimal. Leaf size=196 \[ -\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2} \]
[Out]
27/2*e^5*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^4-8*e^5*(-e*x+d)/d^4/(-e^2*x^2+d^2)^(1/2)-1/5*(-e^2*x^2+d^2)^(1/2)/
x^5+e*(-e^2*x^2+d^2)^(1/2)/d/x^4-13/5*e^2*(-e^2*x^2+d^2)^(1/2)/d^2/x^3+11/2*e^3*(-e^2*x^2+d^2)^(1/2)/d^3/x^2-6
6/5*e^4*(-e^2*x^2+d^2)^(1/2)/d^4/x
________________________________________________________________________________________
Rubi [A] time = 0.52, antiderivative size = 196, 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
= {852, 1805, 1807, 807, 266, 63, 208} \[ -\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4} \]
Antiderivative was successfully verified.
[In]
Int[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]
[Out]
(-8*e^5*(d - e*x))/(d^4*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(5*x^5) + (e*Sqrt[d^2 - e^2*x^2])/(d*x^4) -
(13*e^2*Sqrt[d^2 - e^2*x^2])/(5*d^2*x^3) + (11*e^3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) - (66*e^4*Sqrt[d^2 - e^2*
x^2])/(5*d^4*x) + (27*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^4)
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 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 1807
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*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^6 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3-8 e^4 x^4+\frac {8 e^5 x^5}{d}}{x^6 \sqrt {d^2-e^2 x^2}} \, dx}{d^2}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {\int \frac {-20 d^5 e+39 d^4 e^2 x-40 d^3 e^3 x^2+40 d^2 e^4 x^3-40 d e^5 x^4}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{5 d^4}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {\int \frac {-156 d^6 e^2+220 d^5 e^3 x-160 d^4 e^4 x^2+160 d^3 e^5 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{20 d^6}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {\int \frac {-660 d^7 e^3+792 d^6 e^4 x-480 d^5 e^5 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{60 d^8}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {\int \frac {-1584 d^8 e^4+1620 d^7 e^5 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{120 d^{10}}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}-\frac {\left (27 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {\left (27 e^3\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 )}{2 d^3}\\ &=-\frac {8 e^5 (d-e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{5 x^5}+\frac {e \sqrt {d^2-e^2 x^2}}{d x^4}-\frac {13 e^2 \sqrt {d^2-e^2 x^2}}{5 d^2 x^3}+\frac {11 e^3 \sqrt {d^2-e^2 x^2}}{2 d^3 x^2}-\frac {66 e^4 \sqrt {d^2-e^2 x^2}}{5 d^4 x}+\frac {27 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.33, size = 118, normalized size = 0.60 \[ -\frac {-135 e^5 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}+135 e^5 \log (x)}{10 d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]
[Out]
-1/10*((Sqrt[d^2 - e^2*x^2]*(2*d^5 - 8*d^4*e*x + 16*d^3*e^2*x^2 - 29*d^2*e^3*x^3 + 77*d*e^4*x^4 + 212*e^5*x^5)
)/(x^5*(d + e*x)) + 135*e^5*Log[x] - 135*e^5*Log[d + Sqrt[d^2 - e^2*x^2]])/d^4
________________________________________________________________________________________
fricas [A] time = 1.10, size = 147, normalized size = 0.75 \[ -\frac {80 \, e^{6} x^{6} + 80 \, d e^{5} x^{5} + 135 \, {\left (e^{6} x^{6} + d e^{5} x^{5}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (212 \, e^{5} x^{5} + 77 \, d e^{4} x^{4} - 29 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} - 8 \, d^{4} e x + 2 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (d^{4} e x^{6} + d^{5} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^4,x, algorithm="fricas")
[Out]
-1/10*(80*e^6*x^6 + 80*d*e^5*x^5 + 135*(e^6*x^6 + d*e^5*x^5)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (212*e^5*x^5
+ 77*d*e^4*x^4 - 29*d^2*e^3*x^3 + 16*d^3*e^2*x^2 - 8*d^4*e*x + 2*d^5)*sqrt(-e^2*x^2 + d^2))/(d^4*e*x^6 + d^5*
