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]

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

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Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {7}{40} \log \left (2 x^4+5\right )+\frac {4 \log (x)}{5} \]

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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fricas [A]  time = 0.86, size = 15, normalized size = 0.79 \[ \frac {7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac {4}{5} \, \log \relax (x) \]

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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giac [A]  time = 0.16, size = 17, normalized size = 0.89 \[ \frac {7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac {1}{5} \, \log \left (x^{4}\right ) \]

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)

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maple [A]  time = 0.05, size = 16, normalized size = 0.84 \[ \frac {4 \ln \relax (x )}{5}+\frac {7 \ln \left (2 x^{4}+5\right )}{40} \]

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 = 2.91, size = 15, normalized size = 0.79 \[ \frac {7}{40} \, \log \left (2 \, x^{4} + 5\right ) + \frac {4}{5} \, \log \relax (x) \]

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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mupad [B]  time = 0.17, size = 13, normalized size = 0.68 \[ \frac {7\,\ln \left (x^4+\frac {5}{2}\right )}{40}+\frac {4\,\ln \relax (x)}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^4 + 4)/(5*x + 2*x^5),x)

[Out]

(7*log(x^4 + 5/2))/40 + (4*log(x))/5

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sympy [A]  time = 0.11, size = 17, normalized size = 0.89 \[ \frac {4 \log {\relax (x )}}{5} + \frac {7 \log {\left (2 x^{4} + 5 \right )}}{40} \]

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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