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

Optimal. Leaf size=19 \[ \frac {1}{2} \log \left (a-b x^2-c x^4\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1247, 628} \begin {gather*} \frac {1}{2} \log \left (a-b x^2-c x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a - b*x^2 - c*x^4]/2

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (-a+b x^2+c x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[-a + b*x^2 + c*x^4]/2

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(c*x^4 + b*x^2 - a)

________________________________________________________________________________________

giac [A]  time = 1.62, size = 18, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, \log \left ({\left | c x^{4} + b x^{2} - a \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(c*x^4 + b*x^2 - a))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 18, normalized size = 0.95 \begin {gather*} \frac {\ln \left (c \,x^{4}+b \,x^{2}-a \right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.44, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(c*x^4 + b*x^2 - a)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 17, normalized size = 0.89 \begin {gather*} \frac {\ln \left (c\,x^4+b\,x^2-a\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*x^2 - a + c*x^4)/2

________________________________________________________________________________________

sympy [A]  time = 0.28, size = 14, normalized size = 0.74 \begin {gather*} \frac {\log {\left (- a + b x^{2} + c x^{4} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-a + b*x**2 + c*x**4)/2

________________________________________________________________________________________