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

Optimal. Leaf size=595 \[ -\frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 c^2}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c} \]

[Out]

((3*c*d - 4*b*e)*x*Sqrt[d + e*x^2])/(8*c^2) + (x*(d + e*x^2)^(3/2))/(4*c) - (Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*
c])*e]*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*
d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^3*Sqrt[b - Sqrt[b^2 - 4
*a*c]]) - (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a
*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqr
t[d + e*x^2])])/(2*c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (d*(3*c*d - 4*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])
/(8*c^2*Sqrt[e]) - (Sqrt[e]*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*
c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^3) - (Sqrt[e]*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b
^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 3.28462, antiderivative size = 595, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {1291, 388, 195, 217, 206, 1692, 402, 377, 205} \[ -\frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (-\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (-\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \sqrt{d+e x^2} (3 c d-4 b e)}{8 c^2}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*c*d - 4*b*e)*x*Sqrt[d + e*x^2])/(8*c^2) + (x*(d + e*x^2)^(3/2))/(4*c) - (Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*
c])*e]*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*
d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(2*c^3*Sqrt[b - Sqrt[b^2 - 4
*a*c]]) - (Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a
*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqr
t[d + e*x^2])])/(2*c^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (d*(3*c*d - 4*b*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])
/(8*c^2*Sqrt[e]) - (Sqrt[e]*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*
c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^3) - (Sqrt[e]*(b*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b
^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c^3)

Rule 1291

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[f
^4/c^2, Int[(f*x)^(m - 4)*(c*d - b*e + c*e*x^2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^4/c^2, Int[((f*x)^(m - 4)
*(d + e*x^2)^(q - 1)*Simp[a*(c*d - b*e) + (b*c*d - b^2*e + a*c*e)*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; Free
Q[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 3]

Rule 388

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1692

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

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 205

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

Rubi steps

\begin{align*} \int \frac{x^4 \left (d+e x^2\right )^{3/2}}{a+b x^2+c x^4} \, dx &=\frac{\int \sqrt{d+e x^2} \left (c d-b e+c e x^2\right ) \, dx}{c^2}-\frac{\int \frac{\sqrt{d+e x^2} \left (a (c d-b e)+\left (b c d-b^2 e+a c e\right ) x^2\right )}{a+b x^2+c x^4} \, dx}{c^2}\\ &=\frac{x \left (d+e x^2\right )^{3/2}}{4 c}-\frac{\int \left (\frac{\left (b c d-b^2 e+a c e+\frac{-b^2 c d+2 a c^2 d+b^3 e-3 a b c e}{\sqrt{b^2-4 a c}}\right ) \sqrt{d+e x^2}}{b-\sqrt{b^2-4 a c}+2 c x^2}+\frac{\left (b c d-b^2 e+a c e-\frac{-b^2 c d+2 a c^2 d+b^3 e-3 a b c e}{\sqrt{b^2-4 a c}}\right ) \sqrt{d+e x^2}}{b+\sqrt{b^2-4 a c}+2 c x^2}\right ) \, dx}{c^2}+\frac{(3 c d-4 b e) \int \sqrt{d+e x^2} \, dx}{4 c^2}\\ &=\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 c^2}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c}+\frac{(d (3 c d-4 b e)) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{8 c^2}-\frac{\left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{\sqrt{d+e x^2}}{b-\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c^2}-\frac{\left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \int \frac{\sqrt{d+e x^2}}{b+\sqrt{b^2-4 a c}+2 c x^2} \, dx}{c^2}\\ &=\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 c^2}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c}+\frac{(d (3 c d-4 b e)) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{8 c^2}-\frac{\left (e \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{2 c^3}-\frac{\left (\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{2 c^3}-\frac{\left (e \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{2 c^3}-\frac{\left (\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{2 c^3}\\ &=\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 c^2}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}-\frac{\left (e \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{2 c^3}-\frac{\left (\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{b^2-4 a c}-\left (-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{2 c^3}-\frac{\left (e \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{2 c^3}-\frac{\left (\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{b^2-4 a c}-\left (-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{2 c^3}\\ &=\frac{(3 c d-4 b e) x \sqrt{d+e x^2}}{8 c^2}+\frac{x \left (d+e x^2\right )^{3/2}}{4 c}-\frac{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 c^3 \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{d (3 c d-4 b e) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 c^2 \sqrt{e}}-\frac{\sqrt{e} \left (b c d-b^2 e+a c e-\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^3}-\frac{\sqrt{e} \left (b c d-b^2 e+a c e+\frac{b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 c^3}\\ \end{align*}

