3.13.49 \(\int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx\)

Optimal. Leaf size=72 \[ -\frac {648 x^9}{5}-\frac {13851 x^8}{50}-\frac {40338 x^7}{875}+\frac {331713 x^6}{1250}+\frac {2212083 x^5}{15625}-\frac {5848749 x^4}{62500}-\frac {17453753 x^3}{234375}+\frac {11111259 x^2}{781250}+\frac {41666223 x}{1953125}+\frac {1331 \log (5 x+3)}{9765625} \]

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Rubi [A]  time = 0.03, antiderivative size = 72, 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 {648 x^9}{5}-\frac {13851 x^8}{50}-\frac {40338 x^7}{875}+\frac {331713 x^6}{1250}+\frac {2212083 x^5}{15625}-\frac {5848749 x^4}{62500}-\frac {17453753 x^3}{234375}+\frac {11111259 x^2}{781250}+\frac {41666223 x}{1953125}+\frac {1331 \log (5 x+3)}{9765625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x),x]

[Out]

(41666223*x)/1953125 + (11111259*x^2)/781250 - (17453753*x^3)/234375 - (5848749*x^4)/62500 + (2212083*x^5)/156
25 + (331713*x^6)/1250 - (40338*x^7)/875 - (13851*x^8)/50 - (648*x^9)/5 + (1331*Log[3 + 5*x])/9765625

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)^6}{3+5 x} \, dx &=\int \left (\frac {41666223}{1953125}+\frac {11111259 x}{390625}-\frac {17453753 x^2}{78125}-\frac {5848749 x^3}{15625}+\frac {2212083 x^4}{3125}+\frac {995139 x^5}{625}-\frac {40338 x^6}{125}-\frac {55404 x^7}{25}-\frac {5832 x^8}{5}+\frac {1331}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac {41666223 x}{1953125}+\frac {11111259 x^2}{781250}-\frac {17453753 x^3}{234375}-\frac {5848749 x^4}{62500}+\frac {2212083 x^5}{15625}+\frac {331713 x^6}{1250}-\frac {40338 x^7}{875}-\frac {13851 x^8}{50}-\frac {648 x^9}{5}+\frac {1331 \log (3+5 x)}{9765625}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 57, normalized size = 0.79 \begin {gather*} \frac {-531562500000 x^9-1136214843750 x^8-189084375000 x^7+1088433281250 x^6+580671787500 x^5-383824153125 x^4-305440677500 x^3+58334109750 x^2+87499068300 x+559020 \log (5 x+3)[phone]}{4101562500} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x),x]

[Out]

(18072649071 + 87499068300*x + 58334109750*x^2 - 305440677500*x^3 - 383824153125*x^4 + 580671787500*x^5 + 1088
433281250*x^6 - 189084375000*x^7 - 1136214843750*x^8 - 531562500000*x^9 + 559020*Log[3 + 5*x])/4101562500

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (2+3 x)^6}{3+5 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x),x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^3*(2 + 3*x)^6)/(3 + 5*x), x]

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fricas [A]  time = 1.60, size = 52, normalized size = 0.72 \begin {gather*} -\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \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)^6/(3+5*x),x, algorithm="fricas")

[Out]

-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/15625*x^5 - 5848749/62500*x^4 - 17453753
/234375*x^3 + 11111259/781250*x^2 + 41666223/1953125*x + 1331/9765625*log(5*x + 3)

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giac [A]  time = 0.97, size = 53, normalized size = 0.74 \begin {gather*} -\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \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)^6/(3+5*x),x, algorithm="giac")

[Out]

-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/15625*x^5 - 5848749/62500*x^4 - 17453753
/234375*x^3 + 11111259/781250*x^2 + 41666223/1953125*x + 1331/9765625*log(abs(5*x + 3))

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maple [A]  time = 0.00, size = 53, normalized size = 0.74 \begin {gather*} -\frac {648 x^{9}}{5}-\frac {13851 x^{8}}{50}-\frac {40338 x^{7}}{875}+\frac {331713 x^{6}}{1250}+\frac {2212083 x^{5}}{15625}-\frac {5848749 x^{4}}{62500}-\frac {17453753 x^{3}}{234375}+\frac {11111259 x^{2}}{781250}+\frac {41666223 x}{1953125}+\frac {1331 \ln \left (5 x +3\right )}{9765625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3*x+2)^6/(5*x+3),x)

[Out]

41666223/1953125*x+11111259/781250*x^2-17453753/234375*x^3-5848749/62500*x^4+2212083/15625*x^5+331713/1250*x^6
-40338/875*x^7-13851/50*x^8-648/5*x^9+1331/9765625*ln(5*x+3)

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maxima [A]  time = 0.46, size = 52, normalized size = 0.72 \begin {gather*} -\frac {648}{5} \, x^{9} - \frac {13851}{50} \, x^{8} - \frac {40338}{875} \, x^{7} + \frac {331713}{1250} \, x^{6} + \frac {2212083}{15625} \, x^{5} - \frac {5848749}{62500} \, x^{4} - \frac {17453753}{234375} \, x^{3} + \frac {11111259}{781250} \, x^{2} + \frac {41666223}{1953125} \, x + \frac {1331}{9765625} \, \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)^6/(3+5*x),x, algorithm="maxima")

[Out]

-648/5*x^9 - 13851/50*x^8 - 40338/875*x^7 + 331713/1250*x^6 + 2212083/15625*x^5 - 5848749/62500*x^4 - 17453753
/234375*x^3 + 11111259/781250*x^2 + 41666223/1953125*x + 1331/9765625*log(5*x + 3)

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mupad [B]  time = 0.05, size = 50, normalized size = 0.69 \begin {gather*} \frac {41666223\,x}{1953125}+\frac {1331\,\ln \left (x+\frac {3}{5}\right )}{9765625}+\frac {11111259\,x^2}{781250}-\frac {17453753\,x^3}{234375}-\frac {5848749\,x^4}{62500}+\frac {2212083\,x^5}{15625}+\frac {331713\,x^6}{1250}-\frac {40338\,x^7}{875}-\frac {13851\,x^8}{50}-\frac {648\,x^9}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)^3*(3*x + 2)^6)/(5*x + 3),x)

[Out]

(41666223*x)/1953125 + (1331*log(x + 3/5))/9765625 + (11111259*x^2)/781250 - (17453753*x^3)/234375 - (5848749*
x^4)/62500 + (2212083*x^5)/15625 + (331713*x^6)/1250 - (40338*x^7)/875 - (13851*x^8)/50 - (648*x^9)/5

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sympy [A]  time = 0.12, size = 68, normalized size = 0.94 \begin {gather*} - \frac {648 x^{9}}{5} - \frac {13851 x^{8}}{50} - \frac {40338 x^{7}}{875} + \frac {331713 x^{6}}{1250} + \frac {2212083 x^{5}}{15625} - \frac {5848749 x^{4}}{62500} - \frac {17453753 x^{3}}{234375} + \frac {11111259 x^{2}}{781250} + \frac {41666223 x}{1953125} + \frac {1331 \log {\left (5 x + 3 \right )}}{9765625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**6/(3+5*x),x)

[Out]

-648*x**9/5 - 13851*x**8/50 - 40338*x**7/875 + 331713*x**6/1250 + 2212083*x**5/15625 - 5848749*x**4/62500 - 17
453753*x**3/234375 + 11111259*x**2/781250 + 41666223*x/1953125 + 1331*log(5*x + 3)/9765625

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