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

Optimal. Leaf size=48 \[ \frac {27 x^4}{25}-\frac {36 x^3}{125}-\frac {1449 x^2}{1250}+\frac {2416 x}{3125}-\frac {121}{15625 (5 x+3)}+\frac {209 \log (5 x+3)}{3125} \]

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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} \begin {gather*} \frac {27 x^4}{25}-\frac {36 x^3}{125}-\frac {1449 x^2}{1250}+\frac {2416 x}{3125}-\frac {121}{15625 (5 x+3)}+\frac {209 \log (5 x+3)}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2416*x)/3125 - (1449*x^2)/1250 - (36*x^3)/125 + (27*x^4)/25 - 121/(15625*(3 + 5*x)) + (209*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)^2 (2+3 x)^3}{(3+5 x)^2} \, dx &=\int \left (\frac {2416}{3125}-\frac {1449 x}{625}-\frac {108 x^2}{125}+\frac {108 x^3}{25}+\frac {121}{3125 (3+5 x)^2}+\frac {209}{625 (3+5 x)}\right ) \, dx\\ &=\frac {2416 x}{3125}-\frac {1449 x^2}{1250}-\frac {36 x^3}{125}+\frac {27 x^4}{25}-\frac {121}{15625 (3+5 x)}+\frac {209 \log (3+5 x)}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 1.06 \begin {gather*} \frac {33750 x^5+11250 x^4-41625 x^3+2425 x^2+35715 x+418 (5 x+3) \log (6 (5 x+3))+12683}{6250 (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12683 + 35715*x + 2425*x^2 - 41625*x^3 + 11250*x^4 + 33750*x^5 + 418*(3 + 5*x)*Log[6*(3 + 5*x)])/(6250*(3 + 5
*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2 (2+3 x)^3}{(3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.17, size = 47, normalized size = 0.98 \begin {gather*} \frac {168750 \, x^{5} + 56250 \, x^{4} - 208125 \, x^{3} + 12125 \, x^{2} + 2090 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 72480 \, x - 242}{31250 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31250*(168750*x^5 + 56250*x^4 - 208125*x^3 + 12125*x^2 + 2090*(5*x + 3)*log(5*x + 3) + 72480*x - 242)/(5*x +
3)

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giac [A]  time = 1.11, size = 66, normalized size = 1.38 \begin {gather*} -\frac {1}{31250} \, {\left (5 \, x + 3\right )}^{4} {\left (\frac {720}{5 \, x + 3} - \frac {2115}{{\left (5 \, x + 3\right )}^{2}} - \frac {5750}{{\left (5 \, x + 3\right )}^{3}} - 54\right )} - \frac {121}{15625 \, {\left (5 \, x + 3\right )}} - \frac {209}{3125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31250*(5*x + 3)^4*(720/(5*x + 3) - 2115/(5*x + 3)^2 - 5750/(5*x + 3)^3 - 54) - 121/15625/(5*x + 3) - 209/31
25*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

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maple [A]  time = 0.01, size = 37, normalized size = 0.77 \begin {gather*} \frac {27 x^{4}}{25}-\frac {36 x^{3}}{125}-\frac {1449 x^{2}}{1250}+\frac {2416 x}{3125}+\frac {209 \ln \left (5 x +3\right )}{3125}-\frac {121}{15625 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2416/3125*x-1449/1250*x^2-36/125*x^3+27/25*x^4-121/15625/(5*x+3)+209/3125*ln(5*x+3)

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maxima [A]  time = 0.52, size = 36, normalized size = 0.75 \begin {gather*} \frac {27}{25} \, x^{4} - \frac {36}{125} \, x^{3} - \frac {1449}{1250} \, x^{2} + \frac {2416}{3125} \, x - \frac {121}{15625 \, {\left (5 \, x + 3\right )}} + \frac {209}{3125} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/25*x^4 - 36/125*x^3 - 1449/1250*x^2 + 2416/3125*x - 121/15625/(5*x + 3) + 209/3125*log(5*x + 3)

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mupad [B]  time = 0.03, size = 34, normalized size = 0.71 \begin {gather*} \frac {2416\,x}{3125}+\frac {209\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {121}{78125\,\left (x+\frac {3}{5}\right )}-\frac {1449\,x^2}{1250}-\frac {36\,x^3}{125}+\frac {27\,x^4}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2416*x)/3125 + (209*log(x + 3/5))/3125 - 121/(78125*(x + 3/5)) - (1449*x^2)/1250 - (36*x^3)/125 + (27*x^4)/25

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sympy [A]  time = 0.12, size = 41, normalized size = 0.85 \begin {gather*} \frac {27 x^{4}}{25} - \frac {36 x^{3}}{125} - \frac {1449 x^{2}}{1250} + \frac {2416 x}{3125} + \frac {209 \log {\left (5 x + 3 \right )}}{3125} - \frac {121}{78125 x + 46875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*x**4/25 - 36*x**3/125 - 1449*x**2/1250 + 2416*x/3125 + 209*log(5*x + 3)/3125 - 121/(78125*x + 46875)

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