3.7.96 \(\int \frac {x^4}{a-b x^2+c x^4} \, dx\)

Optimal. Leaf size=179 \[ -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x}{c} \]

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Rubi [A]  time = 0.36, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1122, 1166, 208} \begin {gather*} -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a - b*x^2 + c*x^4),x]

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2
]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sq
rt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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]

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 208, normalized size = 1.16 \begin {gather*} \frac {\left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}-b}}+\frac {\left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}-b}}+\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a - b*x^2 + c*x^4),x]

[Out]

x/c + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*
c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b - Sqrt[b^2 - 4*a*c]]) + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2
]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b + Sqrt[b^2 - 4*a*c]])

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4/(a - b*x^2 + c*x^4),x]

[Out]

IntegrateAlgebraic[x^4/(a - b*x^2 + c*x^4), x]

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fricas [B]  time = 1.16, size = 1051, normalized size = 5.87 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} c \sqrt {\frac {b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x + \sqrt {\frac {1}{2}} {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2} - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {\frac {b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x - \sqrt {\frac {1}{2}} {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2} - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {\frac {b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x + \sqrt {\frac {1}{2}} {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {\frac {b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x - \sqrt {\frac {1}{2}} {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - 2 \, x}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(1/2)*c*sqrt((b^3 - 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^
7)))/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x + sqrt(1/2)*(b^4 - 5*a*b^2*c + 4*a^2*c^2 - (b^3*c^3 - 4*a*b
*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*sqrt((b^3 - 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b
^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))) - sqrt(1/2)*c*sqrt((b^3 - 3*a*b*c + (b^2
*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^
2*c)*x - sqrt(1/2)*(b^4 - 5*a*b^2*c + 4*a^2*c^2 - (b^3*c^3 - 4*a*b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*
c^6 - 4*a*c^7)))*sqrt((b^3 - 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7
)))/(b^2*c^3 - 4*a*c^4))) + sqrt(1/2)*c*sqrt((b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*
c^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x + sqrt(1/2)*(b^4 - 5*a*b^2*c + 4*a^2*
c^2 + (b^3*c^3 - 4*a*b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*sqrt((b^3 - 3*a*b*c - (b^2*
c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))) - sqrt(1/2)*c*sqrt
((b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^
4))*log(-2*(a*b^2 - a^2*c)*x - sqrt(1/2)*(b^4 - 5*a*b^2*c + 4*a^2*c^2 + (b^3*c^3 - 4*a*b*c^4)*sqrt((b^4 - 2*a*
b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*sqrt((b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c
^2)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))) - 2*x)/c

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giac [B]  time = 0.98, size = 2153, normalized size = 12.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c + 1/8*(2*b^5*c^4 - 12*a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c
)*b^5*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^2
*b*c^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(-b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^5
- 2*(b^2 - 4*a*c)*b^3*c^4 + 4*(b^2 - 4*a*c)*a*b*c^5 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*
c)*a*b^3*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b
^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt
(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*c^2 - 2*(sqrt(2)*sqr
t(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 2*sqrt(2)*s
qrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4
- 8*sqrt(2)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + sqrt(2)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 +
16*a^2*b^2*c^4 - 4*sqrt(2)*sqrt(-b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 32*a^3*c^5 + 2*(b^2 - 4*a*c)*a*b^2*c^3 -
8*(b^2 - 4*a*c)*a^2*c^4)*abs(c))*arctan(2*sqrt(1/2)*x/sqrt(-(b*c + sqrt(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3
- 8*a^2*b^2*c^4 + 2*a*b^3*c^4 + 16*a^3*c^5 - 8*a^2*b*c^5 + a*b^2*c^5 - 4*a^2*c^6)*c^2) - 1/8*(2*b^5*c^4 - 12*
a*b^3*c^5 + 16*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 6*sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 -
4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + 4*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c
)*c)*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4
+ 4*(b^2 - 4*a*c)*a*b*c^5 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - s
qrt(b^2 - 4*a*c)*c)*b^5 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 2*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*
a*c)*c)*a^2*b*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*
c)*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*c^2 + 2*(sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)
*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 2*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*
c)*c)*a*b^3*c^3 + 2*a*b^4*c^3 + 16*sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^4 - 8*sqrt(2)*sqrt(-b*c - sq
rt(b^2 - 4*a*c)*c)*a^2*b*c^4 + sqrt(2)*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 16*a^2*b^2*c^4 - 4*sqrt(2)
*sqrt(-b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^5 + 32*a^3*c^5 - 2*(b^2 - 4*a*c)*a*b^2*c^3 + 8*(b^2 - 4*a*c)*a^2*c^4)*
abs(c))*arctan(2*sqrt(1/2)*x/sqrt(-(b*c - sqrt(b^2*c^2 - 4*a*c^3))/c^2))/((a*b^4*c^3 - 8*a^2*b^2*c^4 + 2*a*b^3
*c^4 + 16*a^3*c^5 - 8*a^2*b*c^5 + a*b^2*c^5 - 4*a^2*c^6)*c^2)

