3.2.15 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)} \, dx\) [115]

Optimal. Leaf size=172 \[ -\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}+\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]

[Out]

1/24*e^3*(-e^2*x^2+d^2)^(3/2)/d^2/x^4-1/7*(-e^2*x^2+d^2)^(5/2)/d/x^7+1/6*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^6-2/35*e
^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^5+1/16*e^7*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^3-1/16*e^5*(-e^2*x^2+d^2)^(1/2)/d^2
/x^2

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Rubi [A]
time = 0.10, 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 = {864, 849, 821, 272, 43, 65, 214} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(e^5*Sqrt[d^2 - e^2*x^2])/(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[S
qrt[d^2 - e^2*x^2]/d])/(16*d^3)

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 864

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 \text {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 \text {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 \text {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 \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 )}{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 = 0.57, size = 134, normalized size = 0.78 \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 )-210 e^7 x^7 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{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) - 210*e^7*x^7*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(1680*d^3*x^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1324\) vs. \(2(148)=296\).
time = 0.08, size = 1325, normalized size = 7.70

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (96 e^{6} x^{6}-105 d \,e^{5} x^{5}+48 d^{2} e^{4} x^{4}+490 d^{3} e^{3} x^{3}-384 d^{4} e^{2} x^{2}-280 e \,d^{5} x +240 d^{6}\right )}{1680 x^{7} d^{3}}+\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d^{2} \sqrt {d^{2}}}\)	\(132\)
default	\(\text {Expression too large to display}\)	\(1325\)

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,method=_RETURNVERBOSE)

[Out]

e^7/d^8*(1/5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+d*e*(-1/8*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(3/2)+3/4*d^2*(-1/4*(-2*e^2*(x+d/e)+2*d*e)/e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/2*d^2/(e^2)^(1/2
)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))))+e^2/d^3*(-1/5/d^2/x^5*(-e^2*x^2+d^2)^(7/2)-2/5
*e^2/d^2*(-1/3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/3*e^2/d^2*(-1/d^2/x*(-e^2*x^2+d^2)^(7/2)-6*e^2/d^2*(1/6*x*(-e^2*
x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/(e^2)^(1/2)*arc
tan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))))-e^3/d^4*(-1/4/d^2/x^4*(-e^2*x^2+d^2)^(7/2)-3/4*e^2/d^2*(-1/2/d^2
/x^2*(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d
^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x))))))-e^5/d^6*(-1/2/d^2/x^2*(-e^2*x^
2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2
/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)))))+e^6/d^7*(-1/d^2/x*(-e^2*x^2+d^2)^(7/2)-6*e^2
/d^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d
^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))+e^4/d^5*(-1/3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/3*e
^2/d^2*(-1/d^2/x*(-e^2*x^2+d^2)^(7/2)-6*e^2/d^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2
)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))))-e^7/
d^8*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(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*(-1/6/d^2/x^6*(-e^2*x^2+d^2)^(7/2)-1/6*e^2/d^2*(-1/4/d^2/x^4
*(-e^2*x^2+d^2)^(7/2)-3/4*e^2/d^2*(-1/2/d^2/x^2*(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2
*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1
/2))/x)))))))-1/7/d^3/x^7*(-e^2*x^2+d^2)^(7/2)

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Maxima [A]
time = 0.48, size = 190, normalized size = 1.10 \begin {gather*} \frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{3}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} e^{7}}{16 \, d^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{16 \, d^{4} x^{2}} + \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{35 \, d^{3} x^{3}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{8 \, d^{2} x^{4}} + \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d x^{5}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e}{6 \, x^{6}} - \frac {{\left (-x^{2} e^{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(-x^2*e^2 + d^2)*d/abs(x))/d^3 - 1/16*sqrt(-x^2*e^2 + d^2)*e^7/d^4 - 1/16*(-
x^2*e^2 + d^2)^(3/2)*e^5/(d^4*x^2) + 2/35*(-x^2*e^2 + d^2)^(3/2)*e^4/(d^3*x^3) - 1/8*(-x^2*e^2 + d^2)^(3/2)*e^
3/(d^2*x^4) + 3/35*(-x^2*e^2 + d^2)^(3/2)*e^2/(d*x^5) + 1/6*(-x^2*e^2 + d^2)^(3/2)*e/x^6 - 1/7*(-x^2*e^2 + d^2
)^(3/2)*d/x^7

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Fricas [A]
time = 2.93, size = 112, normalized size = 0.65 \begin {gather*} -\frac {105 \, x^{7} e^{7} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (96 \, x^{6} e^{6} - 105 \, d x^{5} e^{5} + 48 \, d^{2} x^{4} e^{4} + 490 \, d^{3} x^{3} e^{3} - 384 \, d^{4} x^{2} e^{2} - 280 \, d^{5} x e + 240 \, d^{6}\right )} \sqrt {-x^{2} e^{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*x^7*e^7*log(-(d - sqrt(-x^2*e^2 + d^2))/x) + (96*x^6*e^6 - 105*d*x^5*e^5 + 48*d^2*x^4*e^4 + 490*d
^3*x^3*e^3 - 384*d^4*x^2*e^2 - 280*d^5*x*e + 240*d^6)*sqrt(-x^2*e^2 + d^2))/(d^3*x^7)

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Sympy [C] Result contains complex when optimal does not.
time = 10.64, size = 1037, normalized size = 6.03 \begin {gather*} d^{3} \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 ) - d^{2} 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 ) - d 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 ) + e^{3} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) \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)

[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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (139) = 278\).
time = 1.09, size = 492, normalized size = 2.86 \begin {gather*} -\frac {x^{7} {\left (\frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{5}}{x} + \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{3}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e}{x^{3}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-1\right )}}{x^{4}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-3\right )}}{x^{5}} - \frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-5\right )}}{x^{6}} - 15 \, e^{7}\right )} e^{14}}{13440 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{3}} + \frac {e^{7} \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 )}{16 \, d^{3}} - \frac {\frac {315 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{5}}{x} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{3}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e}{x^{3}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{\left (-1\right )}}{x^{4}} - \frac {21 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18} e^{\left (-3\right )}}{x^{5}} - \frac {35 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{\left (-5\right )}}{x^{6}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{\left (-7\right )}}{x^{7}}}{13440 \, d^{21}} \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]

-1/13440*x^7*(35*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^5/x + 21*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^3/x^2 - 105*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)^3*e/x^3 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-1)/x^4 - 105*(d*e + sqrt(-x^2*e^
2 + d^2)*e)^5*e^(-3)/x^5 - 315*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-5)/x^6 - 15*e^7)*e^14/((d*e + sqrt(-x^2*e^
2 + d^2)*e)^7*d^3) + 1/16*e^7*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^3 - 1/13440*(315
*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^18*e^5/x + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^18*e^3/x^2 - 105*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^3*d^18*e/x^3 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^18*e^(-1)/x^4 - 21*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^5*d^18*e^(-3)/x^5 - 35*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^18*e^(-5)/x^6 + 15*(d*e + sqrt(-x^2*
e^2 + d^2)*e)^7*d^18*e^(-7)/x^7)/d^21

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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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