### 3.107 $$\int \frac{x^2 (b+2 c x^3)}{a+b x^3+c x^6} \, dx$$

Optimal. Leaf size=17 $\frac{1}{3} \log \left (a+b x^3+c x^6\right )$

[Out]

Log[a + b*x^3 + c*x^6]/3

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Rubi [A]  time = 0.0237446, antiderivative size = 17, 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 = {1468, 628} $\frac{1}{3} \log \left (a+b x^3+c x^6\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

Log[a + b*x^3 + c*x^6]/3

Rule 1468

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

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^2 \left (b+2 c x^3\right )}{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \log \left (a+b x^3+c x^6\right )\\ \end{align*}

Mathematica [A]  time = 0.0067678, size = 17, normalized size = 1. $\frac{1}{3} \log \left (a+b x^3+c x^6\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x^2*(b + 2*c*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

Log[a + b*x^3 + c*x^6]/3

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a),x)

[Out]

1/3*ln(c*x^6+b*x^3+a)

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Maxima [A]  time = 1.02964, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/3*log(c*x^6 + b*x^3 + a)

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Fricas [A]  time = 0.98849, size = 38, normalized size = 2.24 \begin{align*} \frac{1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/3*log(c*x^6 + b*x^3 + a)

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Sympy [A]  time = 0.529788, size = 14, normalized size = 0.82 \begin{align*} \frac{\log{\left (a + b x^{3} + c x^{6} \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(2*c*x**3+b)/(c*x**6+b*x**3+a),x)

[Out]

log(a + b*x**3 + c*x**6)/3

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Giac [A]  time = 1.36809, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{3} \, \log \left (c x^{6} + b x^{3} + a\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*c*x^3+b)/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/3*log(c*x^6 + b*x^3 + a)