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

Optimal. Leaf size=66 \[ -\frac {1458 x^5}{625}-\frac {12393 x^4}{2500}-\frac {6399 x^3}{3125}+\frac {297 x^2}{125}+\frac {36936 x}{15625}-\frac {196}{390625 (5 x+3)}-\frac {11}{781250 (5 x+3)^2}+\frac {1449 \log (5 x+3)}{390625} \]

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Rubi [A]  time = 0.03, antiderivative size = 66, 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 {1458 x^5}{625}-\frac {12393 x^4}{2500}-\frac {6399 x^3}{3125}+\frac {297 x^2}{125}+\frac {36936 x}{15625}-\frac {196}{390625 (5 x+3)}-\frac {11}{781250 (5 x+3)^2}+\frac {1449 \log (5 x+3)}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(36936*x)/15625 + (297*x^2)/125 - (6399*x^3)/3125 - (12393*x^4)/2500 - (1458*x^5)/625 - 11/(781250*(3 + 5*x)^2
) - 196/(390625*(3 + 5*x)) + (1449*Log[3 + 5*x])/390625

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)^6}{(3+5 x)^3} \, dx &=\int \left (\frac {36936}{15625}+\frac {594 x}{125}-\frac {19197 x^2}{3125}-\frac {12393 x^3}{625}-\frac {1458 x^4}{125}+\frac {11}{78125 (3+5 x)^3}+\frac {196}{78125 (3+5 x)^2}+\frac {1449}{78125 (3+5 x)}\right ) \, dx\\ &=\frac {36936 x}{15625}+\frac {297 x^2}{125}-\frac {6399 x^3}{3125}-\frac {12393 x^4}{2500}-\frac {1458 x^5}{625}-\frac {11}{781250 (3+5 x)^2}-\frac {196}{390625 (3+5 x)}+\frac {1449 \log (3+5 x)}{390625}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.92 \begin {gather*} \frac {-455625000 x^7-1514953125 x^6-1725806250 x^5-364415625 x^4+874597500 x^3+834723225 x^2+302537270 x+28980 (5 x+3)^2 \log (5 x+3)[phone]}{7812500 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(40891591 + 302537270*x + 834723225*x^2 + 874597500*x^3 - 364415625*x^4 - 1725806250*x^5 - 1514953125*x^6 - 45
5625000*x^7 + 28980*(3 + 5*x)^2*Log[3 + 5*x])/(7812500*(3 + 5*x)^2)

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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)^6}{(3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 67, normalized size = 1.02 \begin {gather*} -\frac {91125000 \, x^{7} + 302990625 \, x^{6} + 345161250 \, x^{5} + 72883125 \, x^{4} - 174919500 \, x^{3} - 144220500 \, x^{2} - 5796 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 33238480 \, x + 2374}{1562500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/1562500*(91125000*x^7 + 302990625*x^6 + 345161250*x^5 + 72883125*x^4 - 174919500*x^3 - 144220500*x^2 - 5796
*(25*x^2 + 30*x + 9)*log(5*x + 3) - 33238480*x + 2374)/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.19, size = 47, normalized size = 0.71 \begin {gather*} -\frac {1458}{625} \, x^{5} - \frac {12393}{2500} \, x^{4} - \frac {6399}{3125} \, x^{3} + \frac {297}{125} \, x^{2} + \frac {36936}{15625} \, x - \frac {1960 \, x + 1187}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {1449}{390625} \, \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+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

-1458/625*x^5 - 12393/2500*x^4 - 6399/3125*x^3 + 297/125*x^2 + 36936/15625*x - 1/781250*(1960*x + 1187)/(5*x +
3)^2 + 1449/390625*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 51, normalized size = 0.77 \begin {gather*} -\frac {1458 x^{5}}{625}-\frac {12393 x^{4}}{2500}-\frac {6399 x^{3}}{3125}+\frac {297 x^{2}}{125}+\frac {36936 x}{15625}+\frac {1449 \ln \left (5 x +3\right )}{390625}-\frac {11}{781250 \left (5 x +3\right )^{2}}-\frac {196}{390625 \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)^6/(5*x+3)^3,x)

[Out]

36936/15625*x+297/125*x^2-6399/3125*x^3-12393/2500*x^4-1458/625*x^5-11/781250/(5*x+3)^2-196/390625/(5*x+3)+144
9/390625*ln(5*x+3)

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maxima [A]  time = 0.50, size = 51, normalized size = 0.77 \begin {gather*} -\frac {1458}{625} \, x^{5} - \frac {12393}{2500} \, x^{4} - \frac {6399}{3125} \, x^{3} + \frac {297}{125} \, x^{2} + \frac {36936}{15625} \, x - \frac {1960 \, x + 1187}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1449}{390625} \, \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)^6/(3+5*x)^3,x, algorithm="maxima")

[Out]

-1458/625*x^5 - 12393/2500*x^4 - 6399/3125*x^3 + 297/125*x^2 + 36936/15625*x - 1/781250*(1960*x + 1187)/(25*x^
2 + 30*x + 9) + 1449/390625*log(5*x + 3)

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mupad [B]  time = 0.03, size = 47, normalized size = 0.71 \begin {gather*} \frac {36936\,x}{15625}+\frac {1449\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {196\,x}{1953125}+\frac {1187}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {297\,x^2}{125}-\frac {6399\,x^3}{3125}-\frac {12393\,x^4}{2500}-\frac {1458\,x^5}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(36936*x)/15625 + (1449*log(x + 3/5))/390625 - ((196*x)/1953125 + 1187/19531250)/((6*x)/5 + x^2 + 9/25) + (297
*x^2)/125 - (6399*x^3)/3125 - (12393*x^4)/2500 - (1458*x^5)/625

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sympy [A]  time = 0.14, size = 56, normalized size = 0.85 \begin {gather*} - \frac {1458 x^{5}}{625} - \frac {12393 x^{4}}{2500} - \frac {6399 x^{3}}{3125} + \frac {297 x^{2}}{125} + \frac {36936 x}{15625} - \frac {1960 x + 1187}{19531250 x^{2} + 23437500 x + 7031250} + \frac {1449 \log {\left (5 x + 3 \right )}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1458*x**5/625 - 12393*x**4/2500 - 6399*x**3/3125 + 297*x**2/125 + 36936*x/15625 - (1960*x + 1187)/(19531250*x
**2 + 23437500*x + 7031250) + 1449*log(5*x + 3)/390625

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