Integrand size = 41, antiderivative size = 149 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{3 d (2 c d-b e) \left (d+e x^2\right )^{3/2}}-\frac {(7 c d-2 b e) x}{3 d^2 (2 c d-b e)^2 \sqrt {d+e x^2}}-\frac {c^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{5/2}} \]
-1/3*x/d/(-b*e+2*c*d)/(e*x^2+d)^(3/2)-c^2*arctanh(x*e^(1/2)*(-b*e+2*c*d)^( 1/2)/(-b*e+c*d)^(1/2)/(e*x^2+d)^(1/2))/(-b*e+2*c*d)^(5/2)/e^(1/2)/(-b*e+c* d)^(1/2)-1/3*(-2*b*e+7*c*d)*x/d^2/(-b*e+2*c*d)^2/(e*x^2+d)^(1/2)
Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {\frac {(-2 c d+b e) x \left (-b e \left (3 d+2 e x^2\right )+c d \left (9 d+7 e x^2\right )\right )}{d^2 \left (d+e x^2\right )^{3/2}}+\frac {3 c^2 \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2} \text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{\sqrt {e} (-c d+b e)}}{3 (-2 c d+b e)^3} \]
-1/3*(((-2*c*d + b*e)*x*(-(b*e*(3*d + 2*e*x^2)) + c*d*(9*d + 7*e*x^2)))/(d ^2*(d + e*x^2)^(3/2)) + (3*c^2*Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2]*ArcTa nh[(-(b*e) + c*(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2]))/Sqrt[2*c^2*d^2 - 3 *b*c*d*e + b^2*e^2]])/(Sqrt[e]*(-(c*d) + b*e)))/(-2*c*d + b*e)^3
Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {1387, 316, 25, 27, 402, 27, 291, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (b d e+b e^2 x^2-c d^2+c e^2 x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1387 |
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (\frac {b d e-c d^2}{d}+c e x^2\right )}dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int -\frac {e \left (-2 c e x^2+5 c d-2 b e\right )}{\left (e x^2+d\right )^{3/2} \left (-c e x^2+c d-b e\right )}dx}{3 d e (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {e \left (-2 c e x^2+5 c d-2 b e\right )}{\left (e x^2+d\right )^{3/2} \left (-c e x^2+c d-b e\right )}dx}{3 d e (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-2 c e x^2+5 c d-2 b e}{\left (e x^2+d\right )^{3/2} \left (-c e x^2+c d-b e\right )}dx}{3 d (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\frac {x (7 c d-2 b e)}{d \sqrt {d+e x^2} (2 c d-b e)}-\frac {\int -\frac {3 c^2 d^2 e}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{d e (2 c d-b e)}}{3 d (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 c^2 d \int \frac {1}{\sqrt {e x^2+d} \left (-c e x^2+c d-b e\right )}dx}{2 c d-b e}+\frac {x (7 c d-2 b e)}{d \sqrt {d+e x^2} (2 c d-b e)}}{3 d (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {3 c^2 d \int \frac {1}{-\frac {(c d e+(c d-b e) e) x^2}{e x^2+d}+c d-b e}d\frac {x}{\sqrt {e x^2+d}}}{2 c d-b e}+\frac {x (7 c d-2 b e)}{d \sqrt {d+e x^2} (2 c d-b e)}}{3 d (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {3 c^2 d \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}}+\frac {x (7 c d-2 b e)}{d \sqrt {d+e x^2} (2 c d-b e)}}{3 d (2 c d-b e)}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)}\) |
-1/3*x/(d*(2*c*d - b*e)*(d + e*x^2)^(3/2)) - (((7*c*d - 2*b*e)*x)/(d*(2*c* d - b*e)*Sqrt[d + e*x^2]) + (3*c^2*d*ArcTanh[(Sqrt[e]*Sqrt[2*c*d - b*e]*x) /(Sqrt[c*d - b*e]*Sqrt[d + e*x^2])])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e )^(3/2)))/(3*d*(2*c*d - b*e))
3.3.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)* (x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^ p, x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 0.38 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {c^{2} d^{2} \operatorname {arctanh}\left (\frac {\left (b e -c d \right ) \sqrt {e \,x^{2}+d}}{x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}+x \left (\frac {2 b \,e^{2} x^{2}}{3}+d \left (-\frac {7 c \,x^{2}}{3}+b \right ) e -3 c \,d^{2}\right ) \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}{\sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (b e -2 c d \right )^{2} d^{2}}\) | \(152\) |
| default | \(\text {Expression too large to display}\) | \(1551\) |
(c^2*d^2*arctanh((b*e-c*d)*(e*x^2+d)^(1/2)/x/(e*(b*e-2*c*d)*(b*e-c*d))^(1/ 2))*(e*x^2+d)^(3/2)+x*(2/3*b*e^2*x^2+d*(-7/3*c*x^2+b)*e-3*c*d^2)*(e*(b*e-2 *c*d)*(b*e-c*d))^(1/2))/(e*(b*e-2*c*d)*(b*e-c*d))^(1/2)/(e*x^2+d)^(3/2)/(b *e-2*c*d)^2/d^2
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (127) = 254\).
