3.1282 \(\int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac {125 x^4}{9}-\frac {800 x^3}{81}-\frac {305 x^2}{54}+\frac {1271 x}{243}+\frac {49}{729 (3 x+2)}+\frac {763}{729} \log (3 x+2) \]

[Out]

1271/243*x-305/54*x^2-800/81*x^3+125/9*x^4+49/729/(2+3*x)+763/729*ln(2+3*x)

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Rubi [A]  time = 0.02, antiderivative size = 48, 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} \[ \frac {125 x^4}{9}-\frac {800 x^3}{81}-\frac {305 x^2}{54}+\frac {1271 x}{243}+\frac {49}{729 (3 x+2)}+\frac {763}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1271*x)/243 - (305*x^2)/54 - (800*x^3)/81 + (125*x^4)/9 + 49/(729*(2 + 3*x)) + (763*Log[2 + 3*x])/729

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)^2 (3+5 x)^3}{(2+3 x)^2} \, dx &=\int \left (\frac {1271}{243}-\frac {305 x}{27}-\frac {800 x^2}{27}+\frac {500 x^3}{9}-\frac {49}{243 (2+3 x)^2}+\frac {763}{243 (2+3 x)}\right ) \, dx\\ &=\frac {1271 x}{243}-\frac {305 x^2}{54}-\frac {800 x^3}{81}+\frac {125 x^4}{9}+\frac {49}{729 (2+3 x)}+\frac {763}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 49, normalized size = 1.02 \[ \frac {182250 x^5-8100 x^4-160515 x^3+19224 x^2+50052 x+4578 (3 x+2) \log (30 x+20)+3158}{4374 (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3158 + 50052*x + 19224*x^2 - 160515*x^3 - 8100*x^4 + 182250*x^5 + 4578*(2 + 3*x)*Log[20 + 30*x])/(4374*(2 + 3
*x))

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fricas [A]  time = 0.55, size = 47, normalized size = 0.98 \[ \frac {60750 \, x^{5} - 2700 \, x^{4} - 53505 \, x^{3} + 6408 \, x^{2} + 1526 \, {\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 15252 \, x + 98}{1458 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1458*(60750*x^5 - 2700*x^4 - 53505*x^3 + 6408*x^2 + 1526*(3*x + 2)*log(3*x + 2) + 15252*x + 98)/(3*x + 2)

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giac [A]  time = 1.03, size = 66, normalized size = 1.38 \[ -\frac {1}{4374} \, {\left (3 \, x + 2\right )}^{4} {\left (\frac {7600}{3 \, x + 2} - \frac {24855}{{\left (3 \, x + 2\right )}^{2}} + \frac {24594}{{\left (3 \, x + 2\right )}^{3}} - 750\right )} + \frac {49}{729 \, {\left (3 \, x + 2\right )}} - \frac {763}{729} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4374*(3*x + 2)^4*(7600/(3*x + 2) - 24855/(3*x + 2)^2 + 24594/(3*x + 2)^3 - 750) + 49/729/(3*x + 2) - 763/72
9*log(1/3*abs(3*x + 2)/(3*x + 2)^2)

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maple [A]  time = 0.01, size = 37, normalized size = 0.77 \[ \frac {125 x^{4}}{9}-\frac {800 x^{3}}{81}-\frac {305 x^{2}}{54}+\frac {1271 x}{243}+\frac {763 \ln \left (3 x +2\right )}{729}+\frac {49}{729 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1271/243*x-305/54*x^2-800/81*x^3+125/9*x^4+49/729/(3*x+2)+763/729*ln(3*x+2)

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maxima [A]  time = 0.58, size = 36, normalized size = 0.75 \[ \frac {125}{9} \, x^{4} - \frac {800}{81} \, x^{3} - \frac {305}{54} \, x^{2} + \frac {1271}{243} \, x + \frac {49}{729 \, {\left (3 \, x + 2\right )}} + \frac {763}{729} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/9*x^4 - 800/81*x^3 - 305/54*x^2 + 1271/243*x + 49/729/(3*x + 2) + 763/729*log(3*x + 2)

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mupad [B]  time = 0.03, size = 34, normalized size = 0.71 \[ \frac {1271\,x}{243}+\frac {763\,\ln \left (x+\frac {2}{3}\right )}{729}+\frac {49}{2187\,\left (x+\frac {2}{3}\right )}-\frac {305\,x^2}{54}-\frac {800\,x^3}{81}+\frac {125\,x^4}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1271*x)/243 + (763*log(x + 2/3))/729 + 49/(2187*(x + 2/3)) - (305*x^2)/54 - (800*x^3)/81 + (125*x^4)/9

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sympy [A]  time = 0.12, size = 41, normalized size = 0.85 \[ \frac {125 x^{4}}{9} - \frac {800 x^{3}}{81} - \frac {305 x^{2}}{54} + \frac {1271 x}{243} + \frac {763 \log {\left (3 x + 2 \right )}}{729} + \frac {49}{2187 x + 1458} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*x**4/9 - 800*x**3/81 - 305*x**2/54 + 1271*x/243 + 763*log(3*x + 2)/729 + 49/(2187*x + 1458)

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