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

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

________________________________________________________________________________________

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 {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} \end {gather*}

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 = 0.01, size = 42, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.27, size = 37, normalized size = 0.73 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 1.08, size = 38, normalized size = 0.75 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 37, normalized size = 0.73 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 35, normalized size = 0.69 \begin {gather*} \frac {41223\,x}{15625}+\frac {1331\,\ln \left (x+\frac {3}{5}\right )}{78125}-\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 48, normalized size = 0.94 \begin {gather*} - \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 {gather*}

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

________________________________________________________________________________________