∫ (1−2x)2(2+3x)6 ________________________

3.12.84 \(\int \frac {(1-2 x)^2 (2+3 x)^6}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=73 \[ \frac {486 x^6}{125}+\frac {17496 x^5}{3125}-\frac {23571 x^4}{12500}-\frac {16299 x^3}{3125}+\frac {189 x^2}{15625}+\frac {920502 x}{390625}-\frac {2134}{1953125 (5 x+3)}-\frac {121}{3906250 (5 x+3)^2}+\frac {15547 \log (5 x+3)}{1953125} \]

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Rubi [A] time = 0.04, antiderivative size = 73, 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 {486 x^6}{125}+\frac {17496 x^5}{3125}-\frac {23571 x^4}{12500}-\frac {16299 x^3}{3125}+\frac {189 x^2}{15625}+\frac {920502 x}{390625}-\frac {2134}{1953125 (5 x+3)}-\frac {121}{3906250 (5 x+3)^2}+\frac {15547 \log (5 x+3)}{1953125} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^3,x]

[Out]

(920502*x)/390625 + (189*x^2)/15625 - (16299*x^3)/3125 - (23571*x^4)/12500 + (17496*x^5)/3125 + (486*x^6)/125

- 121/(3906250*(3 + 5*x)^2) - 2134/(1953125*(3 + 5*x)) + (15547*Log[3 + 5*x])/1953125

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)^6}{(3+5 x)^3} \, dx &=\int \left (\frac {920502}{390625}+\frac {378 x}{15625}-\frac {48897 x^2}{3125}-\frac {23571 x^3}{3125}+\frac {17496 x^4}{625}+\frac {2916 x^5}{125}+\frac {121}{390625 (3+5 x)^3}+\frac {2134}{390625 (3+5 x)^2}+\frac {15547}{390625 (3+5 x)}\right ) \, dx\\ &=\frac {920502 x}{390625}+\frac {189 x^2}{15625}-\frac {16299 x^3}{3125}-\frac {23571 x^4}{12500}+\frac {17496 x^5}{3125}+\frac {486 x^6}{125}-\frac {121}{3906250 (3+5 x)^2}-\frac {2134}{1953125 (3+5 x)}+\frac {15547 \log (3+5 x)}{1953125}\\ \end {align*}

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Mathematica [A] time = 0.05, size = 68, normalized size = 0.93 \begin {gather*} \frac {3796875000 x^8+10023750000 x^7+6086390625 x^6-5334918750 x^5-6763246875 x^4+481792500 x^3+3528738675 x^2+1743814610 x+310940 (5 x+3)^2 \log (6 (5 x+3))+274543613}{39062500 (5 x+3)^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^3,x]

[Out]

(274543613 + 1743814610*x + 3528738675*x^2 + 481792500*x^3 - 6763246875*x^4 - 5334918750*x^5 + 6086390625*x^6

+ 10023750000*x^7 + 3796875000*x^8 + 310940*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(39062500*(3 + 5*x)^2)

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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)^6}{(3+5 x)^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^3,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^2*(2 + 3*x)^6)/(3 + 5*x)^3, x]

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fricas [A] time = 1.09, size = 72, normalized size = 0.99 \begin {gather*} \frac {759375000 \, x^{8} + 2004750000 \, x^{7} + 1217278125 \, x^{6} - 1066983750 \, x^{5} - 1352649375 \, x^{4} + 96358500 \, x^{3} + 553151700 \, x^{2} + 62188 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 165647680 \, x - 25850}{7812500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/7812500*(759375000*x^8 + 2004750000*x^7 + 1217278125*x^6 - 1066983750*x^5 - 1352649375*x^4 + 96358500*x^3 +

553151700*x^2 + 62188*(25*x^2 + 30*x + 9)*log(5*x + 3) + 165647680*x - 25850)/(25*x^2 + 30*x + 9)

