3.1423 \(\int \frac {(1-2 x)^3 (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac {24 x^3}{125}+\frac {354 x^2}{625}-\frac {2978 x}{3125}-\frac {1452}{3125 (5 x+3)}-\frac {1331}{31250 (5 x+3)^2}+\frac {1551 \log (5 x+3)}{3125} \]

[Out]

-2978/3125*x+354/625*x^2-24/125*x^3-1331/31250/(3+5*x)^2-1452/3125/(3+5*x)+1551/3125*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \[ -\frac {24 x^3}{125}+\frac {354 x^2}{625}-\frac {2978 x}{3125}-\frac {1452}{3125 (5 x+3)}-\frac {1331}{31250 (5 x+3)^2}+\frac {1551 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2978*x)/3125 + (354*x^2)/625 - (24*x^3)/125 - 1331/(31250*(3 + 5*x)^2) - 1452/(3125*(3 + 5*x)) + (1551*Log[3
+ 5*x])/3125

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^3 (2+3 x)^2}{(3+5 x)^3} \, dx &=\int \left (-\frac {2978}{3125}+\frac {708 x}{625}-\frac {72 x^2}{125}+\frac {1331}{3125 (3+5 x)^3}+\frac {1452}{625 (3+5 x)^2}+\frac {1551}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {2978 x}{3125}+\frac {354 x^2}{625}-\frac {24 x^3}{125}-\frac {1331}{31250 (3+5 x)^2}-\frac {1452}{3125 (3+5 x)}+\frac {1551 \log (3+5 x)}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 53, normalized size = 1.02 \[ -\frac {30000 x^5-52500 x^4+53500 x^3+274500 x^2+221340 x-3102 (5 x+3)^2 \log (6 (5 x+3))+54943}{6250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6250*(54943 + 221340*x + 274500*x^2 + 53500*x^3 - 52500*x^4 + 30000*x^5 - 3102*(3 + 5*x)^2*Log[6*(3 + 5*x)]
)/(3 + 5*x)^2

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fricas [A]  time = 0.80, size = 57, normalized size = 1.10 \[ -\frac {150000 \, x^{5} - 262500 \, x^{4} + 267500 \, x^{3} + 734100 \, x^{2} - 15510 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 340620 \, x + 44891}{31250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31250*(150000*x^5 - 262500*x^4 + 267500*x^3 + 734100*x^2 - 15510*(25*x^2 + 30*x + 9)*log(5*x + 3) + 340620*
x + 44891)/(25*x^2 + 30*x + 9)

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giac [A]  time = 0.88, size = 37, normalized size = 0.71 \[ -\frac {24}{125} \, x^{3} + \frac {354}{625} \, x^{2} - \frac {2978}{3125} \, x - \frac {121 \, {\left (600 \, x + 371\right )}}{31250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {1551}{3125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/125*x^3 + 354/625*x^2 - 2978/3125*x - 121/31250*(600*x + 371)/(5*x + 3)^2 + 1551/3125*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 41, normalized size = 0.79 \[ -\frac {24 x^{3}}{125}+\frac {354 x^{2}}{625}-\frac {2978 x}{3125}+\frac {1551 \ln \left (5 x +3\right )}{3125}-\frac {1331}{31250 \left (5 x +3\right )^{2}}-\frac {1452}{3125 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2978/3125*x+354/625*x^2-24/125*x^3-1331/31250/(5*x+3)^2-1452/3125/(5*x+3)+1551/3125*ln(5*x+3)

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maxima [A]  time = 0.63, size = 41, normalized size = 0.79 \[ -\frac {24}{125} \, x^{3} + \frac {354}{625} \, x^{2} - \frac {2978}{3125} \, x - \frac {121 \, {\left (600 \, x + 371\right )}}{31250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {1551}{3125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/125*x^3 + 354/625*x^2 - 2978/3125*x - 121/31250*(600*x + 371)/(25*x^2 + 30*x + 9) + 1551/3125*log(5*x + 3)

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mupad [B]  time = 0.03, size = 37, normalized size = 0.71 \[ \frac {1551\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {2978\,x}{3125}-\frac {\frac {1452\,x}{15625}+\frac {44891}{781250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {354\,x^2}{625}-\frac {24\,x^3}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1551*log(x + 3/5))/3125 - (2978*x)/3125 - ((1452*x)/15625 + 44891/781250)/((6*x)/5 + x^2 + 9/25) + (354*x^2)/
625 - (24*x^3)/125

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sympy [A]  time = 0.13, size = 42, normalized size = 0.81 \[ - \frac {24 x^{3}}{125} + \frac {354 x^{2}}{625} - \frac {2978 x}{3125} - \frac {72600 x + 44891}{781250 x^{2} + 937500 x + 281250} + \frac {1551 \log {\left (5 x + 3 \right )}}{3125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*x**3/125 + 354*x**2/625 - 2978*x/3125 - (72600*x + 44891)/(781250*x**2 + 937500*x + 281250) + 1551*log(5*x
+ 3)/3125

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