3.1194 \(\int \frac{(1-2 x) (2+3 x)^5}{3+5 x} \, dx\)

Optimal. Leaf size=51 \[ -\frac{81 x^6}{5}-\frac{5427 x^5}{125}-\frac{17469 x^4}{500}+\frac{2469 x^3}{625}+\frac{127779 x^2}{6250}+\frac{166663 x}{15625}+\frac{11 \log (5 x+3)}{78125} \]

[Out]

(166663*x)/15625 + (127779*x^2)/6250 + (2469*x^3)/625 - (17469*x^4)/500 - (5427*x^5)/125 - (81*x^6)/5 + (11*Lo
g[3 + 5*x])/78125

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Rubi [A]  time = 0.0187147, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{81 x^6}{5}-\frac{5427 x^5}{125}-\frac{17469 x^4}{500}+\frac{2469 x^3}{625}+\frac{127779 x^2}{6250}+\frac{166663 x}{15625}+\frac{11 \log (5 x+3)}{78125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(166663*x)/15625 + (127779*x^2)/6250 + (2469*x^3)/625 - (17469*x^4)/500 - (5427*x^5)/125 - (81*x^6)/5 + (11*Lo
g[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} \, dx &=\int \left (\frac{166663}{15625}+\frac{127779 x}{3125}+\frac{7407 x^2}{625}-\frac{17469 x^3}{125}-\frac{5427 x^4}{25}-\frac{486 x^5}{5}+\frac{11}{15625 (3+5 x)}\right ) \, dx\\ &=\frac{166663 x}{15625}+\frac{127779 x^2}{6250}+\frac{2469 x^3}{625}-\frac{17469 x^4}{500}-\frac{5427 x^5}{125}-\frac{81 x^6}{5}+\frac{11 \log (3+5 x)}{78125}\\ \end{align*}

Mathematica [A]  time = 0.0125791, size = 42, normalized size = 0.82 \[ \frac{-25312500 x^6-67837500 x^5-54590625 x^4+6172500 x^3+31944750 x^2+16666300 x+220 \log (5 x+3)+2813811}{1562500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2813811 + 16666300*x + 31944750*x^2 + 6172500*x^3 - 54590625*x^4 - 67837500*x^5 - 25312500*x^6 + 220*Log[3 +
5*x])/1562500

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Maple [A]  time = 0.002, size = 38, normalized size = 0.8 \begin{align*}{\frac{166663\,x}{15625}}+{\frac{127779\,{x}^{2}}{6250}}+{\frac{2469\,{x}^{3}}{625}}-{\frac{17469\,{x}^{4}}{500}}-{\frac{5427\,{x}^{5}}{125}}-{\frac{81\,{x}^{6}}{5}}+{\frac{11\,\ln \left ( 3+5\,x \right ) }{78125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

166663/15625*x+127779/6250*x^2+2469/625*x^3-17469/500*x^4-5427/125*x^5-81/5*x^6+11/78125*ln(3+5*x)

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Maxima [A]  time = 1.11821, size = 50, normalized size = 0.98 \begin{align*} -\frac{81}{5} \, x^{6} - \frac{5427}{125} \, x^{5} - \frac{17469}{500} \, x^{4} + \frac{2469}{625} \, x^{3} + \frac{127779}{6250} \, x^{2} + \frac{166663}{15625} \, x + \frac{11}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/5*x^6 - 5427/125*x^5 - 17469/500*x^4 + 2469/625*x^3 + 127779/6250*x^2 + 166663/15625*x + 11/78125*log(5*x
+ 3)

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Fricas [A]  time = 1.43624, size = 157, normalized size = 3.08 \begin{align*} -\frac{81}{5} \, x^{6} - \frac{5427}{125} \, x^{5} - \frac{17469}{500} \, x^{4} + \frac{2469}{625} \, x^{3} + \frac{127779}{6250} \, x^{2} + \frac{166663}{15625} \, x + \frac{11}{78125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/5*x^6 - 5427/125*x^5 - 17469/500*x^4 + 2469/625*x^3 + 127779/6250*x^2 + 166663/15625*x + 11/78125*log(5*x
+ 3)

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Sympy [A]  time = 0.093182, size = 48, normalized size = 0.94 \begin{align*} - \frac{81 x^{6}}{5} - \frac{5427 x^{5}}{125} - \frac{17469 x^{4}}{500} + \frac{2469 x^{3}}{625} + \frac{127779 x^{2}}{6250} + \frac{166663 x}{15625} + \frac{11 \log{\left (5 x + 3 \right )}}{78125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x**6/5 - 5427*x**5/125 - 17469*x**4/500 + 2469*x**3/625 + 127779*x**2/6250 + 166663*x/15625 + 11*log(5*x +
3)/78125

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Giac [A]  time = 2.68069, size = 51, normalized size = 1. \begin{align*} -\frac{81}{5} \, x^{6} - \frac{5427}{125} \, x^{5} - \frac{17469}{500} \, x^{4} + \frac{2469}{625} \, x^{3} + \frac{127779}{6250} \, x^{2} + \frac{166663}{15625} \, x + \frac{11}{78125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/5*x^6 - 5427/125*x^5 - 17469/500*x^4 + 2469/625*x^3 + 127779/6250*x^2 + 166663/15625*x + 11/78125*log(abs(
5*x + 3))