Mathematica [B]  time = 6.51279, size = 18689, normalized size = 31.41 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [C]  time = 0.034, size = 516, normalized size = 0.9 \begin{align*}{\frac{x}{4\,c} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,dx}{8\,c}\sqrt{e{x}^{2}+d}}+{\frac{3\,{d}^{2}}{8\,c}\ln \left ( \sqrt{e}x+\sqrt{e{x}^{2}+d} \right ){\frac{1}{\sqrt{e}}}}+{\frac{b{x}^{2}}{4\,{c}^{2}}{e}^{{\frac{3}{2}}}}-{\frac{xeb}{4\,{c}^{2}}\sqrt{e{x}^{2}+d}}+{\frac{bd}{8\,{c}^{2}}\sqrt{e}}+{\frac{a}{{c}^{2}}{e}^{{\frac{3}{2}}}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) }-{\frac{{b}^{2}}{{c}^{3}}{e}^{{\frac{3}{2}}}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) }+{\frac{3\,bd}{2\,{c}^{2}}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) }-{\frac{b{d}^{2}}{8\,{c}^{2}}\sqrt{e} \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{-2}}-{\frac{1}{2\,{c}^{3}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( 2\,ab{e}^{2}c-2\,a{c}^{2}de-{b}^{3}{e}^{2}+2\,{b}^{2}dec-b{c}^{2}{d}^{2} \right ){{\it \_R}}^{2}+2\, \left ( 2\,{a}^{2}c{e}^{3}-2\,a{b}^{2}{e}^{3}+2\,abcd{e}^{2}+{b}^{3}d{e}^{2}-2\,{b}^{2}{d}^{2}ec+b{c}^{2}{d}^{3} \right ){\it \_R}+2\,abc{d}^{2}{e}^{2}-2\,a{c}^{2}{d}^{3}e-{b}^{3}{d}^{2}{e}^{2}+2\,{b}^{2}c{d}^{3}e-{c}^{2}{d}^{4}b}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{2}-{\it \_R} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x*(e*x^2+d)^(3/2)/c+3/8/c*d*x*(e*x^2+d)^(1/2)+3/8/c*d^2/e^(1/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))+1/4/c^2*e^(3
/2)*b*x^2-1/4/c^2*e*b*(e*x^2+d)^(1/2)*x+1/8/c^2*e^(1/2)*b*d+1/c^2*e^(3/2)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)*a-1/c^
3*e^(3/2)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)*b^2+3/2/c^2*e^(1/2)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)*b*d-1/8/c^2*e^(1/2)*
b*d^2/((e*x^2+d)^(1/2)-e^(1/2)*x)^2-1/2/c^3*e^(1/2)*sum(((2*a*b*c*e^2-2*a*c^2*d*e-b^3*e^2+2*b^2*c*d*e-b*c^2*d^
2)*_R^2+2*(2*a^2*c*e^3-2*a*b^2*e^3+2*a*b*c*d*e^2+b^3*d*e^2-2*b^2*c*d^2*e+b*c^2*d^3)*_R+2*a*b*c*d^2*e^2-2*a*c^2
*d^3*e-b^3*d^2*e^2+2*b^2*c*d^3*e-c^2*d^4*b)/(_R^3*c+3*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d
^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-e^(1/2)*x)^2-_R),_R=RootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2
)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{4}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [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(x^4*(e*x^2+d)^(3/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (d + e x^{2}\right )^{\frac{3}{2}}}{a + b x^{2} + c x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4*(d + e*x**2)**(3/2)/(a + b*x**2 + c*x**4), x)

________________________________________________________________________________________

Giac [A]  time = 1.43415, size = 140, normalized size = 0.24 \begin{align*} \frac{1}{8} \, \sqrt{x^{2} e + d}{\left (\frac{2 \, x^{2} e}{c} + \frac{{\left (5 \, c^{5} d e^{2} - 4 \, b c^{4} e^{3}\right )} e^{\left (-2\right )}}{c^{6}}\right )} x - \frac{{\left (3 \, c^{2} d^{2} - 12 \, b c d e + 8 \, b^{2} e^{2} - 8 \, a c e^{2}\right )} e^{\left (-\frac{1}{2}\right )} \log \left ({\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2}\right )}{16 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(x^2*e + d)*(2*x^2*e/c + (5*c^5*d*e^2 - 4*b*c^4*e^3)*e^(-2)/c^6)*x - 1/16*(3*c^2*d^2 - 12*b*c*d*e + 8*
b^2*e^2 - 8*a*c*e^2)*e^(-1/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c^3