3.1391 \(\int \frac{(1-2 x)^3 (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=51 \[ -\frac{36 x^6}{5}+\frac{108 x^5}{125}+\frac{2313 x^4}{250}-\frac{5003 x^3}{1875}-\frac{26241 x^2}{6250}+\frac{41223 x}{15625}+\frac{1331 \log (5 x+3)}{78125} \]

[Out]

(41223*x)/15625 - (26241*x^2)/6250 - (5003*x^3)/1875 + (2313*x^4)/250 + (108*x^5)/125 - (36*x^6)/5 + (1331*Log
[3 + 5*x])/78125

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Rubi [A]  time = 0.0225145, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{36 x^6}{5}+\frac{108 x^5}{125}+\frac{2313 x^4}{250}-\frac{5003 x^3}{1875}-\frac{26241 x^2}{6250}+\frac{41223 x}{15625}+\frac{1331 \log (5 x+3)}{78125} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^3)/(3 + 5*x),x]

[Out]

(41223*x)/15625 - (26241*x^2)/6250 - (5003*x^3)/1875 + (2313*x^4)/250 + (108*x^5)/125 - (36*x^6)/5 + (1331*Log
[3 + 5*x])/78125

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^3 (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac{41223}{15625}-\frac{26241 x}{3125}-\frac{5003 x^2}{625}+\frac{4626 x^3}{125}+\frac{108 x^4}{25}-\frac{216 x^5}{5}+\frac{1331}{15625 (3+5 x)}\right ) \, dx\\ &=\frac{41223 x}{15625}-\frac{26241 x^2}{6250}-\frac{5003 x^3}{1875}+\frac{2313 x^4}{250}+\frac{108 x^5}{125}-\frac{36 x^6}{5}+\frac{1331 \log (3+5 x)}{78125}\\ \end{align*}

Mathematica [A]  time = 0.0107995, size = 42, normalized size = 0.82 \[ \frac{-16875000 x^6+2025000 x^5+21684375 x^4-6253750 x^3-9840375 x^2+6183450 x+39930 \log (5 x+3)+4036284}{2343750} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^3)/(3 + 5*x),x]

[Out]

(4036284 + 6183450*x - 9840375*x^2 - 6253750*x^3 + 21684375*x^4 + 2025000*x^5 - 16875000*x^6 + 39930*Log[3 + 5
*x])/2343750

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*}{\frac{41223\,x}{15625}}-{\frac{26241\,{x}^{2}}{6250}}-{\frac{5003\,{x}^{3}}{1875}}+{\frac{2313\,{x}^{4}}{250}}+{\frac{108\,{x}^{5}}{125}}-{\frac{36\,{x}^{6}}{5}}+{\frac{1331\,\ln \left ( 3+5\,x \right ) }{78125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^3/(3+5*x),x)

[Out]

41223/15625*x-26241/6250*x^2-5003/1875*x^3+2313/250*x^4+108/125*x^5-36/5*x^6+1331/78125*ln(3+5*x)

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Maxima [A]  time = 2.31948, size = 50, normalized size = 0.98 \begin{align*} -\frac{36}{5} \, x^{6} + \frac{108}{125} \, x^{5} + \frac{2313}{250} \, x^{4} - \frac{5003}{1875} \, x^{3} - \frac{26241}{6250} \, x^{2} + \frac{41223}{15625} \, x + \frac{1331}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^3/(3+5*x),x, algorithm="maxima")

[Out]

-36/5*x^6 + 108/125*x^5 + 2313/250*x^4 - 5003/1875*x^3 - 26241/6250*x^2 + 41223/15625*x + 1331/78125*log(5*x +
3)

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Fricas [A]  time = 1.48089, size = 155, normalized size = 3.04 \begin{align*} -\frac{36}{5} \, x^{6} + \frac{108}{125} \, x^{5} + \frac{2313}{250} \, x^{4} - \frac{5003}{1875} \, x^{3} - \frac{26241}{6250} \, x^{2} + \frac{41223}{15625} \, x + \frac{1331}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^3/(3+5*x),x, algorithm="fricas")

[Out]

-36/5*x^6 + 108/125*x^5 + 2313/250*x^4 - 5003/1875*x^3 - 26241/6250*x^2 + 41223/15625*x + 1331/78125*log(5*x +
3)

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Sympy [A]  time = 0.092036, size = 48, normalized size = 0.94 \begin{align*} - \frac{36 x^{6}}{5} + \frac{108 x^{5}}{125} + \frac{2313 x^{4}}{250} - \frac{5003 x^{3}}{1875} - \frac{26241 x^{2}}{6250} + \frac{41223 x}{15625} + \frac{1331 \log{\left (5 x + 3 \right )}}{78125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**3/(3+5*x),x)

[Out]

-36*x**6/5 + 108*x**5/125 + 2313*x**4/250 - 5003*x**3/1875 - 26241*x**2/6250 + 41223*x/15625 + 1331*log(5*x +
3)/78125

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Giac [A]  time = 3.14661, size = 51, normalized size = 1. \begin{align*} -\frac{36}{5} \, x^{6} + \frac{108}{125} \, x^{5} + \frac{2313}{250} \, x^{4} - \frac{5003}{1875} \, x^{3} - \frac{26241}{6250} \, x^{2} + \frac{41223}{15625} \, x + \frac{1331}{78125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^3/(3+5*x),x, algorithm="giac")

[Out]

-36/5*x^6 + 108/125*x^5 + 2313/250*x^4 - 5003/1875*x^3 - 26241/6250*x^2 + 41223/15625*x + 1331/78125*log(abs(5
*x + 3))