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maple [B]  time = 0.03, size = 343, normalized size = 1.92 \begin {gather*} \frac {\sqrt {2}\, a \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {2}\, b^{2} \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}-\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}-\frac {\sqrt {2}\, b \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}+\frac {\sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}\, c}+\frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(c*x^4-b*x^2+a),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x}{c} + \frac {-\frac {{\left (\sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c + 2 \, \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c + 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} - 8 \, \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} - 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} + 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} a b c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c - \sqrt {b^{2} - 4 \, a c} c} b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{4 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + b^{2} c^{2} - 4 \, a c^{3}\right )} c} - \frac {{\left (\sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c + 2 \, \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} - 8 \, \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} + 16 \, a b^{2} c^{2} + 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 32 \, a^{2} c^{3} - 8 \, a b c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {-b c + \sqrt {b^{2} - 4 \, a c} c} b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{4 \, {\left (b^{4} - 8 \, a b^{2} c + 2 \, b^{3} c + 16 \, a^{2} c^{2} - 8 \, a b c^{2} + b^{2} c^{2} - 4 \, a c^{3}\right )} c}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c + integrate((b*x^2 - a)/(c*x^4 - b*x^2 + a), x)/c

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mupad [B]  time = 0.67, size = 3000, normalized size = 16.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a - b*x^2 + c*x^4),x)

[Out]

x/c + atan(((((16*a^2*c^3 - 4*a*b^2*c^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2
) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/
c)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*
c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(
1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2
)*1i - (((16*a^2*c^3 - 4*a*b^2*c^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 1
2*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*((
b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 +
b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)
/((((16*a^2*c^3 - 4*a*b^2*c^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2
*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*((b^5 +
b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*
c^3 - 8*a*b^2*c^4)))^(1/2) + (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (((16*
a^2*c^3 - 4*a*b^2*c^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 -
7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*((b^5 + b^2*(-(
4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*
a*b^2*c^4)))^(1/2) - (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^
2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*a^2*b)/c))*
((b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5
+ b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i + atan(((((16*a^2*c^3 - 4*a*b^2*c^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((
b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 +
b^4*c^3 - 8*a*b^2*c^4)))^(1/2))/c)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4
*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)
*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^
5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((16*a^2*c^3 - 4*a*b^2*c^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 -
b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*
c^3 - 8*a*b^2*c^4)))^(1/2))/c)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c
- b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^
5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b
^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((16*a^2*c^3 - 4*a*b^2*c^2)/c - (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 - b^2*
(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 -
8*a*b^2*c^4)))^(1/2))/c)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2
)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 - b
^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^
3 - 8*a*b^2*c^4)))^(1/2) + (((16*a^2*c^3 - 4*a*b^2*c^2)/c + (2*x*(4*b^3*c^3 - 16*a*b*c^4)*((b^5 - b^2*(-(4*a*c
- b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2
*c^4)))^(1/2))/c)*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/
2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*x*(b^4 + 2*a^2*c^2 - 4*a*b^2*c))/c)*((b^5 - b^2*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*
b^2*c^4)))^(1/2) - (2*a^2*b)/c))*((b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*
c - b^2)^3)^(1/2))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i

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sympy [A]  time = 2.78, size = 129, normalized size = 0.72 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} c^{5} - 128 a b^{2} c^{4} + 16 b^{4} c^{3}\right ) + t^{2} \left (- 48 a^{2} b c^{2} + 28 a b^{3} c - 4 b^{5}\right ) + a^{3}, \left (t \mapsto t \log {\left (x + \frac {- 32 t^{3} a b c^{4} + 8 t^{3} b^{3} c^{3} - 4 t a^{2} c^{2} + 8 t a b^{2} c - 2 t b^{4}}{a^{2} c - a b^{2}} \right )} \right )\right )} + \frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(c*x**4-b*x**2+a),x)

[Out]

RootSum(_t**4*(256*a**2*c**5 - 128*a*b**2*c**4 + 16*b**4*c**3) + _t**2*(-48*a**2*b*c**2 + 28*a*b**3*c - 4*b**5
) + a**3, Lambda(_t, _t*log(x + (-32*_t**3*a*b*c**4 + 8*_t**3*b**3*c**3 - 4*_t*a**2*c**2 + 8*_t*a*b**2*c - 2*_
t*b**4)/(a**2*c - a*b**2)))) + x/c

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