Time = 0.51 (sec) , antiderivative size = 1063, normalized size of antiderivative = 7.13 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{4} + 2 \, c^{2} d^{3} e x^{2} + c^{2} d^{4}\right )} \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} - 4 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left ({\left (3 \, c d e - 2 \, b e^{2}\right )} x^{3} + {\left (c d^{2} - b d e\right )} x\right )} \sqrt {e x^{2} + d}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right ) - 4 \, {\left ({\left (14 \, c^{3} d^{3} e^{2} - 25 \, b c^{2} d^{2} e^{3} + 13 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{3} + 3 \, {\left (6 \, c^{3} d^{4} e - 11 \, b c^{2} d^{3} e^{2} + 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \sqrt {e x^{2} + d}}{12 \, {\left (8 \, c^{4} d^{8} e - 20 \, b c^{3} d^{7} e^{2} + 18 \, b^{2} c^{2} d^{6} e^{3} - 7 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5} + {\left (8 \, c^{4} d^{6} e^{3} - 20 \, b c^{3} d^{5} e^{4} + 18 \, b^{2} c^{2} d^{4} e^{5} - 7 \, b^{3} c d^{3} e^{6} + b^{4} d^{2} e^{7}\right )} x^{4} + 2 \, {\left (8 \, c^{4} d^{7} e^{2} - 20 \, b c^{3} d^{6} e^{3} + 18 \, b^{2} c^{2} d^{5} e^{4} - 7 \, b^{3} c d^{4} e^{5} + b^{4} d^{3} e^{6}\right )} x^{2}\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{4} + 2 \, c^{2} d^{3} e x^{2} + c^{2} d^{4}\right )} \sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} \arctan \left (-\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left ({\left (14 \, c^{3} d^{3} e^{2} - 25 \, b c^{2} d^{2} e^{3} + 13 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{3} + 3 \, {\left (6 \, c^{3} d^{4} e - 11 \, b c^{2} d^{3} e^{2} + 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x\right )} \sqrt {e x^{2} + d}}{6 \, {\left (8 \, c^{4} d^{8} e - 20 \, b c^{3} d^{7} e^{2} + 18 \, b^{2} c^{2} d^{6} e^{3} - 7 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5} + {\left (8 \, c^{4} d^{6} e^{3} - 20 \, b c^{3} d^{5} e^{4} + 18 \, b^{2} c^{2} d^{4} e^{5} - 7 \, b^{3} c d^{3} e^{6} + b^{4} d^{2} e^{7}\right )} x^{4} + 2 \, {\left (8 \, c^{4} d^{7} e^{2} - 20 \, b c^{3} d^{6} e^{3} + 18 \, b^{2} c^{2} d^{5} e^{4} - 7 \, b^{3} c d^{4} e^{5} + b^{4} d^{3} e^{6}\right )} x^{2}\right )}}\right ] \]
