3.1210 \(\int \frac{(1-2 x) (2+3 x)^4}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{81 x^4}{50}-\frac{261 x^3}{125}+\frac{378 x^2}{625}+\frac{5511 x}{3125}-\frac{11}{15625 (5 x+3)}+\frac{26 \log (5 x+3)}{3125} \]

[Out]

(5511*x)/3125 + (378*x^2)/625 - (261*x^3)/125 - (81*x^4)/50 - 11/(15625*(3 + 5*x)) + (26*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0219556, antiderivative size = 48, 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} \[ -\frac{81 x^4}{50}-\frac{261 x^3}{125}+\frac{378 x^2}{625}+\frac{5511 x}{3125}-\frac{11}{15625 (5 x+3)}+\frac{26 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5511*x)/3125 + (378*x^2)/625 - (261*x^3)/125 - (81*x^4)/50 - 11/(15625*(3 + 5*x)) + (26*Log[3 + 5*x])/3125

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)^2} \, dx &=\int \left (\frac{5511}{3125}+\frac{756 x}{625}-\frac{783 x^2}{125}-\frac{162 x^3}{25}+\frac{11}{3125 (3+5 x)^2}+\frac{26}{625 (3+5 x)}\right ) \, dx\\ &=\frac{5511 x}{3125}+\frac{378 x^2}{625}-\frac{261 x^3}{125}-\frac{81 x^4}{50}-\frac{11}{15625 (3+5 x)}+\frac{26 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0156897, size = 49, normalized size = 1.02 \[ \frac{-50625 x^5-95625 x^4-20250 x^3+66450 x^2+51795 x+52 (5 x+3) \log (5 x+3)+11233}{6250 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11233 + 51795*x + 66450*x^2 - 20250*x^3 - 95625*x^4 - 50625*x^5 + 52*(3 + 5*x)*Log[3 + 5*x])/(6250*(3 + 5*x))

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Maple [A]  time = 0.006, size = 37, normalized size = 0.8 \begin{align*}{\frac{5511\,x}{3125}}+{\frac{378\,{x}^{2}}{625}}-{\frac{261\,{x}^{3}}{125}}-{\frac{81\,{x}^{4}}{50}}-{\frac{11}{46875+78125\,x}}+{\frac{26\,\ln \left ( 3+5\,x \right ) }{3125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5511/3125*x+378/625*x^2-261/125*x^3-81/50*x^4-11/15625/(3+5*x)+26/3125*ln(3+5*x)

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Maxima [A]  time = 1.01558, size = 49, normalized size = 1.02 \begin{align*} -\frac{81}{50} \, x^{4} - \frac{261}{125} \, x^{3} + \frac{378}{625} \, x^{2} + \frac{5511}{3125} \, x - \frac{11}{15625 \,{\left (5 \, x + 3\right )}} + \frac{26}{3125} \, \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)^2,x, algorithm="maxima")

[Out]

-81/50*x^4 - 261/125*x^3 + 378/625*x^2 + 5511/3125*x - 11/15625/(5*x + 3) + 26/3125*log(5*x + 3)

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Fricas [A]  time = 1.56189, size = 158, normalized size = 3.29 \begin{align*} -\frac{253125 \, x^{5} + 478125 \, x^{4} + 101250 \, x^{3} - 332250 \, x^{2} - 260 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 165330 \, x + 22}{31250 \,{\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)^2,x, algorithm="fricas")

[Out]

-1/31250*(253125*x^5 + 478125*x^4 + 101250*x^3 - 332250*x^2 - 260*(5*x + 3)*log(5*x + 3) - 165330*x + 22)/(5*x
+ 3)

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Sympy [A]  time = 0.103385, size = 41, normalized size = 0.85 \begin{align*} - \frac{81 x^{4}}{50} - \frac{261 x^{3}}{125} + \frac{378 x^{2}}{625} + \frac{5511 x}{3125} + \frac{26 \log{\left (5 x + 3 \right )}}{3125} - \frac{11}{78125 x + 46875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x**4/50 - 261*x**3/125 + 378*x**2/625 + 5511*x/3125 + 26*log(5*x + 3)/3125 - 11/(78125*x + 46875)

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Giac [A]  time = 2.92302, size = 89, normalized size = 1.85 \begin{align*} \frac{3}{31250} \,{\left (5 \, x + 3\right )}^{4}{\left (\frac{150}{5 \, x + 3} + \frac{360}{{\left (5 \, x + 3\right )}^{2}} + \frac{380}{{\left (5 \, x + 3\right )}^{3}} - 27\right )} - \frac{11}{15625 \,{\left (5 \, x + 3\right )}} - \frac{26}{3125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\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)^2,x, algorithm="giac")

[Out]

3/31250*(5*x + 3)^4*(150/(5*x + 3) + 360/(5*x + 3)^2 + 380/(5*x + 3)^3 - 27) - 11/15625/(5*x + 3) - 26/3125*lo
g(1/5*abs(5*x + 3)/(5*x + 3)^2)