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

Optimal. Leaf size=66 \[ \frac {972 x^5}{625}+\frac {648 x^4}{625}-\frac {5499 x^3}{3125}-\frac {5301 x^2}{6250}+\frac {17796 x}{15625}-\frac {1771}{390625 (5 x+3)}-\frac {121}{781250 (5 x+3)^2}+\frac {10234 \log (5 x+3)}{390625} \]

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Rubi [A]  time = 0.03, antiderivative size = 66, 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 {972 x^5}{625}+\frac {648 x^4}{625}-\frac {5499 x^3}{3125}-\frac {5301 x^2}{6250}+\frac {17796 x}{15625}-\frac {1771}{390625 (5 x+3)}-\frac {121}{781250 (5 x+3)^2}+\frac {10234 \log (5 x+3)}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(17796*x)/15625 - (5301*x^2)/6250 - (5499*x^3)/3125 + (648*x^4)/625 + (972*x^5)/625 - 121/(781250*(3 + 5*x)^2)
- 1771/(390625*(3 + 5*x)) + (10234*Log[3 + 5*x])/390625

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)^5}{(3+5 x)^3} \, dx &=\int \left (\frac {17796}{15625}-\frac {5301 x}{3125}-\frac {16497 x^2}{3125}+\frac {2592 x^3}{625}+\frac {972 x^4}{125}+\frac {121}{78125 (3+5 x)^3}+\frac {1771}{78125 (3+5 x)^2}+\frac {10234}{78125 (3+5 x)}\right ) \, dx\\ &=\frac {17796 x}{15625}-\frac {5301 x^2}{6250}-\frac {5499 x^3}{3125}+\frac {648 x^4}{625}+\frac {972 x^5}{625}-\frac {121}{781250 (3+5 x)^2}-\frac {1771}{390625 (3+5 x)}+\frac {10234 \log (3+5 x)}{390625}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 63, normalized size = 0.95 \begin {gather*} \frac {151875000 x^7+283500000 x^6+4331250 x^5-252590625 x^4-50032500 x^3+161774550 x^2+109699660 x+102340 (5 x+3)^2 \log (6 (5 x+3))[phone]}{3906250 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20870428 + 109699660*x + 161774550*x^2 - 50032500*x^3 - 252590625*x^4 + 4331250*x^5 + 283500000*x^6 + 1518750
00*x^7 + 102340*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(3906250*(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)^5}{(3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 67, normalized size = 1.02 \begin {gather*} \frac {30375000 \, x^{7} + 56700000 \, x^{6} + 866250 \, x^{5} - 50518125 \, x^{4} - 10006500 \, x^{3} + 20730375 \, x^{2} + 20468 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 7990490 \, x - 10747}{781250 \, {\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)^5/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/781250*(30375000*x^7 + 56700000*x^6 + 866250*x^5 - 50518125*x^4 - 10006500*x^3 + 20730375*x^2 + 20468*(25*x^
2 + 30*x + 9)*log(5*x + 3) + 7990490*x - 10747)/(25*x^2 + 30*x + 9)

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giac [A]  time = 0.86, size = 47, normalized size = 0.71 \begin {gather*} \frac {972}{625} \, x^{5} + \frac {648}{625} \, x^{4} - \frac {5499}{3125} \, x^{3} - \frac {5301}{6250} \, x^{2} + \frac {17796}{15625} \, x - \frac {11 \, {\left (1610 \, x + 977\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {10234}{390625} \, \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)^5/(3+5*x)^3,x, algorithm="giac")

[Out]

972/625*x^5 + 648/625*x^4 - 5499/3125*x^3 - 5301/6250*x^2 + 17796/15625*x - 11/781250*(1610*x + 977)/(5*x + 3)
^2 + 10234/390625*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 51, normalized size = 0.77 \begin {gather*} \frac {972 x^{5}}{625}+\frac {648 x^{4}}{625}-\frac {5499 x^{3}}{3125}-\frac {5301 x^{2}}{6250}+\frac {17796 x}{15625}+\frac {10234 \ln \left (5 x +3\right )}{390625}-\frac {121}{781250 \left (5 x +3\right )^{2}}-\frac {1771}{390625 \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)^5/(5*x+3)^3,x)

[Out]

17796/15625*x-5301/6250*x^2-5499/3125*x^3+648/625*x^4+972/625*x^5-121/781250/(5*x+3)^2-1771/390625/(5*x+3)+102
34/390625*ln(5*x+3)

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maxima [A]  time = 0.50, size = 51, normalized size = 0.77 \begin {gather*} \frac {972}{625} \, x^{5} + \frac {648}{625} \, x^{4} - \frac {5499}{3125} \, x^{3} - \frac {5301}{6250} \, x^{2} + \frac {17796}{15625} \, x - \frac {11 \, {\left (1610 \, x + 977\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {10234}{390625} \, \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)^5/(3+5*x)^3,x, algorithm="maxima")

[Out]

972/625*x^5 + 648/625*x^4 - 5499/3125*x^3 - 5301/6250*x^2 + 17796/15625*x - 11/781250*(1610*x + 977)/(25*x^2 +
30*x + 9) + 10234/390625*log(5*x + 3)

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mupad [B]  time = 0.03, size = 47, normalized size = 0.71 \begin {gather*} \frac {17796\,x}{15625}+\frac {10234\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {\frac {1771\,x}{1953125}+\frac {10747}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {5301\,x^2}{6250}-\frac {5499\,x^3}{3125}+\frac {648\,x^4}{625}+\frac {972\,x^5}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(17796*x)/15625 + (10234*log(x + 3/5))/390625 - ((1771*x)/1953125 + 10747/19531250)/((6*x)/5 + x^2 + 9/25) - (
5301*x^2)/6250 - (5499*x^3)/3125 + (648*x^4)/625 + (972*x^5)/625

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sympy [A]  time = 0.14, size = 58, normalized size = 0.88 \begin {gather*} \frac {972 x^{5}}{625} + \frac {648 x^{4}}{625} - \frac {5499 x^{3}}{3125} - \frac {5301 x^{2}}{6250} + \frac {17796 x}{15625} + \frac {- 17710 x - 10747}{19531250 x^{2} + 23437500 x + 7031250} + \frac {10234 \log {\left (5 x + 3 \right )}}{390625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

972*x**5/625 + 648*x**4/625 - 5499*x**3/3125 - 5301*x**2/6250 + 17796*x/15625 + (-17710*x - 10747)/(19531250*x
**2 + 23437500*x + 7031250) + 10234*log(5*x + 3)/390625

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