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

Optimal. Leaf size=55 \[ -\frac {486 x^5}{125}-\frac {3969 x^4}{500}-\frac {1854 x^3}{625}+\frac {24093 x^2}{6250}+\frac {444 x}{125}-\frac {11}{78125 (5 x+3)}+\frac {163 \log (5 x+3)}{78125} \]

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Rubi [A]  time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\frac {486 x^5}{125}-\frac {3969 x^4}{500}-\frac {1854 x^3}{625}+\frac {24093 x^2}{6250}+\frac {444 x}{125}-\frac {11}{78125 (5 x+3)}+\frac {163 \log (5 x+3)}{78125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(444*x)/125 + (24093*x^2)/6250 - (1854*x^3)/625 - (3969*x^4)/500 - (486*x^5)/125 - 11/(78125*(3 + 5*x)) + (163
*Log[3 + 5*x])/78125

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx &=\int \left (\frac {444}{125}+\frac {24093 x}{3125}-\frac {5562 x^2}{625}-\frac {3969 x^3}{125}-\frac {486 x^4}{25}+\frac {11}{15625 (3+5 x)^2}+\frac {163}{15625 (3+5 x)}\right ) \, dx\\ &=\frac {444 x}{125}+\frac {24093 x^2}{6250}-\frac {1854 x^3}{625}-\frac {3969 x^4}{500}-\frac {486 x^5}{125}-\frac {11}{78125 (3+5 x)}+\frac {163 \log (3+5 x)}{78125}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.87 \begin {gather*} \frac {-3645000 x^5-7441875 x^4-2781000 x^3+3613950 x^2+3330000 x-\frac {132}{5 x+3}+1956 \log (-3 (5 x+3))+779800}{937500} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(779800 + 3330000*x + 3613950*x^2 - 2781000*x^3 - 7441875*x^4 - 3645000*x^5 - 132/(3 + 5*x) + 1956*Log[-3*(3 +
5*x)])/937500

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.21, size = 52, normalized size = 0.95 \begin {gather*} -\frac {6075000 \, x^{6} + 16048125 \, x^{5} + 12076875 \, x^{4} - 3242250 \, x^{3} - 9163950 \, x^{2} - 652 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 3330000 \, x + 44}{312500 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/312500*(6075000*x^6 + 16048125*x^5 + 12076875*x^4 - 3242250*x^3 - 9163950*x^2 - 652*(5*x + 3)*log(5*x + 3)
- 3330000*x + 44)/(5*x + 3)

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giac [A]  time = 1.20, size = 75, normalized size = 1.36 \begin {gather*} \frac {3}{1562500} \, {\left (5 \, x + 3\right )}^{5} {\left (\frac {3105}{5 \, x + 3} + \frac {8700}{{\left (5 \, x + 3\right )}^{2}} + \frac {9300}{{\left (5 \, x + 3\right )}^{3}} + \frac {6400}{{\left (5 \, x + 3\right )}^{4}} - 648\right )} - \frac {11}{78125 \, {\left (5 \, x + 3\right )}} - \frac {163}{78125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1562500*(5*x + 3)^5*(3105/(5*x + 3) + 8700/(5*x + 3)^2 + 9300/(5*x + 3)^3 + 6400/(5*x + 3)^4 - 648) - 11/781
25/(5*x + 3) - 163/78125*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

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maple [A]  time = 0.01, size = 42, normalized size = 0.76 \begin {gather*} -\frac {486 x^{5}}{125}-\frac {3969 x^{4}}{500}-\frac {1854 x^{3}}{625}+\frac {24093 x^{2}}{6250}+\frac {444 x}{125}+\frac {163 \ln \left (5 x +3\right )}{78125}-\frac {11}{78125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

444/125*x+24093/6250*x^2-1854/625*x^3-3969/500*x^4-486/125*x^5-11/78125/(5*x+3)+163/78125*ln(5*x+3)

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maxima [A]  time = 0.50, size = 41, normalized size = 0.75 \begin {gather*} -\frac {486}{125} \, x^{5} - \frac {3969}{500} \, x^{4} - \frac {1854}{625} \, x^{3} + \frac {24093}{6250} \, x^{2} + \frac {444}{125} \, x - \frac {11}{78125 \, {\left (5 \, x + 3\right )}} + \frac {163}{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+3*x)^5/(3+5*x)^2,x, algorithm="maxima")

[Out]

-486/125*x^5 - 3969/500*x^4 - 1854/625*x^3 + 24093/6250*x^2 + 444/125*x - 11/78125/(5*x + 3) + 163/78125*log(5
*x + 3)

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mupad [B]  time = 0.03, size = 39, normalized size = 0.71 \begin {gather*} \frac {444\,x}{125}+\frac {163\,\ln \left (x+\frac {3}{5}\right )}{78125}-\frac {11}{390625\,\left (x+\frac {3}{5}\right )}+\frac {24093\,x^2}{6250}-\frac {1854\,x^3}{625}-\frac {3969\,x^4}{500}-\frac {486\,x^5}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(444*x)/125 + (163*log(x + 3/5))/78125 - 11/(390625*(x + 3/5)) + (24093*x^2)/6250 - (1854*x^3)/625 - (3969*x^4
)/500 - (486*x^5)/125

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sympy [A]  time = 0.12, size = 48, normalized size = 0.87 \begin {gather*} - \frac {486 x^{5}}{125} - \frac {3969 x^{4}}{500} - \frac {1854 x^{3}}{625} + \frac {24093 x^{2}}{6250} + \frac {444 x}{125} + \frac {163 \log {\left (5 x + 3 \right )}}{78125} - \frac {11}{390625 x + 234375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-486*x**5/125 - 3969*x**4/500 - 1854*x**3/625 + 24093*x**2/6250 + 444*x/125 + 163*log(5*x + 3)/78125 - 11/(390
625*x + 234375)

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