### 3.106 $$\int \frac{x (b+2 c x^2)}{a+b x^2+c x^4} \, dx$$

Optimal. Leaf size=17 $\frac{1}{2} \log \left (a+b x^2+c x^4\right )$

[Out]

Log[a + b*x^2 + c*x^4]/2

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Rubi [A]  time = 0.0186135, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, 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 = 0.0060359, size = 17, 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., size = 16, normalized size = 0.9 \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 = 1.13879, size = 20, normalized size = 1.18 \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 = 0.995402, size = 38, normalized size = 2.24 \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 = 0.417882, size = 14, normalized size = 0.82 \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 = 1.13661, size = 20, normalized size = 1.18 \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)