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giac [A] time = 0.89, size = 52, normalized size = 0.71 \begin {gather*} \frac {486}{125} \, x^{6} + \frac {17496}{3125} \, x^{5} - \frac {23571}{12500} \, x^{4} - \frac {16299}{3125} \, x^{3} + \frac {189}{15625} \, x^{2} + \frac {920502}{390625} \, x - \frac {11 \, {\left (388 \, x + 235\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {15547}{1953125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

486/125*x^6 + 17496/3125*x^5 - 23571/12500*x^4 - 16299/3125*x^3 + 189/15625*x^2 + 920502/390625*x - 11/781250*

(388*x + 235)/(5*x + 3)^2 + 15547/1953125*log(abs(5*x + 3))

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maple [A] time = 0.01, size = 56, normalized size = 0.77 \begin {gather*} \frac {486 x^{6}}{125}+\frac {17496 x^{5}}{3125}-\frac {23571 x^{4}}{12500}-\frac {16299 x^{3}}{3125}+\frac {189 x^{2}}{15625}+\frac {920502 x}{390625}+\frac {15547 \ln \left (5 x +3\right )}{1953125}-\frac {121}{3906250 \left (5 x +3\right )^{2}}-\frac {2134}{1953125 \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)^6/(5*x+3)^3,x)

[Out]

920502/390625*x+189/15625*x^2-16299/3125*x^3-23571/12500*x^4+17496/3125*x^5+486/125*x^6-121/3906250/(5*x+3)^2-

2134/1953125/(5*x+3)+15547/1953125*ln(5*x+3)

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maxima [A] time = 0.51, size = 56, normalized size = 0.77 \begin {gather*} \frac {486}{125} \, x^{6} + \frac {17496}{3125} \, x^{5} - \frac {23571}{12500} \, x^{4} - \frac {16299}{3125} \, x^{3} + \frac {189}{15625} \, x^{2} + \frac {920502}{390625} \, x - \frac {11 \, {\left (388 \, x + 235\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {15547}{1953125} \, \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)^6/(3+5*x)^3,x, algorithm="maxima")

[Out]

486/125*x^6 + 17496/3125*x^5 - 23571/12500*x^4 - 16299/3125*x^3 + 189/15625*x^2 + 920502/390625*x - 11/781250*

(388*x + 235)/(25*x^2 + 30*x + 9) + 15547/1953125*log(5*x + 3)

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mupad [B] time = 0.04, size = 52, normalized size = 0.71 \begin {gather*} \frac {920502\,x}{390625}+\frac {15547\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {\frac {2134\,x}{9765625}+\frac {517}{3906250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {189\,x^2}{15625}-\frac {16299\,x^3}{3125}-\frac {23571\,x^4}{12500}+\frac {17496\,x^5}{3125}+\frac {486\,x^6}{125} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(3*x + 2)^6)/(5*x + 3)^3,x)

[Out]

(920502*x)/390625 + (15547*log(x + 3/5))/1953125 - ((2134*x)/9765625 + 517/3906250)/((6*x)/5 + x^2 + 9/25) + (

189*x^2)/15625 - (16299*x^3)/3125 - (23571*x^4)/12500 + (17496*x^5)/3125 + (486*x^6)/125

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sympy [A] time = 0.14, size = 65, normalized size = 0.89 \begin {gather*} \frac {486 x^{6}}{125} + \frac {17496 x^{5}}{3125} - \frac {23571 x^{4}}{12500} - \frac {16299 x^{3}}{3125} + \frac {189 x^{2}}{15625} + \frac {920502 x}{390625} + \frac {- 4268 x - 2585}{19531250 x^{2} + 23437500 x + 7031250} + \frac {15547 \log {\left (5 x + 3 \right )}}{1953125} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(2+3*x)**6/(3+5*x)**3,x)

[Out]

486*x**6/125 + 17496*x**5/3125 - 23571*x**4/12500 - 16299*x**3/3125 + 189*x**2/15625 + 920502*x/390625 + (-426

8*x - 2585)/(19531250*x**2 + 23437500*x + 7031250) + 15547*log(5*x + 3)/1953125

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