3.1292 \(\int \frac{(1-2 x)^2 (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=51 \[ \frac{54 x^6}{5}+\frac{1728 x^5}{125}-\frac{3159 x^4}{500}-\frac{7841 x^3}{625}+\frac{5569 x^2}{6250}+\frac{83293 x}{15625}+\frac{121 \log (5 x+3)}{78125} \]

[Out]

(83293*x)/15625 + (5569*x^2)/6250 - (7841*x^3)/625 - (3159*x^4)/500 + (1728*x^5)/125 + (54*x^6)/5 + (121*Log[3
+ 5*x])/78125

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Rubi [A]  time = 0.0218496, 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} \[ \frac{54 x^6}{5}+\frac{1728 x^5}{125}-\frac{3159 x^4}{500}-\frac{7841 x^3}{625}+\frac{5569 x^2}{6250}+\frac{83293 x}{15625}+\frac{121 \log (5 x+3)}{78125} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(83293*x)/15625 + (5569*x^2)/6250 - (7841*x^3)/625 - (3159*x^4)/500 + (1728*x^5)/125 + (54*x^6)/5 + (121*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)^2 (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac{83293}{15625}+\frac{5569 x}{3125}-\frac{23523 x^2}{625}-\frac{3159 x^3}{125}+\frac{1728 x^4}{25}+\frac{324 x^5}{5}+\frac{121}{15625 (3+5 x)}\right ) \, dx\\ &=\frac{83293 x}{15625}+\frac{5569 x^2}{6250}-\frac{7841 x^3}{625}-\frac{3159 x^4}{500}+\frac{1728 x^5}{125}+\frac{54 x^6}{5}+\frac{121 \log (3+5 x)}{78125}\\ \end{align*}

Mathematica [A]  time = 0.0120571, size = 42, normalized size = 0.82 \[ \frac{16875000 x^6+21600000 x^5-9871875 x^4-19602500 x^3+1392250 x^2+8329300 x+2420 \log (5 x+3)+2433921}{1562500} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(2433921 + 8329300*x + 1392250*x^2 - 19602500*x^3 - 9871875*x^4 + 21600000*x^5 + 16875000*x^6 + 2420*Log[3 + 5
*x])/1562500

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Maple [A]  time = 0.002, size = 38, normalized size = 0.8 \begin{align*}{\frac{83293\,x}{15625}}+{\frac{5569\,{x}^{2}}{6250}}-{\frac{7841\,{x}^{3}}{625}}-{\frac{3159\,{x}^{4}}{500}}+{\frac{1728\,{x}^{5}}{125}}+{\frac{54\,{x}^{6}}{5}}+{\frac{121\,\ln \left ( 3+5\,x \right ) }{78125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(2+3*x)^4/(3+5*x),x)

[Out]

83293/15625*x+5569/6250*x^2-7841/625*x^3-3159/500*x^4+1728/125*x^5+54/5*x^6+121/78125*ln(3+5*x)

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Maxima [A]  time = 1.00674, size = 50, normalized size = 0.98 \begin{align*} \frac{54}{5} \, x^{6} + \frac{1728}{125} \, x^{5} - \frac{3159}{500} \, x^{4} - \frac{7841}{625} \, x^{3} + \frac{5569}{6250} \, x^{2} + \frac{83293}{15625} \, x + \frac{121}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

54/5*x^6 + 1728/125*x^5 - 3159/500*x^4 - 7841/625*x^3 + 5569/6250*x^2 + 83293/15625*x + 121/78125*log(5*x + 3)

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Fricas [A]  time = 1.42184, size = 151, normalized size = 2.96 \begin{align*} \frac{54}{5} \, x^{6} + \frac{1728}{125} \, x^{5} - \frac{3159}{500} \, x^{4} - \frac{7841}{625} \, x^{3} + \frac{5569}{6250} \, x^{2} + \frac{83293}{15625} \, x + \frac{121}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

54/5*x^6 + 1728/125*x^5 - 3159/500*x^4 - 7841/625*x^3 + 5569/6250*x^2 + 83293/15625*x + 121/78125*log(5*x + 3)

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Sympy [A]  time = 0.090107, size = 48, normalized size = 0.94 \begin{align*} \frac{54 x^{6}}{5} + \frac{1728 x^{5}}{125} - \frac{3159 x^{4}}{500} - \frac{7841 x^{3}}{625} + \frac{5569 x^{2}}{6250} + \frac{83293 x}{15625} + \frac{121 \log{\left (5 x + 3 \right )}}{78125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(2+3*x)**4/(3+5*x),x)

[Out]

54*x**6/5 + 1728*x**5/125 - 3159*x**4/500 - 7841*x**3/625 + 5569*x**2/6250 + 83293*x/15625 + 121*log(5*x + 3)/
78125

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Giac [A]  time = 1.96071, size = 51, normalized size = 1. \begin{align*} \frac{54}{5} \, x^{6} + \frac{1728}{125} \, x^{5} - \frac{3159}{500} \, x^{4} - \frac{7841}{625} \, x^{3} + \frac{5569}{6250} \, x^{2} + \frac{83293}{15625} \, x + \frac{121}{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)^2*(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

54/5*x^6 + 1728/125*x^5 - 3159/500*x^4 - 7841/625*x^3 + 5569/6250*x^2 + 83293/15625*x + 121/78125*log(abs(5*x
+ 3))