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

Optimal. Leaf size=149 \[ -\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{5/2}}-\frac{x (7 c d-2 b e)}{3 d^2 \sqrt{d+e x^2} (2 c d-b e)^2}-\frac{x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

[Out]

-x/(3*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))

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Rubi [A]  time = 0.268589, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {1149, 414, 527, 12, 377, 208} \[ -\frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} (2 c d-b e)^{5/2}}-\frac{x (7 c d-2 b e)}{3 d^2 \sqrt{d+e x^2} (2 c d-b e)^2}-\frac{x}{3 d \left (d+e x^2\right )^{3/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[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]

-x/(3*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 1149

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*x^2)/e)^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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 12

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

Rule 377

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \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 )^{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 \operatorname{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 [C]  time = 3.49204, size = 1058, normalized size = 7.1 \[ -\frac{x \left (-\frac{56 c^2 e^2 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2} x^4}{(c d-b e)^2}+\frac{168 c^2 e^2 \tanh ^{-1}\left (\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right ) x^4}{(c d-b e)^2}+\frac{36 c^2 e^2 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^4}{(c d-b e)^2}+\frac{12 c^2 e^2 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{2}\right \},\left \{1,\frac{9}{2}\right \},\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^4}{(c d-b e)^2}-\frac{168 c^2 e^2 \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}} x^4}{(c d-b e)^2}+\frac{140 c e \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2} x^2}{c d-b e}-\frac{420 c e \tanh ^{-1}\left (\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right ) x^2}{c d-b e}-\frac{84 c e \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^2}{c d-b e}-\frac{24 c e \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{2}\right \},\left \{1,\frac{9}{2}\right \},\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right ) x^2}{c d-b e}+\frac{420 c e \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}} x^2}{c d-b e}-105 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{3/2}+315 \tanh ^{-1}\left (\sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right )+48 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )+12 \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{7/2} \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{2}\right \},\left \{1,\frac{9}{2}\right \},\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )-315 \sqrt{\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}}\right )}{63 (c d-b e) \left (\frac{e (b e-2 c d) x^2}{(b e-c d) \left (e x^2+d\right )}\right )^{5/2} \left (e x^2+d\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[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]

-(x*(-315*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] + (420*c*e*x^2*Sqrt[(e*(-2*c*d + b*e)*x^2)
/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) - (168*c^2*e^2*x^4*Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d
+ e*x^2))])/(c*d - b*e)^2 - 105*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2) + (140*c*e*x^2*((e
*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e) - (56*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2
)/((-(c*d) + b*e)*(d + e*x^2)))^(3/2))/(c*d - b*e)^2 + 315*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)
*(d + e*x^2))]] - (420*c*e*x^2*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e)
+ (168*c^2*e^2*x^4*ArcTanh[Sqrt[(e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))]])/(c*d - b*e)^2 + 48*((e
*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)
/((-(c*d) + b*e)*(d + e*x^2))] - (84*c*e*x^2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hyper
geometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (36*c^2*e^2*x^4*
((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (e*(-2*c*d + b*e)*x
^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e)^2 + 12*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7
/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))] - (24*c*e*x^
2*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*
c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2))])/(c*d - b*e) + (12*c^2*e^2*x^4*((e*(-2*c*d + b*e)*x^2)/((-(c*d)
+ b*e)*(d + e*x^2)))^(7/2)*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, (e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d
+ e*x^2))])/(c*d - b*e)^2))/(63*(c*d - b*e)*((e*(-2*c*d + b*e)*x^2)/((-(c*d) + b*e)*(d + e*x^2)))^(5/2)*(d + e
*x^2)^(5/2))

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Maple [B]  time = 0.02, size = 1637, normalized size = 11. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

-1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(
1/2)/(b*e-2*c*d)/((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)
-(b*e-2*c*d)/c)^(1/2)+1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2
))/(b*e-2*c*d)/d/((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)
-(b*e-2*c*d)/c)^(1/2)*x+1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1
/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c
*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e
)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))-1/6*c/d/((-d*e)
^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(x+(-d*e)^(1/2)/e)/((x+(-d*e)^(1/2)/
e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)-1/3*c*e/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(
1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/2)/e))^(1/2)*x-1/6*c/d/((-d
*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(x-(-d*e)^(1/2)/e)/((x-(-d*e)^(1/
2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)-1/3*c*e/d^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e
)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/((x-(-d*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)*x+1/2*c^3*e
/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(b*e-
2*c*d)/((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*
d)/c)^(1/2)+1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(b*e-2*
c*d)/d/((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*
d)/c)^(1/2)*x-1/2*c^3*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*
e-c*d)*c*e)^(1/2)/(b*e-2*c*d)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e
-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*
(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 5.15639, size = 2169, normalized size = 14.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x^{2}\right )^{\frac{5}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out