### 3.114 $$\int \frac{x (b+2 c x^2)}{-a+b x^2+c x^4} \, dx$$

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

[Out]

Log[a - b*x^2 - c*x^4]/2

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Rubi [A]  time = 0.0193331, antiderivative size = 19, normalized size of antiderivative = 1., 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} $\frac{1}{2} \log \left (a-b x^2-c x^4\right )$

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 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]

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]

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 = 0.006661, size = 19, normalized size = 1. $\frac{1}{2} \log \left (-a+b x^2+c x^4\right )$

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

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Maple [A]  time = 0.003, size = 18, normalized size = 1. \begin{align*}{\frac{\ln \left ( c{x}^{4}+b{x}^{2}-a \right ) }{2}} \end{align*}

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)

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Maxima [A]  time = 1.01428, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}

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)

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Fricas [A]  time = 1.03838, size = 38, normalized size = 2. \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}

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)

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Sympy [A]  time = 0.418333, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (- a + b x^{2} + c x^{4} \right )}}{2} \end{align*}

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

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Giac [A]  time = 1.158, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}

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(c*x^4 + b*x^2 - a)