3.1195 \(\int \frac{(1-2 x) (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=44 \[ -\frac{162 x^5}{25}-\frac{1269 x^4}{100}-\frac{531 x^3}{125}+\frac{7779 x^2}{1250}+\frac{16663 x}{3125}+\frac{11 \log (5 x+3)}{15625} \]

[Out]

(16663*x)/3125 + (7779*x^2)/1250 - (531*x^3)/125 - (1269*x^4)/100 - (162*x^5)/25 + (11*Log[3 + 5*x])/15625

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Rubi [A]  time = 0.0168164, antiderivative size = 44, 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} \[ -\frac{162 x^5}{25}-\frac{1269 x^4}{100}-\frac{531 x^3}{125}+\frac{7779 x^2}{1250}+\frac{16663 x}{3125}+\frac{11 \log (5 x+3)}{15625} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(16663*x)/3125 + (7779*x^2)/1250 - (531*x^3)/125 - (1269*x^4)/100 - (162*x^5)/25 + (11*Log[3 + 5*x])/15625

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)^4}{3+5 x} \, dx &=\int \left (\frac{16663}{3125}+\frac{7779 x}{625}-\frac{1593 x^2}{125}-\frac{1269 x^3}{25}-\frac{162 x^4}{5}+\frac{11}{3125 (3+5 x)}\right ) \, dx\\ &=\frac{16663 x}{3125}+\frac{7779 x^2}{1250}-\frac{531 x^3}{125}-\frac{1269 x^4}{100}-\frac{162 x^5}{25}+\frac{11 \log (3+5 x)}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0127459, size = 37, normalized size = 0.84 \[ \frac{-2025000 x^5-3965625 x^4-1327500 x^3+1944750 x^2+1666300 x+220 \log (5 x+3)+369411}{312500} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(369411 + 1666300*x + 1944750*x^2 - 1327500*x^3 - 3965625*x^4 - 2025000*x^5 + 220*Log[3 + 5*x])/312500

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Maple [A]  time = 0.003, size = 33, normalized size = 0.8 \begin{align*}{\frac{16663\,x}{3125}}+{\frac{7779\,{x}^{2}}{1250}}-{\frac{531\,{x}^{3}}{125}}-{\frac{1269\,{x}^{4}}{100}}-{\frac{162\,{x}^{5}}{25}}+{\frac{11\,\ln \left ( 3+5\,x \right ) }{15625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^4/(3+5*x),x)

[Out]

16663/3125*x+7779/1250*x^2-531/125*x^3-1269/100*x^4-162/25*x^5+11/15625*ln(3+5*x)

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Maxima [A]  time = 1.32061, size = 43, normalized size = 0.98 \begin{align*} -\frac{162}{25} \, x^{5} - \frac{1269}{100} \, x^{4} - \frac{531}{125} \, x^{3} + \frac{7779}{1250} \, x^{2} + \frac{16663}{3125} \, x + \frac{11}{15625} \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

-162/25*x^5 - 1269/100*x^4 - 531/125*x^3 + 7779/1250*x^2 + 16663/3125*x + 11/15625*log(5*x + 3)

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Fricas [A]  time = 1.56556, size = 131, normalized size = 2.98 \begin{align*} -\frac{162}{25} \, x^{5} - \frac{1269}{100} \, x^{4} - \frac{531}{125} \, x^{3} + \frac{7779}{1250} \, x^{2} + \frac{16663}{3125} \, x + \frac{11}{15625} \, \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)^4/(3+5*x),x, algorithm="fricas")

[Out]

-162/25*x^5 - 1269/100*x^4 - 531/125*x^3 + 7779/1250*x^2 + 16663/3125*x + 11/15625*log(5*x + 3)

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Sympy [A]  time = 0.088346, size = 41, normalized size = 0.93 \begin{align*} - \frac{162 x^{5}}{25} - \frac{1269 x^{4}}{100} - \frac{531 x^{3}}{125} + \frac{7779 x^{2}}{1250} + \frac{16663 x}{3125} + \frac{11 \log{\left (5 x + 3 \right )}}{15625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**4/(3+5*x),x)

[Out]

-162*x**5/25 - 1269*x**4/100 - 531*x**3/125 + 7779*x**2/1250 + 16663*x/3125 + 11*log(5*x + 3)/15625

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Giac [A]  time = 2.3878, size = 45, normalized size = 1.02 \begin{align*} -\frac{162}{25} \, x^{5} - \frac{1269}{100} \, x^{4} - \frac{531}{125} \, x^{3} + \frac{7779}{1250} \, x^{2} + \frac{16663}{3125} \, x + \frac{11}{15625} \, \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)^4/(3+5*x),x, algorithm="giac")

[Out]

-162/25*x^5 - 1269/100*x^4 - 531/125*x^3 + 7779/1250*x^2 + 16663/3125*x + 11/15625*log(abs(5*x + 3))