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

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

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2 (2+3 x)^4}{3+5 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.28, size = 37, normalized size = 0.73 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.93, size = 38, normalized size = 0.75 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.00, size = 38, normalized size = 0.75 \begin {gather*} \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 \ln \left (5 x +3\right )}{78125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.49, size = 37, normalized size = 0.73 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 35, normalized size = 0.69 \begin {gather*} \frac {83293\,x}{15625}+\frac {121\,\ln \left (x+\frac {3}{5}\right )}{78125}+\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(83293*x)/15625 + (121*log(x + 3/5))/78125 + (5569*x^2)/6250 - (7841*x^3)/625 - (3159*x^4)/500 + (1728*x^5)/12
5 + (54*x^6)/5

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sympy [A]  time = 0.10, size = 48, normalized size = 0.94 \begin {gather*} \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 {gather*}

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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