x^5)
________________________________________________________________________________________
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^6/(e*x+d)^4,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 1/480*((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^7*(247680*exp(1)^20*exp(2)^2+414720*exp(1)^18*exp(2)^3+115200*exp(1)^16*exp(2)^4+45
600*exp(1)^14*exp(2)^5+43920*exp(1)^12*exp(2)^6-106200*exp(1)^10*exp(2)^7-10680*exp(1)^8*exp(2)^8+15720*exp(1)
^6*exp(2)^9-1080*exp(1)^4*exp(2)^10+1200*exp(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2)
)^8*(138240*exp(1)^20*exp(2)^2+203200*exp(1)^18*exp(2)^3+342720*exp(1)^16*exp(2)^4+46320*exp(1)^14*exp(2)^5-15
8400*exp(1)^12*exp(2)^6+12440*exp(1)^10*exp(2)^7-360*exp(1)^8*exp(2)^8-3965*exp(1)^6*exp(2)^9+4505*exp(1)^4*ex
p(2)^10+220*exp(2)^12+93440*exp(1)^22*exp(2))+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^9*(9
9840*exp(1)^20*exp(2)^2+144000*exp(1)^18*exp(2)^3+27360*exp(1)^16*exp(2)^4+56160*exp(1)^14*exp(2)^5+39840*exp(
1)^12*exp(2)^6-60120*exp(1)^10*exp(2)^7-5460*exp(1)^8*exp(2)^8+8340*exp(1)^6*exp(2)^9-540*exp(1)^4*exp(2)^10+6
60*exp(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^10*(26880*exp(1)^18*exp(2)^3+37440*e
xp(1)^16*exp(2)^4-6000*exp(1)^14*exp(2)^5-5040*exp(1)^12*exp(2)^6+10440*exp(1)^10*exp(2)^7-6180*exp(1)^8*exp(2
)^8-1110*exp(1)^6*exp(2)^9+1350*exp(1)^4*exp(2)^10+60*exp(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^6*(204480*exp(1)^18*exp(2)^3+362880*exp(1)^16*exp(2)^4+30960*exp(1)^14*exp(2)^5-144240*exp(1)^1
2*exp(2)^6+3600*exp(1)^10*exp(2)^7+840*exp(1)^8*exp(2)^8-3432*exp(1)^6*exp(2)^9+3384*exp(1)^4*exp(2)^10+312*ex
p(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*(60960*exp(1)^16*exp(2)^4+112320*exp(1)
^14*exp(2)^5+9120*exp(1)^12*exp(2)^6-56840*exp(1)^10*exp(2)^7-4512*exp(1)^8*exp(2)^8+6704*exp(1)^6*exp(2)^9-57
6*exp(1)^4*exp(2)^10+408*exp(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*(3360*exp(1)
^14*exp(2)^5+6480*exp(1)^12*exp(2)^6+936*exp(1)^10*exp(2)^7-3192*exp(1)^8*exp(2)^8-558*exp(1)^6*exp(2)^9+198*e
xp(1)^4*exp(2)^10+216*exp(2)^12)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*(-336*exp(1)^12
*exp(2)^6-648*exp(1)^10*exp(2)^7-72*exp(1)^8*exp(2)^8+312*exp(1)^6*exp(2)^9-72*exp(1)^4*exp(2)^10-144*exp(2)^1
2)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*(56*exp(1)^10*exp(2)^7+108*exp(1)^8*exp(2)^8+
22*exp(1)^6*exp(2)^9-22*exp(1)^4*exp(2)^10+76*exp(2)^12)+3*exp(1)^6*exp(2)^9+9*exp(1)^4*exp(2)^10+12*exp(2)^12
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*(-12*exp(1)^8*exp(2)^8-36*exp(1)^6*exp(2)^9-36*exp(1)^4*exp(2
)^10-12*exp(2)^12)/x/exp(2))/d^4/(2*exp(2))^3/(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5/((
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(2)-(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)/x+exp(2))^3/exp(1)^7+1/33554432*(83886080*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2
*exp(1)^34*exp(2)^23-41943040/3*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^32*e
xp(2)^24+2097152*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^30*exp(2)^25-104857
6/5*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^28*exp(2)^26-50331648*d^16*(-1/2
*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^32*exp(2)^24+4194304*d^16*(-1/2*(-2*d*exp(1)-2
*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^30*exp(2)^25+16777216*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*
exp(2))*exp(1))/x/exp(2))^2*exp(1)^30*exp(2)^25-5242880/3*d^16*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1
))/x/exp(2))^3*exp(1)^28*exp(2)^26+5242880*d^16*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^28*exp(2)^2
6/x/exp(2)-18874368*d^16*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^30*exp(2)^25/x/exp(2)+88080384*d^1
