3.1226 \(\int \frac{(1-2 x) (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{27 x^2}{125}+\frac{81 x}{625}-\frac{97}{3125 (5 x+3)}-\frac{11}{6250 (5 x+3)^2}+\frac{279 \log (5 x+3)}{3125} \]

[Out]

(81*x)/625 - (27*x^2)/125 - 11/(6250*(3 + 5*x)^2) - 97/(3125*(3 + 5*x)) + (279*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0203152, antiderivative size = 45, 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} \[ -\frac{27 x^2}{125}+\frac{81 x}{625}-\frac{97}{3125 (5 x+3)}-\frac{11}{6250 (5 x+3)^2}+\frac{279 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(81*x)/625 - (27*x^2)/125 - 11/(6250*(3 + 5*x)^2) - 97/(3125*(3 + 5*x)) + (279*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)^3}{(3+5 x)^3} \, dx &=\int \left (\frac{81}{625}-\frac{54 x}{125}+\frac{11}{625 (3+5 x)^3}+\frac{97}{625 (3+5 x)^2}+\frac{279}{625 (3+5 x)}\right ) \, dx\\ &=\frac{81 x}{625}-\frac{27 x^2}{125}-\frac{11}{6250 (3+5 x)^2}-\frac{97}{3125 (3+5 x)}+\frac{279 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0144965, size = 48, normalized size = 1.07 \[ \frac{-33750 x^4-20250 x^3+40650 x^2+40520 x+558 (5 x+3)^2 \log (-3 (5 x+3))+9667}{6250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9667 + 40520*x + 40650*x^2 - 20250*x^3 - 33750*x^4 + 558*(3 + 5*x)^2*Log[-3*(3 + 5*x)])/(6250*(3 + 5*x)^2)

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \begin{align*}{\frac{81\,x}{625}}-{\frac{27\,{x}^{2}}{125}}-{\frac{11}{6250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{97}{9375+15625\,x}}+{\frac{279\,\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)^3/(3+5*x)^3,x)

[Out]

81/625*x-27/125*x^2-11/6250/(3+5*x)^2-97/3125/(3+5*x)+279/3125*ln(3+5*x)

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Maxima [A]  time = 1.0455, size = 49, normalized size = 1.09 \begin{align*} -\frac{27}{125} \, x^{2} + \frac{81}{625} \, x - \frac{970 \, x + 593}{6250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{279}{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)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-27/125*x^2 + 81/625*x - 1/6250*(970*x + 593)/(25*x^2 + 30*x + 9) + 279/3125*log(5*x + 3)

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Fricas [A]  time = 1.51482, size = 161, normalized size = 3.58 \begin{align*} -\frac{33750 \, x^{4} + 20250 \, x^{3} - 12150 \, x^{2} - 558 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 6320 \, x + 593}{6250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6250*(33750*x^4 + 20250*x^3 - 12150*x^2 - 558*(25*x^2 + 30*x + 9)*log(5*x + 3) - 6320*x + 593)/(25*x^2 + 30
*x + 9)

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Sympy [A]  time = 0.117339, size = 36, normalized size = 0.8 \begin{align*} - \frac{27 x^{2}}{125} + \frac{81 x}{625} - \frac{970 x + 593}{156250 x^{2} + 187500 x + 56250} + \frac{279 \log{\left (5 x + 3 \right )}}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*x**2/125 + 81*x/625 - (970*x + 593)/(156250*x**2 + 187500*x + 56250) + 279*log(5*x + 3)/3125

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Giac [A]  time = 3.05293, size = 43, normalized size = 0.96 \begin{align*} -\frac{27}{125} \, x^{2} + \frac{81}{625} \, x - \frac{970 \, x + 593}{6250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{279}{3125} \, \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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-27/125*x^2 + 81/625*x - 1/6250*(970*x + 593)/(5*x + 3)^2 + 279/3125*log(abs(5*x + 3))