3.278 \(\int \frac{4+3 x^4}{5 x+2 x^5} \, dx\)

Optimal. Leaf size=19 \[ \frac{7}{40} \log \left (2 x^4+5\right )+\frac{4 \log (x)}{5} \]

[Out]

(4*Log[x])/5 + (7*Log[5 + 2*x^4])/40

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Rubi [A]  time = 0.0257862, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1593, 446, 72} \[ \frac{7}{40} \log \left (2 x^4+5\right )+\frac{4 \log (x)}{5} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 3*x^4)/(5*x + 2*x^5),x]

[Out]

(4*Log[x])/5 + (7*Log[5 + 2*x^4])/40

Rule 1593

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

Rubi steps

\begin{align*} \int \frac{4+3 x^4}{5 x+2 x^5} \, dx &=\int \frac{4+3 x^4}{x \left (5+2 x^4\right )} \, dx\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{4+3 x}{x (5+2 x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{4}{5 x}+\frac{7}{5 (5+2 x)}\right ) \, dx,x,x^4\right )\\ &=\frac{4 \log (x)}{5}+\frac{7}{40} \log \left (5+2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0052056, size = 19, normalized size = 1. \[ \frac{7}{40} \log \left (2 x^4+5\right )+\frac{4 \log (x)}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 3*x^4)/(5*x + 2*x^5),x]

[Out]

(4*Log[x])/5 + (7*Log[5 + 2*x^4])/40

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Maple [A]  time = 0.049, size = 16, normalized size = 0.8 \begin{align*}{\frac{4\,\ln \left ( x \right ) }{5}}+{\frac{7\,\ln \left ( 2\,{x}^{4}+5 \right ) }{40}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4+4)/(2*x^5+5*x),x)

[Out]

4/5*ln(x)+7/40*ln(2*x^4+5)

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Maxima [A]  time = 1.69763, size = 20, normalized size = 1.05 \begin{align*} \frac{7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac{4}{5} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+4)/(2*x^5+5*x),x, algorithm="maxima")

[Out]

7/40*log(2*x^4 + 5) + 4/5*log(x)

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Fricas [A]  time = 2.01541, size = 46, normalized size = 2.42 \begin{align*} \frac{7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac{4}{5} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+4)/(2*x^5+5*x),x, algorithm="fricas")

[Out]

7/40*log(2*x^4 + 5) + 4/5*log(x)

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Sympy [A]  time = 0.099904, size = 17, normalized size = 0.89 \begin{align*} \frac{4 \log{\left (x \right )}}{5} + \frac{7 \log{\left (2 x^{4} + 5 \right )}}{40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**4+4)/(2*x**5+5*x),x)

[Out]

4*log(x)/5 + 7*log(2*x**4 + 5)/40

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Giac [A]  time = 1.20917, size = 23, normalized size = 1.21 \begin{align*} \frac{7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac{1}{5} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^4+4)/(2*x^5+5*x),x, algorithm="giac")

[Out]

7/40*log(2*x^4 + 5) + 1/5*log(x^4)