3.244 \(\int x^3 \log (\frac{7+5 x}{x^2}) \, dx\)

Optimal. Leaf size=54 \[ \frac{x^4}{16}+\frac{7 x^3}{60}-\frac{49 x^2}{200}+\frac{1}{4} x^4 \log \left (\frac{5 x+7}{x^2}\right )+\frac{343 x}{500}-\frac{2401 \log (5 x+7)}{2500} \]

[Out]

(343*x)/500 - (49*x^2)/200 + (7*x^3)/60 + x^4/16 - (2401*Log[7 + 5*x])/2500 + (x^4*Log[(7 + 5*x)/x^2])/4

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Rubi [A]  time = 0.0392872, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2495, 30, 43} \[ \frac{x^4}{16}+\frac{7 x^3}{60}-\frac{49 x^2}{200}+\frac{1}{4} x^4 \log \left (\frac{5 x+7}{x^2}\right )+\frac{343 x}{500}-\frac{2401 \log (5 x+7)}{2500} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[(7 + 5*x)/x^2],x]

[Out]

(343*x)/500 - (49*x^2)/200 + (7*x^3)/60 + x^4/16 - (2401*Log[7 + 5*x])/2500 + (x^4*Log[(7 + 5*x)/x^2])/4

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
 x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 x^3 \log \left (\frac{7+5 x}{x^2}\right ) \, dx &=\frac{1}{4} x^4 \log \left (\frac{7+5 x}{x^2}\right )+\frac{\int x^3 \, dx}{2}-\frac{5}{4} \int \frac{x^4}{7+5 x} \, dx\\ &=\frac{x^4}{8}+\frac{1}{4} x^4 \log \left (\frac{7+5 x}{x^2}\right )-\frac{5}{4} \int \left (-\frac{343}{625}+\frac{49 x}{125}-\frac{7 x^2}{25}+\frac{x^3}{5}+\frac{2401}{625 (7+5 x)}\right ) \, dx\\ &=\frac{343 x}{500}-\frac{49 x^2}{200}+\frac{7 x^3}{60}+\frac{x^4}{16}-\frac{2401 \log (7+5 x)}{2500}+\frac{1}{4} x^4 \log \left (\frac{7+5 x}{x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0127958, size = 54, normalized size = 1. \[ \frac{x^4}{16}+\frac{7 x^3}{60}-\frac{49 x^2}{200}+\frac{1}{4} x^4 \log \left (\frac{5 x+7}{x^2}\right )+\frac{343 x}{500}-\frac{2401 \log (5 x+7)}{2500} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Log[(7 + 5*x)/x^2],x]

[Out]

(343*x)/500 - (49*x^2)/200 + (7*x^3)/60 + x^4/16 - (2401*Log[7 + 5*x])/2500 + (x^4*Log[(7 + 5*x)/x^2])/4

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Maple [A]  time = 0.016, size = 53, normalized size = 1. \begin{align*}{\frac{{x}^{4}}{4}\ln \left ({\frac{1}{x} \left ( 7\,{x}^{-1}+5 \right ) } \right ) }-{\frac{2401}{2500}\ln \left ( 7\,{x}^{-1}+5 \right ) }+{\frac{{x}^{4}}{16}}+{\frac{7\,{x}^{3}}{60}}-{\frac{49\,{x}^{2}}{200}}+{\frac{343\,x}{500}}+{\frac{2401\,\ln \left ({x}^{-1} \right ) }{2500}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln((7+5*x)/x^2),x)

[Out]

1/4*x^4*ln((7/x+5)/x)-2401/2500*ln(7/x+5)+1/16*x^4+7/60*x^3-49/200*x^2+343/500*x+2401/2500*ln(1/x)

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Maxima [A]  time = 1.05635, size = 57, normalized size = 1.06 \begin{align*} \frac{1}{4} \, x^{4} \log \left (\frac{5 \, x + 7}{x^{2}}\right ) + \frac{1}{16} \, x^{4} + \frac{7}{60} \, x^{3} - \frac{49}{200} \, x^{2} + \frac{343}{500} \, x - \frac{2401}{2500} \, \log \left (5 \, x + 7\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log((7+5*x)/x^2),x, algorithm="maxima")

[Out]

1/4*x^4*log((5*x + 7)/x^2) + 1/16*x^4 + 7/60*x^3 - 49/200*x^2 + 343/500*x - 2401/2500*log(5*x + 7)

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Fricas [A]  time = 1.86796, size = 135, normalized size = 2.5 \begin{align*} \frac{1}{4} \, x^{4} \log \left (\frac{5 \, x + 7}{x^{2}}\right ) + \frac{1}{16} \, x^{4} + \frac{7}{60} \, x^{3} - \frac{49}{200} \, x^{2} + \frac{343}{500} \, x - \frac{2401}{2500} \, \log \left (5 \, x + 7\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log((7+5*x)/x^2),x, algorithm="fricas")

[Out]

1/4*x^4*log((5*x + 7)/x^2) + 1/16*x^4 + 7/60*x^3 - 49/200*x^2 + 343/500*x - 2401/2500*log(5*x + 7)

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Sympy [A]  time = 0.149822, size = 48, normalized size = 0.89 \begin{align*} \frac{x^{4} \log{\left (\frac{5 x + 7}{x^{2}} \right )}}{4} + \frac{x^{4}}{16} + \frac{7 x^{3}}{60} - \frac{49 x^{2}}{200} + \frac{343 x}{500} - \frac{2401 \log{\left (5 x + 7 \right )}}{2500} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*ln((7+5*x)/x**2),x)

[Out]

x**4*log((5*x + 7)/x**2)/4 + x**4/16 + 7*x**3/60 - 49*x**2/200 + 343*x/500 - 2401*log(5*x + 7)/2500

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Giac [A]  time = 1.27232, size = 58, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} \log \left (\frac{5 \, x + 7}{x^{2}}\right ) + \frac{1}{16} \, x^{4} + \frac{7}{60} \, x^{3} - \frac{49}{200} \, x^{2} + \frac{343}{500} \, x - \frac{2401}{2500} \, \log \left ({\left | 5 \, x + 7 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log((7+5*x)/x^2),x, algorithm="giac")

[Out]

1/4*x^4*log((5*x + 7)/x^2) + 1/16*x^4 + 7/60*x^3 - 49/200*x^2 + 343/500*x - 2401/2500*log(abs(5*x + 7))