∫ x2 _________________________________________√d + ex2 (a+bx2+cx4) dx [389]

3.4.89 $\int \frac {x^2}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx$ [389]

Optimal. Leaf size=240 \[ -\frac {\sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-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-(-4*a*c+b^2)

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

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

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Rubi [A]
time = 0.21, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 29, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.103, Rules used = {1317, 385, 211} \begin {gather*} \frac {\sqrt {\sqrt {b^2-4 a c}+b} \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 )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \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 )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

- 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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 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 1317

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Exp

andIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2

- 4*a*c, 0] && !IntegerQ[q] && IntegerQ[m]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 5.21, size = 250, normalized size = 1.04 \begin {gather*} \frac {1}{2} \sqrt {e} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}^2+4 b d^2 e \text {$\#$1}^2+6 c d^2 \text {$\#$1}^4-8 b d e \text {$\#$1}^4+16 a e^2 \text {$\#$1}^4-4 c d \text {$\#$1}^6+4 b e \text {$\#$1}^6+c \text {$\#$1}^8\&,\frac {d^2 \log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right )-2 d \log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-\sqrt {e} x+\sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^4}{c d^3-b d^2 e-3 c d^2 \text {$\#$1}^2+4 b d e \text {$\#$1}^2-8 a e^2 \text {$\#$1}^2+3 c d \text {$\#$1}^4-3 b e \text {$\#$1}^4-c \text {$\#$1}^6}\&\right ] \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[e]*RootSum[c*d^4 - 4*c*d^3*#1^2 + 4*b*d^2*e*#1^2 + 6*c*d^2*#1^4 - 8*b*d*e*#1^4 + 16*a*e^2*#1^4 - 4*c*d*#

1^6 + 4*b*e*#1^6 + c*#1^8 & , (d^2*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2] - #1] - 2*d*Log[-(Sqrt[e]*x) + Sqrt[d +

e*x^2] - #1]*#1^2 + Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2] - #1]*#1^4)/(c*d^3 - b*d^2*e - 3*c*d^2*#1^2 + 4*b*d*e*#

1^2 - 8*a*e^2*#1^2 + 3*c*d*#1^4 - 3*b*e*#1^4 - c*#1^6) & ])/2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 161, normalized size = 0.67


method	result	size

default	$-\frac {\sqrt {e}\, \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\left (4 e b -4 c d \right ) \textit {\_Z}^{3}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z} +d^{4} c \right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} d +d^{2}\right ) \ln \left (\left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x \right )^{2}-\textit {\_R} \right )}{c \,\textit {\_R}^{3}+3 \textit {\_R}^{2} b e -3 \textit {\_R}^{2} c d +8 \textit {\_R} a \,e^{2}-4 \textit {\_R} b d e +3 c \,d^{2} \textit {\_R} +d^{2} e b -c \,d^{3}}\right )}{2}$	$161$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3417 vs. $2 (206) = 412$.
time = 2.78, size = 3417, normalized size = 14.24 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

3*x - (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*x*e + 3*(a*b^3 - 4*a^2*b*c)*d*x*e^2 - 2*(a^2*b^2 - 4*a^3*c)*x*e^3)*sqr

t(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^

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

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

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

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

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

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

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

^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c

)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*lo

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

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

c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*

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

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

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

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

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

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

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

)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a

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

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

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

3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (

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

(a*b^2 - 4*a^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*

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

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

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

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

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

^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a

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

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

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

b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*

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

^2*c)*e^2)*sqrt(d^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2

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

(a*b^2 - 4*a^2*c)*e^2)) + ((b^2*c - 4*a*c^2)*d...

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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}} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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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}\,\left (c\,x^4+b\,x^2+a\right )} \,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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3.4.89 \(\int \frac {x^2}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [389]