3.2.13 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^6 (d+e x)} \, dx\) [113]

Optimal. Leaf size=108 \[ -\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d} \]

[Out]

1/4*e*(-e^2*x^2+d^2)^(3/2)/x^4-1/5*(-e^2*x^2+d^2)^(5/2)/d/x^5+3/8*e^5*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d-3/8*e^
3*(-e^2*x^2+d^2)^(1/2)/x^2

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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 821, 272, 43, 65, 214} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}+\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d}-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)),x]

[Out]

(-3*e^3*Sqrt[d^2 - e^2*x^2])/(8*x^2) + (e*(d^2 - e^2*x^2)^(3/2))/(4*x^4) - (d^2 - e^2*x^2)^(5/2)/(5*d*x^5) + (
3*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d)

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 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^6 (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {1}{2} e \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {1}{8} \left (3 e^3\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}-\frac {1}{16} \left (3 e^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )\\ &=-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {1}{8} \left (3 e^3\right ) \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 )\\ &=-\frac {3 e^3 \sqrt {d^2-e^2 x^2}}{8 x^2}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}+\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 151, normalized size = 1.40 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-8 d^4+10 d^3 e x+16 d^2 e^2 x^2-25 d e^3 x^3-8 e^4 x^4\right )-15 e^5 x^5 \log \left (d \left (-d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )\right )+15 e^5 x^5 \log \left (d-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{40 d x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-8*d^4 + 10*d^3*e*x + 16*d^2*e^2*x^2 - 25*d*e^3*x^3 - 8*e^4*x^4) - 15*e^5*x^5*Log[d*(-d
- Sqrt[-e^2]*x + Sqrt[d^2 - e^2*x^2])] + 15*e^5*x^5*Log[d - Sqrt[-e^2]*x + Sqrt[d^2 - e^2*x^2]])/(40*d*x^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1099\) vs. \(2(92)=184\).
time = 0.07, size = 1100, normalized size = 10.19

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (8 e^{4} x^{4}+25 d \,e^{3} x^{3}-16 d^{2} x^{2} e^{2}-10 d^{3} e x +8 d^{4}\right )}{40 x^{5} d}+\frac {3 e^{5} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\)	\(107\)
default	\(\text {Expression too large to display}\)	\(1100\)

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),x,method=_RETURNVERBOSE)

[Out]

e^5/d^6*(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)))))+1/d*(-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)*arctan(
(e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))))-e/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))))))-e^3/d^4*(-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^4/d^5*(-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^2/d^3*(-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^5/d^6*(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))))

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Maxima [A]
time = 0.47, size = 144, normalized size = 1.33 \begin {gather*} \frac {3 \, e^{5} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d} - \frac {3 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{5}}{8 \, d^{2}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{8 \, d^{2} x^{2}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{5 \, d x^{3}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e}{4 \, x^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d}{5 \, x^{5}} \end {gather*}

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),x, algorithm="maxima")

[Out]

3/8*e^5*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x))/d - 3/8*sqrt(-x^2*e^2 + d^2)*e^5/d^2 - 3/8*(-x^2*e
^2 + d^2)^(3/2)*e^3/(d^2*x^2) + 1/5*(-x^2*e^2 + d^2)^(3/2)*e^2/(d*x^3) + 1/4*(-x^2*e^2 + d^2)^(3/2)*e/x^4 - 1/
5*(-x^2*e^2 + d^2)^(3/2)*d/x^5

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Fricas [A]
time = 2.58, size = 92, normalized size = 0.85 \begin {gather*} -\frac {15 \, x^{5} e^{5} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (8 \, x^{4} e^{4} + 25 \, d x^{3} e^{3} - 16 \, d^{2} x^{2} e^{2} - 10 \, d^{3} x e + 8 \, d^{4}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{40 \, d x^{5}} \end {gather*}

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),x, algorithm="fricas")

[Out]

-1/40*(15*x^5*e^5*log(-(d - sqrt(-x^2*e^2 + d^2))/x) + (8*x^4*e^4 + 25*d*x^3*e^3 - 16*d^2*x^2*e^2 - 10*d^3*x*e
+ 8*d^4)*sqrt(-x^2*e^2 + d^2))/(d*x^5)

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Sympy [C] Result contains complex when optimal does not.
time = 5.62, size = 774, normalized size = 7.17 \begin {gather*} d^{3} \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 ) - d^{2} e \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 ) - d e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \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**6/(e*x+d),x)

[Out]

d**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), 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 + 15*d*e**2*x**7), True)) - d**2*e*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)) - d*e**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**
2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(
e**2*x**2) + 1)/(3*d**2), True)) + e**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*x) + e**2*acosh(d/(e*x))/(
2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x
**2) + 1)) - I*e**2*asin(d/(e*x))/(2*d), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (87) = 174\).
time = 1.08, size = 366, normalized size = 3.39 \begin {gather*} -\frac {x^{5} {\left (\frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{3}}{x} + \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e}{x^{2}} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{\left (-1\right )}}{x^{3}} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-3\right )}}{x^{4}} - 2 \, e^{5}\right )} e^{10}}{320 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d} + \frac {3 \, e^{5} \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 )}{8 \, d} - \frac {\frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{3}}{x} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e}{x^{2}} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{\left (-1\right )}}{x^{3}} - \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{\left (-3\right )}}{x^{4}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{4} e^{\left (-5\right )}}{x^{5}}}{320 \, d^{5}} \end {gather*}

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),x, algorithm="giac")

[Out]

-1/320*x^5*(5*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^3/x + 10*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e/x^2 - 40*(d*e + sqr
t(-x^2*e^2 + d^2)*e)^3*e^(-1)/x^3 - 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^(-3)/x^4 - 2*e^5)*e^10/((d*e + sqrt(
-x^2*e^2 + d^2)*e)^5*d) + 3/8*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d - 1/320*(20*
(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*e^3/x + 40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^4*e/x^2 - 10*(d*e + sqrt(-x^2
*e^2 + d^2)*e)^3*d^4*e^(-1)/x^3 - 5*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^4*e^(-3)/x^4 + 2*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^5*d^4*e^(-5)/x^5)/d^5

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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^6\,\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^6*(d + e*x)),x)

[Out]

int((d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)), x)

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