\(\int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx\) [101]

Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 849, 821, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=-\frac {3 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}-\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2} \]

[In]

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

[Out]

-1/4*Sqrt[d^2 - e^2*x^2]/(d*x^4) + (e*Sqrt[d^2 - e^2*x^2])/(3*d^2*x^3) - (3*e^2*Sqrt[d^2 - e^2*x^2])/(8*d^3*x^
2) + (2*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^4*x) - (3*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

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*} \text {integral}& = \int \frac {d-e x}{x^5 \sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}-\frac {\int \frac {4 d^2 e-3 d e^2 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}+\frac {\int \frac {9 d^3 e^2-8 d^2 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\int \frac {16 d^4 e^3-9 d^3 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^6} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}+\frac {\left (3 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {\left (3 e^2\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 )}{8 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 d x^4}+\frac {e \sqrt {d^2-e^2 x^2}}{3 d^2 x^3}-\frac {3 e^2 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {2 e^3 \sqrt {d^2-e^2 x^2}}{3 d^4 x}-\frac {3 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3+8 d^2 e x-9 d e^2 x^2+16 e^3 x^3\right )}{24 d^4 x^4}-\frac {3 \sqrt {d^2} e^4 \log (x)}{8 d^5}+\frac {3 \sqrt {d^2} e^4 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{8 d^5} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^3 + 8*d^2*e*x - 9*d*e^2*x^2 + 16*e^3*x^3))/(24*d^4*x^4) - (3*Sqrt[d^2]*e^4*Log[x])/
(8*d^5) + (3*Sqrt[d^2]*e^4*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(8*d^5)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-16 e^{3} x^{3}+9 d \,e^{2} x^{2}-8 d^{2} e x +6 d^{3}\right )}{24 d^{4} x^{4}}-\frac {3 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{3} \sqrt {d^{2}}}\)	\(99\)
default	\(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 d^{2} x^{4}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{4 d^{2}}}{d}+\frac {e^{4} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{5}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{4}}+\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{4} x^{3}}-\frac {e^{4} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{5}}\)	\(477\)

[In]

int((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

-1/24*(-e^2*x^2+d^2)^(1/2)*(-16*e^3*x^3+9*d*e^2*x^2-8*d^2*e*x+6*d^3)/d^4/x^4-3/8/d^3*e^4/(d^2)^(1/2)*ln((2*d^2
+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\frac {9 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{3} x^{3} - 9 \, d e^{2} x^{2} + 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, d^{4} x^{4}} \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x, algorithm="fricas")

[Out]

1/24*(9*e^4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (16*e^3*x^3 - 9*d*e^2*x^2 + 8*d^2*e*x - 6*d^3)*sqrt(-e^2*
x^2 + d^2))/(d^4*x^4)

Sympy [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{5} \left (d + e x\right )}\, dx \]

[In]

integrate((-e**2*x**2+d**2)**(1/2)/x**5/(e*x+d),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**5*(d + e*x)), x)

Maxima [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{5}} \,d x } \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^5), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (123) = 246\).

Time = 0.28 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\frac {{\left (3 \, e^{5} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} + \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} - \frac {72 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}}\right )} e^{8} x^{4}}{192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{4} {\left | e \right |}} - \frac {3 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, d^{4} {\left | e \right |}} + \frac {\frac {72 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{12} e^{5} {\left | e \right |}}{x} - \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{12} e^{3} {\left | e \right |}}{x^{2}} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{12} e {\left | e \right |}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{12} {\left | e \right |}}{e x^{4}}}{192 \, d^{16} e^{4}} \]

[In]

integrate((-e^2*x^2+d^2)^(1/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

1/192*(3*e^5 - 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3/x + 24*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e/x^2 -
72*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e*x^3))*e^8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^4*abs(e)) -
3/8*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^4*abs(e)) + 1/192*(72*(d*e + sqr
t(-e^2*x^2 + d^2)*abs(e))*d^12*e^5*abs(e)/x - 24*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^12*e^3*abs(e)/x^2 + 8
*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^12*e*abs(e)/x^3 - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^12*abs(e)
/(e*x^4))/(d^16*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^5 (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^5\,\left (d+e\,x\right )} \,d x \]

[In]

int((d^2 - e^2*x^2)^(1/2)/(x^5*(d + e*x)),x)

[Out]

int((d^2 - e^2*x^2)^(1/2)/(x^5*(d + e*x)), x)