3.4.62 \(\int \frac {x^2 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) [362]

Optimal. Leaf size=324 \[ \frac {\left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c} \]

[Out]

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

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Rubi [A]
time = 1.02, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1307, 223, 212, 1706, 385, 211} \begin {gather*} \frac {\left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+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 )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+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 )}{c \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c*d - b*e - (b*c*d - b^2*e + 2*a*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])])/(c*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*
a*c])*e]) + ((c*d - b*e + (b*c*d - b^2*e + 2*a*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])])/(c*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + S
qrt[b^2 - 4*a*c])*e]) + (Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/c

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 1307

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e
*(f^2/c), Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)*(S
imp[a*e - (c*d - b*e)*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[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^2 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx &=-\frac {\int \frac {a e-(c d-b e) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c}+\frac {e \int \frac {1}{\sqrt {d+e x^2}} \, dx}{c}\\ &=-\frac {\int \left (\frac {-c d+b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {-c d+b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{c}+\frac {e \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c}\\ &=\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c}-\frac {\left (-c d+b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c}\\ &=\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\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 )}{c}-\frac {\left (-c d+b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\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 )}{c}\\ &=\frac {\left (c d-b e-\frac {b c d-b^2 e+2 a 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 )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (c d-b e+\frac {b c d-b^2 e+2 a 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 )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4331\) vs. \(2(324)=648\).
time = 16.03, size = 4331, normalized size = 13.37 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e
)*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[-(Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2]) + x]) + Sqrt[-((b
+ Sqrt[b^2 - 4*a*c])/c)]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*Sqrt[2*d -
((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2] + x] + b*c*Sqrt[(-b + Sqrt[b^2 - 4
*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x]
+ c*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sq
rt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] - b^2*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + 2*a*c*Sqrt[(-b + Sqrt[b^2 - 4*a
*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] -
b*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[-(Sqrt
[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] - b*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt
[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - c*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[
b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]
+ x] + b^2*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt
[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - 2*a*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*
c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] + b*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c
])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] + 2*c*
Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*Sqrt[e]*Sqrt[(2*c*d - b*
e + Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]] - b*c
*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + S
qrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + c*Sqrt[b^2 - 4*a
*c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b
+ Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + b^2*Sqrt[-((b
+ Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 -
4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - 2*a*c*Sqrt[-((b + Sqrt[b^2
- 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*
e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - b*Sqrt[b^2 - 4*a*c]*Sqrt[-((b + Sqrt[
b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d - Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/
c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + b*c*Sqrt[-((b + Sqrt[b^2 - 4*a*c])
/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt
[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - c*Sqrt[b^2 - 4*a*c]*Sqrt[-((b + Sqrt[b^2 - 4*a*
c])/c)]*d*Sqrt[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + S
qrt[(4*c*d - 2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] - b^2*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqr
t[2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d -
2*b*e + 2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + 2*a*c*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d - ((b
+ Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e + 2*S
qrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + b*Sqrt[b^2 - 4*a*c]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*Sqrt[2*d -
((b + Sqrt[b^2 - 4*a*c])*e)/c]*Log[2*d + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*x + Sqrt[(4*c*d - 2*b*e +
2*Sqrt[b^2 - 4*a*c]*e)/c]*Sqrt[d + e*x^2]] + b*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[b
^2 - 4*a*c]*e)/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a
*c])*e)/c]*Sqrt[d + e*x^2]] + c*Sqrt[b^2 - 4*a*c]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*d*Sqrt[(2*c*d - b*e + Sqrt[
b^2 - 4*a*c]*e)/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*
a*c])*e)/c]*Sqrt[d + e*x^2]] - b^2*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)
/c]*Log[2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x + Sqrt[4*d - (2*(b + Sqrt[b^2 - 4*a*c])*e)/c]*Sqr
t[d + e*x^2]] + 2*a*c*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*e*Sqrt[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/c]*Log[2*d -
Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*x...

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.13, size = 222, normalized size = 0.69

method	result	size
default	\(-\sqrt {e}\, \left (-\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\left (e b -c d \right ) \textit {\_R}^{2}+2 \left (2 a \,e^{2}-d e b +c \,d^{2}\right ) \textit {\_R} +d^{2} e b -c \,d^{3}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}}{2 c}+\frac {\ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )}{c}\right )\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1663 vs. \(2 (289) = 578\).
time = 2.66, size = 1663, normalized size = 5.13 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} c \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (\frac {b c d^{2} x^{2} + 4 \, a b x^{2} e^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{2} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}} - 2 \, a c d^{2} + 2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} x e + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {x^{2} e + d} \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} - {\left ({\left (b^{2} + 4 \, a c\right )} d x^{2} - 2 \, a b d\right )} e}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (\frac {b c d^{2} x^{2} + 4 \, a b x^{2} e^{2} + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{2} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}} - 2 \, a c d^{2} - 2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} x e + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {x^{2} e + d} \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} - {\left ({\left (b^{2} + 4 \, a c\right )} d x^{2} - 2 \, a b d\right )} e}{x^{2}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (\frac {b c d^{2} x^{2} + 4 \, a b x^{2} e^{2} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{2} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}} - 2 \, a c d^{2} + 2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} x e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {x^{2} e + d} \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} - {\left ({\left (b^{2} + 4 \, a c\right )} d x^{2} - 2 \, a b d\right )} e}{x^{2}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} \log \left (\frac {b c d^{2} x^{2} + 4 \, a b x^{2} e^{2} - {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{2} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}} - 2 \, a c d^{2} - 2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (b^{3} - 4 \, a b c\right )} x e - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}\right )} \sqrt {x^{2} e + d} \sqrt {-\frac {b c d - {\left (b^{2} - 2 \, a c\right )} e + {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {\frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}}{b^{2} c^{4} - 4 \, a c^{5}}}}{b^{2} c^{2} - 4 \, a c^{3}}} - {\left ({\left (b^{2} + 4 \, a c\right )} d x^{2} - 2 \, a b d\right )} e}{x^{2}}\right ) - 2 \, e^{\frac {1}{2}} \log \left (-2 \, x^{2} e - 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - d\right )}{4 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.85, size = 27, normalized size = 0.08 \begin {gather*} -\frac {e^{\frac {1}{2}} \log \left ({\left (x e^{\frac {1}{2}} - \sqrt {x^{2} e + d}\right )}^{2}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(1/2)*log((x*e^(1/2) - sqrt(x^2*e + d))^2)/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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