\(\int \frac {1}{(d+e x^2)^{3/2} (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\) [224]

Optimal result

Integrand size = 41, antiderivative size = 149 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{3 d (2 c d-b e) \left (d+e x^2\right )^{3/2}}-\frac {(7 c d-2 b e) x}{3 d^2 (2 c d-b e)^2 \sqrt {d+e x^2}}-\frac {c^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{5/2}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1163, 425, 541, 12, 385, 214} \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {c^2 \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{5/2}}-\frac {x (7 c d-2 b e)}{3 d^2 \sqrt {d+e x^2} (2 c d-b e)^2}-\frac {x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

[In]

Int[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-1/3*x/(d*(2*c*d - b*e)*(d + e*x^2)^(3/2)) - ((7*c*d - 2*b*e)*x)/(3*d^2*(2*c*d - b*e)^2*Sqrt[d + e*x^2]) - (c^
2*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)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1163

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx \\ & = -\frac {x}{3 d (2 c d-b e) \left (d+e x^2\right )^{3/2}}+\frac {\int \frac {e (5 c d-2 b e)-2 c e^2 x^2}{\left (d+e x^2\right )^{3/2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{3 d e (2 c d-b e)} \\ & = -\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 {\int \frac {3 c^2 d^2 e^2}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{3 d^2 e^2 (2 c d-b e)^2} \\ & = -\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 \int \frac {1}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{(2 c d-b e)^2} \\ & = -\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 {Subst}\left (\int \frac {1}{\frac {-c d^2+b d e}{d}-\left (-c d e+\frac {e \left (-c d^2+b d e\right )}{d}\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{(2 c d-b e)^2} \\ & = -\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 \tanh ^{-1}\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {\frac {(-2 c d+b e) x \left (-b e \left (3 d+2 e x^2\right )+c d \left (9 d+7 e x^2\right )\right )}{d^2 \left (d+e x^2\right )^{3/2}}+\frac {3 c^2 \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2} \text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{\sqrt {e} (-c d+b e)}}{3 (-2 c d+b e)^3} \]

[In]

Integrate[1/((d + e*x^2)^(3/2)*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-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]*ArcTanh[(-(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

Maple [A] (verified)

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	\(\text {Expression too large to display}\)	\(1551\)

[In]

int(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x,method=_RETURNVERBOSE)

[Out]

(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

Fricas [B] (verification not implemented)

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 ] \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[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)*arctan(-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*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)]

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int \frac {1}{\left (d + e x^{2}\right )^{\frac {5}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]

[In]

integrate(1/(e*x**2+d)**(3/2)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int { \frac {1}{{\left (c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

integrate(1/((c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e)*(e*x^2 + d)^(3/2)), x)

Giac [B] (verification not implemented)

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 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {c^{2} \sqrt {e} \arctan \left (\frac {{\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right )}{{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}} - \frac {{\left (\frac {{\left (28 \, c^{3} d^{3} e^{2} - 36 \, b c^{2} d^{2} e^{3} + 15 \, b^{2} c d e^{4} - 2 \, b^{3} e^{5}\right )} x^{2}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}} + \frac {3 \, {\left (12 \, c^{3} d^{4} e - 16 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )}}{16 \, c^{4} d^{6} e - 32 \, b c^{3} d^{5} e^{2} + 24 \, b^{2} c^{2} d^{4} e^{3} - 8 \, b^{3} c d^{3} e^{4} + b^{4} d^{2} e^{5}}\right )} x}{3 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(e*x^2+d)^(3/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

-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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,\left (-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2\right )} \,d x \]

[In]

int(1/((d + e*x^2)^(3/2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

int(1/((d + e*x^2)^(3/2)*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)), x)