6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^32*exp(2)^24/x/exp(2)-251658240*d^16*(-2*d*exp(1)-2*sqrt(
d^2-x^2*exp(2))*exp(1))*exp(1)^34*exp(2)^23/x/exp(2)+293601280*d^16*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1)
)*exp(1)^36*exp(2)^22/x/exp(2))/d^20/exp(1)^35/exp(2)^20+1/2*(-120*exp(1)^6*exp(2)^2+44*exp(1)^4*exp(2)^3-9*ex
p(2)^5+112*exp(1)^8*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^4/exp(1)^4/
exp(1)+1/2*(-432*exp(1)^10*exp(2)^2+72*exp(1)^8*exp(2)^3+472*exp(1)^6*exp(2)^4+90*exp(1)^4*exp(2)^5-104*exp(2)
^7-224*exp(1)^12*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))/sqrt(-exp(1)^4+exp(2)^2)/(d^4*exp(1)^7+3*d^4*exp(1)^5*exp(2)+4*d^4*exp(1)*exp(2)^3)
________________________________________________________________________________________
maple [B] time = 0.02, size = 628, normalized size = 3.20 \[ \frac {27 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}\, d^{3}}+\frac {333 e^{6} \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 {333 e^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {333 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x}{8 d^{6}}+\frac {333 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{6} x}{8 d^{6}}-\frac {27 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}{2 d^{5}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6} x}{4 d^{8}}+\frac {111 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{6} x}{4 d^{8}}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}{2 d^{7}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6} x}{5 d^{10}}-\frac {27 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}{10 d^{9}}+\frac {111 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{5}}{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 {7}{2}} e}{\left (x +\frac {d}{e}\right )^{4} d^{7}}+\frac {3 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{2}}{\left (x +\frac {d}{e}\right )^{3} d^{8}}+\frac {11 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{3}}{\left (x +\frac {d}{e}\right )^{2} d^{9}}-\frac {111 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{5 d^{10} x}+\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{2 d^{9} x^{2}}-\frac {16 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d^{8} x^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{d^{7} x^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^4,x)
[Out]
1/d^7*e/x^4*(-e^2*x^2+d^2)^(7/2)-16/5/d^8*e^2/x^3*(-e^2*x^2+d^2)^(7/2)-111/5/d^10*e^4/x*(-e^2*x^2+d^2)^(7/2)-1
11/5/d^10*e^6*x*(-e^2*x^2+d^2)^(5/2)-111/4/d^8*e^6*x*(-e^2*x^2+d^2)^(3/2)-333/8/d^6*e^6*x*(-e^2*x^2+d^2)^(1/2)
-333/8/d^4*e^6/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+17/2/d^9*e^3/x^2*(-e^2*x^2+d^2)^(7/2)+33
3/8/d^4*e^6/(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/d^7*e/(x+d/e)^4*(2*(x+d/e)
*d*e-(x+d/e)^2*e^2)^(7/2)+3/d^8*e^2/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)+11/d^9*e^3/(x+d/e)^2*(2*(x+d
/e)*d*e-(x+d/e)^2*e^2)^(7/2)+111/4/d^8*e^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x+333/8/d^6*e^6*(2*(x+d/e)*d*e-
(x+d/e)^2*e^2)^(1/2)*x+27/2/(d^2)^(1/2)/d^3*e^5*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/5/d^6/x^5*(
-e^2*x^2+d^2)^(7/2)-27/10/d^9*e^5*(-e^2*x^2+d^2)^(5/2)-9/2/d^7*e^5*(-e^2*x^2+d^2)^(3/2)-27/2/d^5*e^5*(-e^2*x^2
+d^2)^(1/2)+111/5/d^9*e^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^4,x, algorithm="maxima")
[Out]
integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^6), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^6\,{\left (d+e\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x)
[Out]
int((d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{6} \left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e**2*x**2+d**2)**(5/2)/x**6/(e*x+d)**4,x)
[Out]
Integral((-(-d + e*x)*(d + e*x))**(5/2)/(x**6*(d + e*x)**4), x)
________________________________________________________________________________________