[1/12*(3*(c^2*d^2*e^2*x^4 + 2*c^2*d^3*e*x^2 + c^2*d^4)*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2* d^2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 - 4*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*((3*c*d*e - 2*b*e^2)*x^3 + (c*d^2 - b*d*e)*x)*sqrt(e*x^2 + d))/(c^2*e^2*x^4 + c^2*d ^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x^2)) - 4*((14*c^3*d^3*e^ 2 - 25*b*c^2*d^2*e^3 + 13*b^2*c*d*e^4 - 2*b^3*e^5)*x^3 + 3*(6*c^3*d^4*e - 11*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x^2 + d))/(8*c^4 *d^8*e - 20*b*c^3*d^7*e^2 + 18*b^2*c^2*d^6*e^3 - 7*b^3*c*d^5*e^4 + b^4*d^4 *e^5 + (8*c^4*d^6*e^3 - 20*b*c^3*d^5*e^4 + 18*b^2*c^2*d^4*e^5 - 7*b^3*c*d^ 3*e^6 + b^4*d^2*e^7)*x^4 + 2*(8*c^4*d^7*e^2 - 20*b*c^3*d^6*e^3 + 18*b^2*c^ 2*d^5*e^4 - 7*b^3*c*d^4*e^5 + b^4*d^3*e^6)*x^2), -1/6*(3*(c^2*d^2*e^2*x^4 + 2*c^2*d^3*e*x^2 + c^2*d^4)*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*ar ctan(-1/2*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*(c*d^2 - b*d*e + (3*c *d*e - 2*b*e^2)*x^2)*sqrt(e*x^2 + d)/((2*c^2*d^2*e^2 - 3*b*c*d*e^3 + b^2*e ^4)*x^3 + (2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x)) + 2*((14*c^3*d^3*e ^2 - 25*b*c^2*d^2*e^3 + 13*b^2*c*d*e^4 - 2*b^3*e^5)*x^3 + 3*(6*c^3*d^4*e - 11*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e^3 - b^3*d*e^4)*x)*sqrt(e*x^2 + d))/(8*c^ 4*d^8*e - 20*b*c^3*d^7*e^2 + 18*b^2*c^2*d^6*e^3 - 7*b^3*c*d^5*e^4 + b^4*d^ 4*e^5 + (8*c^4*d^6*e^3 - 20*b*c^3*d^5*e^4 + 18*b^2*c^2*d^4*e^5 - 7*b^3*...
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int \frac {1}{\left (d + e x^{2}\right )^{\frac {5}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]
\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int { \frac {1}{{\left (c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.30 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {c^{2} \sqrt {e} \arctan \left (\frac {{\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right )}{{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}} - \frac {{\left (\frac {{\left (28 \, c^{3} d^{3} e^{2} - 36 \, b c^{2} d^{2} e^{3} + 15 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{2}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}} + \frac {3 \, {\left (12 \, c^{3} d^{4} e - 16 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}}\right )} x}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \]
-c^2*sqrt(e)*arctan(1/2*((sqrt(e)*x - sqrt(e*x^2 + d))^2*c - 3*c*d + 2*b*e )/sqrt(-2*c^2*d^2 + 3*b*c*d*e - b^2*e^2))/((4*c^2*d^2*e - 4*b*c*d*e^2 + b^ 2*e^3)*sqrt(-2*c^2*d^2 + 3*b*c*d*e - b^2*e^2)) - 1/3*((28*c^3*d^3*e^2 - 36 *b*c^2*d^2*e^3 + 15*b^2*c*d*e^4 - 2*b^3*e^5)*x^2/(16*c^4*d^6*e - 32*b*c^3* d^5*e^2 + 24*b^2*c^2*d^4*e^3 - 8*b^3*c*d^3*e^4 + b^4*d^2*e^5) + 3*(12*c^3* d^4*e - 16*b*c^2*d^3*e^2 + 7*b^2*c*d^2*e^3 - b^3*d*e^4)/(16*c^4*d^6*e - 32 *b*c^3*d^5*e^2 + 24*b^2*c^2*d^4*e^3 - 8*b^3*c*d^3*e^4 + b^4*d^2*e^5))*x/(e *x^2 + d)^(3/2)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2\right )} \,d x \]