3.1420 \(\int \frac {(1-2 x)^3 (2+3 x)^5}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=73 \[ -\frac {324 x^6}{125}-\frac {324 x^5}{3125}+\frac {22977 x^4}{6250}-\frac {393 x^3}{625}-\frac {62097 x^2}{31250}+\frac {424432 x}{390625}-\frac {19239}{1953125 (5 x+3)}-\frac {1331}{3906250 (5 x+3)^2}+\frac {109032 \log (5 x+3)}{1953125} \]

[Out]

424432/390625*x-62097/31250*x^2-393/625*x^3+22977/6250*x^4-324/3125*x^5-324/125*x^6-1331/3906250/(3+5*x)^2-192
39/1953125/(3+5*x)+109032/1953125*ln(3+5*x)

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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} \[ -\frac {324 x^6}{125}-\frac {324 x^5}{3125}+\frac {22977 x^4}{6250}-\frac {393 x^3}{625}-\frac {62097 x^2}{31250}+\frac {424432 x}{390625}-\frac {19239}{1953125 (5 x+3)}-\frac {1331}{3906250 (5 x+3)^2}+\frac {109032 \log (5 x+3)}{1953125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(424432*x)/390625 - (62097*x^2)/31250 - (393*x^3)/625 + (22977*x^4)/6250 - (324*x^5)/3125 - (324*x^6)/125 - 13
31/(3906250*(3 + 5*x)^2) - 19239/(1953125*(3 + 5*x)) + (109032*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)^3 (2+3 x)^5}{(3+5 x)^3} \, dx &=\int \left (\frac {424432}{390625}-\frac {62097 x}{15625}-\frac {1179 x^2}{625}+\frac {45954 x^3}{3125}-\frac {324 x^4}{625}-\frac {1944 x^5}{125}+\frac {1331}{390625 (3+5 x)^3}+\frac {19239}{390625 (3+5 x)^2}+\frac {109032}{390625 (3+5 x)}\right ) \, dx\\ &=\frac {424432 x}{390625}-\frac {62097 x^2}{31250}-\frac {393 x^3}{625}+\frac {22977 x^4}{6250}-\frac {324 x^5}{3125}-\frac {324 x^6}{125}-\frac {1331}{3906250 (3+5 x)^2}-\frac {19239}{1953125 (3+5 x)}+\frac {109032 \log (3+5 x)}{1953125}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.93 \[ \frac {-1265625000 x^8-1569375000 x^7+1278703125 x^6+1828837500 x^5-692475000 x^4-744310000 x^3+711123525 x^2+698557830 x+1090320 (5 x+3)^2 \log (6 (5 x+3))[phone]}{19531250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(151973789 + 698557830*x + 711123525*x^2 - 744310000*x^3 - 692475000*x^4 + 1828837500*x^5 + 1278703125*x^6 - 1
569375000*x^7 - 1265625000*x^8 + 1090320*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(19531250*(3 + 5*x)^2)

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fricas [A]  time = 0.57, size = 72, normalized size = 0.99 \[ -\frac {253125000 \, x^{8} + 313875000 \, x^{7} - 255740625 \, x^{6} - 365767500 \, x^{5} + 138495000 \, x^{4} + 148862000 \, x^{3} - 57470475 \, x^{2} - 218064 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 38006490 \, x + 116765}{3906250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3906250*(253125000*x^8 + 313875000*x^7 - 255740625*x^6 - 365767500*x^5 + 138495000*x^4 + 148862000*x^3 - 57
470475*x^2 - 218064*(25*x^2 + 30*x + 9)*log(5*x + 3) - 38006490*x + 116765)/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.09, size = 52, normalized size = 0.71 \[ -\frac {324}{125} \, x^{6} - \frac {324}{3125} \, x^{5} + \frac {22977}{6250} \, x^{4} - \frac {393}{625} \, x^{3} - \frac {62097}{31250} \, x^{2} + \frac {424432}{390625} \, x - \frac {121 \, {\left (318 \, x + 193\right )}}{781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {109032}{1953125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-324/125*x^6 - 324/3125*x^5 + 22977/6250*x^4 - 393/625*x^3 - 62097/31250*x^2 + 424432/390625*x - 121/781250*(3
18*x + 193)/(5*x + 3)^2 + 109032/1953125*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 56, normalized size = 0.77 \[ -\frac {324 x^{6}}{125}-\frac {324 x^{5}}{3125}+\frac {22977 x^{4}}{6250}-\frac {393 x^{3}}{625}-\frac {62097 x^{2}}{31250}+\frac {424432 x}{390625}+\frac {109032 \ln \left (5 x +3\right )}{1953125}-\frac {1331}{3906250 \left (5 x +3\right )^{2}}-\frac {19239}{1953125 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

424432/390625*x-62097/31250*x^2-393/625*x^3+22977/6250*x^4-324/3125*x^5-324/125*x^6-1331/3906250/(5*x+3)^2-192
39/1953125/(5*x+3)+109032/1953125*ln(5*x+3)

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maxima [A]  time = 0.50, size = 56, normalized size = 0.77 \[ -\frac {324}{125} \, x^{6} - \frac {324}{3125} \, x^{5} + \frac {22977}{6250} \, x^{4} - \frac {393}{625} \, x^{3} - \frac {62097}{31250} \, x^{2} + \frac {424432}{390625} \, x - \frac {121 \, {\left (318 \, x + 193\right )}}{781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {109032}{1953125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-324/125*x^6 - 324/3125*x^5 + 22977/6250*x^4 - 393/625*x^3 - 62097/31250*x^2 + 424432/390625*x - 121/781250*(3
18*x + 193)/(25*x^2 + 30*x + 9) + 109032/1953125*log(5*x + 3)

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mupad [B]  time = 0.04, size = 52, normalized size = 0.71 \[ \frac {424432\,x}{390625}+\frac {109032\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {\frac {19239\,x}{9765625}+\frac {23353}{19531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {62097\,x^2}{31250}-\frac {393\,x^3}{625}+\frac {22977\,x^4}{6250}-\frac {324\,x^5}{3125}-\frac {324\,x^6}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(424432*x)/390625 + (109032*log(x + 3/5))/1953125 - ((19239*x)/9765625 + 23353/19531250)/((6*x)/5 + x^2 + 9/25
) - (62097*x^2)/31250 - (393*x^3)/625 + (22977*x^4)/6250 - (324*x^5)/3125 - (324*x^6)/125

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sympy [A]  time = 0.15, size = 63, normalized size = 0.86 \[ - \frac {324 x^{6}}{125} - \frac {324 x^{5}}{3125} + \frac {22977 x^{4}}{6250} - \frac {393 x^{3}}{625} - \frac {62097 x^{2}}{31250} + \frac {424432 x}{390625} - \frac {38478 x + 23353}{19531250 x^{2} + 23437500 x + 7031250} + \frac {109032 \log {\left (5 x + 3 \right )}}{1953125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-324*x**6/125 - 324*x**5/3125 + 22977*x**4/6250 - 393*x**3/625 - 62097*x**2/31250 + 424432*x/390625 - (38478*x
+ 23353)/(19531250*x**2 + 23437500*x + 7031250) + 109032*log(5*x + 3)/1953125

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