\(\int \frac {(d+e x)^3}{x^3 (d^2-e^2 x^2)^{7/2}} \, dx\) [91]

Optimal result

Integrand size = 27, antiderivative size = 182 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {13 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \]

[Out]

4/5*e^2*(e*x+d)/d^2/(-e^2*x^2+d^2)^(5/2)+1/15*e^2*(31*e*x+25*d)/d^4/(-e^2*x^2+d^2)^(3/2)-13/2*e^2*arctanh((-e^
2*x^2+d^2)^(1/2)/d)/d^6+1/15*e^2*(107*e*x+90*d)/d^6/(-e^2*x^2+d^2)^(1/2)-1/2*(-e^2*x^2+d^2)^(1/2)/d^5/x^2-3*e*
(-e^2*x^2+d^2)^(1/2)/d^6/x

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 182, 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 = {1819, 1821, 821, 272, 65, 214} \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {13 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[(d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(4*e^2*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (e^2*(25*d + 31*e*x))/(15*d^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*
(90*d + 107*e*x))/(15*d^6*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(2*d^5*x^2) - (3*e*Sqrt[d^2 - e^2*x^2])/(
d^6*x) - (13*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^6)

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 1819

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 1821

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*} \text {integral}& = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^3-15 d^2 e x-20 d e^2 x^2-16 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^3+45 d^2 e x+75 d e^2 x^2+62 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^3-45 d^2 e x-90 d e^2 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {\int \frac {90 d^4 e+195 d^3 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^8} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {\left (13 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^5} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {\left (13 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^5} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {13 \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 )}{2 d^5} \\ & = \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {13 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (15 d^4+45 d^3 e x-479 d^2 e^2 x^2+717 d e^3 x^3-304 e^4 x^4\right )}{x^2 (-d+e x)^3}-195 \sqrt {d^2} e^2 \log (x)+195 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{30 d^7} \]

[In]

Integrate[(d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((d*Sqrt[d^2 - e^2*x^2]*(15*d^4 + 45*d^3*e*x - 479*d^2*e^2*x^2 + 717*d*e^3*x^3 - 304*e^4*x^4))/(x^2*(-d + e*x)
^3) - 195*Sqrt[d^2]*e^2*Log[x] + 195*Sqrt[d^2]*e^2*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(30*d^7)

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.18

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (6 e x +d \right )}{2 d^{6} x^{2}}-\frac {13 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{5} \sqrt {d^{2}}}-\frac {107 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 d^{6} \left (x -\frac {d}{e}\right )}+\frac {17 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 d^{5} \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d^{4} e \left (x -\frac {d}{e}\right )^{3}}\)	\(214\)
default	\(e^{3} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+d^{3} \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )+3 d^{2} e \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )\)	\(436\)

[In]

int((e*x+d)^3/x^3/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-e^2*x^2+d^2)^(1/2)*(6*e*x+d)/d^6/x^2-13/2/d^5*e^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1
/2))/x)-107/15/d^6*e/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)+17/15/d^5/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x
-d/e))^(1/2)-1/5/d^4/e/(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 = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {254 \, e^{5} x^{5} - 762 \, d e^{4} x^{4} + 762 \, d^{2} e^{3} x^{3} - 254 \, d^{3} e^{2} x^{2} + 195 \, {\left (e^{5} x^{5} - 3 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (304 \, e^{4} x^{4} - 717 \, d e^{3} x^{3} + 479 \, d^{2} e^{2} x^{2} - 45 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{6} e^{3} x^{5} - 3 \, d^{7} e^{2} x^{4} + 3 \, d^{8} e x^{3} - d^{9} x^{2}\right )}} \]

[In]

integrate((e*x+d)^3/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(254*e^5*x^5 - 762*d*e^4*x^4 + 762*d^2*e^3*x^3 - 254*d^3*e^2*x^2 + 195*(e^5*x^5 - 3*d*e^4*x^4 + 3*d^2*e^3
*x^3 - d^3*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (304*e^4*x^4 - 717*d*e^3*x^3 + 479*d^2*e^2*x^2 - 45*d
^3*e*x - 15*d^4)*sqrt(-e^2*x^2 + d^2))/(d^6*e^3*x^5 - 3*d^7*e^2*x^4 + 3*d^8*e*x^3 - d^9*x^2)

Sympy [F]

\[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate((e*x+d)**3/x**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

Integral((d + e*x)**3/(x**3*(-(-d + e*x)*(d + e*x))**(7/2)), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {19 \, e^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}} + \frac {13 \, e^{2}}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {76 \, e^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {13 \, e^{2}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {3 \, e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x} + \frac {152 \, e^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} - \frac {13 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{6}} + \frac {13 \, e^{2}}{2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}} - \frac {d}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2}} \]

[In]

integrate((e*x+d)^3/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

19/5*e^3*x/((-e^2*x^2 + d^2)^(5/2)*d^2) + 13/10*e^2/((-e^2*x^2 + d^2)^(5/2)*d) + 76/15*e^3*x/((-e^2*x^2 + d^2)
^(3/2)*d^4) + 13/6*e^2/((-e^2*x^2 + d^2)^(3/2)*d^3) - 3*e/((-e^2*x^2 + d^2)^(5/2)*x) + 152/15*e^3*x/(sqrt(-e^2
*x^2 + d^2)*d^6) - 13/2*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^6 + 13/2*e^2/(sqrt(-e^2*x^2
+ d^2)*d^5) - 1/2*d/((-e^2*x^2 + d^2)^(5/2)*x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (160) = 320\).

Time = 0.31 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {13 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{6} {\left | e \right |}} - \frac {{\left (15 \, e^{3} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x} - \frac {2782 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e x^{2}} + \frac {9410 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{3} x^{3}} - \frac {13645 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{5} x^{4}} + \frac {9285 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{7} x^{5}} - \frac {2580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{9} x^{6}}\right )} e^{4} x^{2}}{120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} - \frac {\frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} {\left | e \right |}}{e x^{2}}}{8 \, d^{12} e^{2}} \]

[In]

integrate((e*x+d)^3/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-13/2*e^3*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^6*abs(e)) - 1/120*(15*e^3 + 105
*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e/x - 2782*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e*x^2) + 9410*(d*e + sq
rt(-e^2*x^2 + d^2)*abs(e))^3/(e^3*x^3) - 13645*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^5*x^4) + 9285*(d*e + s
qrt(-e^2*x^2 + d^2)*abs(e))^5/(e^7*x^5) - 2580*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^9*x^6))*e^4*x^2/((d*e
+ sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^6*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e)) - 1/8*(12*(d*
e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^6*e*abs(e)/x + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^6*abs(e)/(e*x^2))/(d
^12*e^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{x^3\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[In]

int((d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)),x)

[Out]

int((d + e*x)^3/(x^3*(d^2 - e^2*x^2)^(7/2)), x)