3.2.1 \(\int \frac {x^2 (b+2 c x^3)}{b x^3+c x^6} \, dx\)

Optimal. Leaf size=16 \[ \frac {1}{3} \log \left (b x^3+c x^6\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1584, 446, 72} \begin {gather*} \frac {1}{3} \log \left (b+c x^3\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[b + c*x^3]/3

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} \frac {1}{3} \log \left (b+c x^3\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[x] + Log[b + c*x^3]/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (b+2 c x^3\right )}{b x^3+c x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.87, size = 13, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, \log \left (c x^{3} + b\right ) + \log \relax (x) \end {gather*}

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),x, algorithm="fricas")

[Out]

1/3*log(c*x^3 + b) + log(x)

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giac [A]  time = 0.34, size = 15, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, \log \left ({\left | c x^{6} + b x^{3} \right |}\right ) \end {gather*}

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),x, algorithm="giac")

[Out]

1/3*log(abs(c*x^6 + b*x^3))

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maple [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} \ln \relax (x )+\frac {\ln \left (c \,x^{3}+b \right )}{3} \end {gather*}

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),x)

[Out]

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

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maxima [A]  time = 0.43, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{3} \, \log \left (c x^{3} + b\right ) + \frac {1}{3} \, \log \left (x^{3}\right ) \end {gather*}

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),x, algorithm="maxima")

[Out]

1/3*log(c*x^3 + b) + 1/3*log(x^3)

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mupad [B]  time = 1.99, size = 13, normalized size = 0.81 \begin {gather*} \frac {\ln \left (c\,x^3+b\right )}{3}+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b + c*x^3)/3 + log(x)

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sympy [A]  time = 0.20, size = 12, normalized size = 0.75 \begin {gather*} \log {\relax (x )} + \frac {\log {\left (\frac {b}{c} + x^{3} \right )}}{3} \end {gather*}

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),x)

[Out]

log(x) + log(b/c + x**3)/3

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