### 3.115 $$\int \frac{x^2 (b+2 c x^3)}{-a+b x^3+c x^6} \, dx$$

Optimal. Leaf size=19 $\frac{1}{3} \log \left (a-b x^3-c x^6\right )$

[Out]

Log[a - b*x^3 - c*x^6]/3

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Rubi [A]  time = 0.0240185, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, 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 = 0.0064442, size = 19, 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., size = 18, normalized size = 1. \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 = 0.978929, size = 23, normalized size = 1.21 \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 = 1.01625, size = 38, normalized size = 2. \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 = 0.512, size = 14, normalized size = 0.74 \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 = 1.33209, size = 23, normalized size = 1.21 \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)