\(\int \frac {1}{x^2 (d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\) [145]

Optimal result

Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7} \]

[Out]

1/15*(-5*e*x+6*d)/d^4/x/(-e^2*x^2+d^2)^(3/2)+1/5/d^2/x/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+e*arctanh((-e^2*x^2+d^2)^(
1/2)/d)/d^7+1/5*(-5*e*x+8*d)/d^6/x/(-e^2*x^2+d^2)^(1/2)-16/5*(-e^2*x^2+d^2)^(1/2)/d^7/x

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {871, 837, 821, 272, 65, 214} \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}+\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(6*d - 5*e*x)/(15*d^4*x*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (8*d - 5*e*x)/(
5*d^6*x*Sqrt[d^2 - e^2*x^2]) - (16*Sqrt[d^2 - e^2*x^2])/(5*d^7*x) + (e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^7

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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 871

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[d*(f + g*x)^
(n + 1)*((a + c*x^2)^(p + 1)/(2*a*p*(e*f - d*g)*(d + e*x))), x] + Dist[1/(p*(2*c*d)*(e*f - d*g)), Int[(f + g*x
)^n*(a + c*x^2)^p*(c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, c, d, e
, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] &
&  !IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-6 d e^2+5 e^3 x}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-24 d^3 e^4+15 d^2 e^5 x}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-48 d^5 e^6+15 d^4 e^7 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}-\frac {e \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^6} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^6} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {\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 )}{d^6 e} \\ & = \frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (15 d^5+38 d^4 e x-52 d^3 e^2 x^2-87 d^2 e^3 x^3+33 d e^4 x^4+48 e^5 x^5\right )}{x (d-e x)^2 (d+e x)^3}-15 \sqrt {d^2} e \log (x)+15 \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{15 d^8} \]

[In]

Integrate[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-1/15*((d*Sqrt[d^2 - e^2*x^2]*(15*d^5 + 38*d^4*e*x - 52*d^3*e^2*x^2 - 87*d^2*e^3*x^3 + 33*d*e^4*x^4 + 48*e^5*x
^5))/(x*(d - e*x)^2*(d + e*x)^3) - 15*Sqrt[d^2]*e*Log[x] + 15*Sqrt[d^2]*e*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]]
)/d^8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(136)=272\).

Time = 0.44 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.89

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{7} x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{6} \sqrt {d^{2}}}-\frac {17 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{60 d^{6} e \left (x +\frac {d}{e}\right )^{2}}-\frac {413 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 d^{7} \left (x +\frac {d}{e}\right )}+\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{24 d^{6} e \left (x -\frac {d}{e}\right )^{2}}-\frac {23 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{48 d^{7} \left (x -\frac {d}{e}\right )}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{5} e^{2} \left (x +\frac {d}{e}\right )^{3}}\)	\(291\)
default	\(\frac {-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}}{d}-\frac {e \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{2}}+\frac {e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{2}}\)	\(333\)

[In]

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

[Out]

-(-e^2*x^2+d^2)^(1/2)/d^7/x+1/d^6*e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-17/60/d^6/e/(
x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-413/240/d^7/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/24/d^
6/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-23/48/d^7/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/20
/d^5/e^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.72 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {23 \, e^{6} x^{6} + 23 \, d e^{5} x^{5} - 46 \, d^{2} e^{4} x^{4} - 46 \, d^{3} e^{3} x^{3} + 23 \, d^{4} e^{2} x^{2} + 23 \, d^{5} e x + 15 \, {\left (e^{6} x^{6} + d e^{5} x^{5} - 2 \, d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2} x^{2} + d^{5} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (48 \, e^{5} x^{5} + 33 \, d e^{4} x^{4} - 87 \, d^{2} e^{3} x^{3} - 52 \, d^{3} e^{2} x^{2} + 38 \, d^{4} e x + 15 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{7} e^{5} x^{6} + d^{8} e^{4} x^{5} - 2 \, d^{9} e^{3} x^{4} - 2 \, d^{10} e^{2} x^{3} + d^{11} e x^{2} + d^{12} x\right )}} \]

[In]

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

[Out]

-1/15*(23*e^6*x^6 + 23*d*e^5*x^5 - 46*d^2*e^4*x^4 - 46*d^3*e^3*x^3 + 23*d^4*e^2*x^2 + 23*d^5*e*x + 15*(e^6*x^6
+ d*e^5*x^5 - 2*d^2*e^4*x^4 - 2*d^3*e^3*x^3 + d^4*e^2*x^2 + d^5*e*x)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (48
*e^5*x^5 + 33*d*e^4*x^4 - 87*d^2*e^3*x^3 - 52*d^3*e^2*x^2 + 38*d^4*e*x + 15*d^5)*sqrt(-e^2*x^2 + d^2))/(d^7*e^
5*x^6 + d^8*e^4*x^5 - 2*d^9*e^3*x^4 - 2*d^10*e^2*x^3 + d^11*e*x^2 + d^12*x)

Sympy [F]

\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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