3.688 \(\int \frac{x^7}{2+3 x^4} \, dx\)

Optimal. Leaf size=20 \[ \frac{x^4}{12}-\frac{1}{18} \log \left (3 x^4+2\right ) \]

[Out]

x^4/12 - Log[2 + 3*x^4]/18

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Rubi [A]  time = 0.0127968, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{x^4}{12}-\frac{1}{18} \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(2 + 3*x^4),x]

[Out]

x^4/12 - Log[2 + 3*x^4]/18

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^7}{2+3 x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{2+3 x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{1}{3}-\frac{2}{3 (2+3 x)}\right ) \, dx,x,x^4\right )\\ &=\frac{x^4}{12}-\frac{1}{18} \log \left (2+3 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0049126, size = 21, normalized size = 1.05 \[ \frac{1}{36} \left (3 x^4-2 \log \left (3 x^4+2\right )+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(2 + 3*x^4),x]

[Out]

(2 + 3*x^4 - 2*Log[2 + 3*x^4])/36

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Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{12}}-{\frac{\ln \left ( 3\,{x}^{4}+2 \right ) }{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(3*x^4+2),x)

[Out]

1/12*x^4-1/18*ln(3*x^4+2)

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Maxima [A]  time = 0.989227, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{12} \, x^{4} - \frac{1}{18} \, \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(3*x^4+2),x, algorithm="maxima")

[Out]

1/12*x^4 - 1/18*log(3*x^4 + 2)

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Fricas [A]  time = 1.63523, size = 43, normalized size = 2.15 \begin{align*} \frac{1}{12} \, x^{4} - \frac{1}{18} \, \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(3*x^4+2),x, algorithm="fricas")

[Out]

1/12*x^4 - 1/18*log(3*x^4 + 2)

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Sympy [A]  time = 0.10083, size = 14, normalized size = 0.7 \begin{align*} \frac{x^{4}}{12} - \frac{\log{\left (3 x^{4} + 2 \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(3*x**4+2),x)

[Out]

x**4/12 - log(3*x**4 + 2)/18

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Giac [A]  time = 1.11947, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{12} \, x^{4} - \frac{1}{18} \, \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(3*x^4+2),x, algorithm="giac")

[Out]

1/12*x^4 - 1/18*log(3*x^4 + 2)