3.4.71 \(\int \frac {x^4 (d+e x^2)^{3/2}}{a+b x^2+c x^4} \, dx\) [371]

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

[Out]

1/4*x*(e*x^2+d)^(3/2)/c+1/8*d*(-4*b*e+3*c*d)*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/c^2/e^(1/2)-1/2*arctanh(x*e^(1
/2)/(e*x^2+d)^(1/2))*(b*c*d-b^2*e+a*c*e+(-3*a*b*c*e+2*a*c^2*d+b^3*e-b^2*c*d)/(-4*a*c+b^2)^(1/2))*e^(1/2)/c^3-1
/2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2
))*e^(1/2)/c^3+1/8*(-4*b*e+3*c*d)*x*(e*x^2+d)^(1/2)/c^2-1/2*arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(e
*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b*c*d-b^2*e+a*c*e+(-3*a*b*c*e+2*a*c^2*d+b^3*e-b^2*c*d)/(-4*a*c+b^
2)^(1/2))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/c^3/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctan(x*(2*c*d-e*(b+(-4
*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e-2*a*c^2*d-
b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/c^3/(b+(-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 2.38, antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1305, 396, 201, 223, 212, 1706, 399, 385, 211} \begin {gather*} -\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^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {ArcTan}\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^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {ArcTan}\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^3 (-e)+b^2 c d}{\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^3 (-e)+b^2 c d}{\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} \end {gather*}

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 201

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 211

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

Rule 212

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

Rule 223

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 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 396

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 399

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

Rule 1305

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 1706

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]

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)) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(18689\) vs. \(2(595)=1190\).
time = 16.35, size = 18689, normalized size = 31.41 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 459, normalized size = 0.77

method	result	size
risch	\(-\frac {x \left (-2 c e \,x^{2}+4 e b -5 c d \right ) \sqrt {e \,x^{2}+d}}{8 c^{2}}-\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) e^{\frac {3}{2}} a}{c^{2}}+\frac {\ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) e^{\frac {3}{2}} b^{2}}{c^{3}}-\frac {3 \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) \sqrt {e}\, b d}{2 c^{2}}+\frac {3 \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right ) d^{2}}{8 c \sqrt {e}}-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (2 a b c \,e^{2}-2 a \,c^{2} d e -b^{3} e^{2}+2 b^{2} c d e -b \,c^{2} d^{2}\right ) \textit {\_R}^{2}+2 \left (2 e^{3} c \,a^{2}-2 a \,b^{2} e^{3}+2 a b c d \,e^{2}+b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) \textit {\_R} +2 a b c \,d^{2} e^{2}-2 a \,c^{2} d^{3} e -b^{3} d^{2} e^{2}+2 b^{2} c \,d^{3} e -b \,c^{2} d^{4}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2 c^{3}}\)	\(444\)
default	\(\frac {\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e}\, x +\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}}{c}+\frac {\sqrt {e}\, \left (\frac {b \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}}{2 c}+\frac {\left (4 a c e -4 b^{2} e +6 b c d \right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{c^{2}}-\frac {b \,d^{2}}{2 c \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}}-\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (2 a b c \,e^{2}-2 a \,c^{2} d e -b^{3} e^{2}+2 b^{2} c d e -b \,c^{2} d^{2}\right ) \textit {\_R}^{2}+2 \left (2 e^{3} c \,a^{2}-2 a \,b^{2} e^{3}+2 a b c d \,e^{2}+b^{3} d \,e^{2}-2 b^{2} c \,d^{2} e +b \,c^{2} d^{3}\right ) \textit {\_R} +2 a b c \,d^{2} e^{2}-2 a \,c^{2} d^{3} e -b^{3} d^{2} e^{2}+2 b^{2} c \,d^{3} e -b \,c^{2} d^{4}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{c^{2}}\right )}{4 c}\)	\(459\)

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,method=_RETURNVERBOSE)

[Out]

1/c*(1/4*x*(e*x^2+d)^(3/2)+3/4*d*(1/2*x*(e*x^2+d)^(1/2)+1/2*d/e^(1/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))))+1/4/c*e^
(1/2)*(1/2*b*((e*x^2+d)^(1/2)-e^(1/2)*x)^2/c+1/c^2*(4*a*c*e-4*b^2*e+6*b*c*d)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)-1/2
/c*b*d^2/((e*x^2+d)^(1/2)-e^(1/2)*x)^2-2/c^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-b*c^2*d^4)/(_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+d^4*c)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}

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)

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Giac [A]
time = 6.19, size = 104, normalized size = 0.17 \begin {gather*} \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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,{\left (e\,x^2+d\right )}^{3/2}}{c\,x^4+b\,x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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