3.16.6 \(\int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^3} \, dx\)

Optimal. Leaf size=80 \[ -\frac {54675 x^7}{56}-\frac {268515 x^6}{32}-\frac {2798631 x^5}{80}-\frac {12299769 x^4}{128}-\frac {25895367 x^3}{128}-\frac {190742391 x^2}{512}-\frac {48280011 x}{64}-\frac {389535839}{1024 (1-2 x)}+\frac {99648703}{2048 (1-2 x)^2}-\frac {84589631}{128} \log (1-2 x) \]

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Rubi [A]  time = 0.05, antiderivative size = 80, 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 {54675 x^7}{56}-\frac {268515 x^6}{32}-\frac {2798631 x^5}{80}-\frac {12299769 x^4}{128}-\frac {25895367 x^3}{128}-\frac {190742391 x^2}{512}-\frac {48280011 x}{64}-\frac {389535839}{1024 (1-2 x)}+\frac {99648703}{2048 (1-2 x)^2}-\frac {84589631}{128} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^7*(3 + 5*x)^2)/(1 - 2*x)^3,x]

[Out]

99648703/(2048*(1 - 2*x)^2) - 389535839/(1024*(1 - 2*x)) - (48280011*x)/64 - (190742391*x^2)/512 - (25895367*x
^3)/128 - (12299769*x^4)/128 - (2798631*x^5)/80 - (268515*x^6)/32 - (54675*x^7)/56 - (84589631*Log[1 - 2*x])/1
28

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 {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^3} \, dx &=\int \left (-\frac {48280011}{64}-\frac {190742391 x}{256}-\frac {77686101 x^2}{128}-\frac {12299769 x^3}{32}-\frac {2798631 x^4}{16}-\frac {805545 x^5}{16}-\frac {54675 x^6}{8}-\frac {99648703}{512 (-1+2 x)^3}-\frac {389535839}{512 (-1+2 x)^2}-\frac {84589631}{64 (-1+2 x)}\right ) \, dx\\ &=\frac {99648703}{2048 (1-2 x)^2}-\frac {389535839}{1024 (1-2 x)}-\frac {48280011 x}{64}-\frac {190742391 x^2}{512}-\frac {25895367 x^3}{128}-\frac {12299769 x^4}{128}-\frac {2798631 x^5}{80}-\frac {268515 x^6}{32}-\frac {54675 x^7}{56}-\frac {84589631}{128} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 71, normalized size = 0.89 \begin {gather*} -\frac {34992000 x^9+265744800 x^8+961797888 x^7+2265332832 x^6+4120214112 x^5+6962248440 x^4+15497514480 x^3-41720946264 x^2+17964456304 x+5921274170 (1-2 x)^2 \log (1-2 x)-1533057471}{8960 (1-2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^7*(3 + 5*x)^2)/(1 - 2*x)^3,x]

[Out]

-1/8960*(-1533057471 + 17964456304*x - 41720946264*x^2 + 15497514480*x^3 + 6962248440*x^4 + 4120214112*x^5 + 2
265332832*x^6 + 961797888*x^7 + 265744800*x^8 + 34992000*x^9 + 5921274170*(1 - 2*x)^2*Log[1 - 2*x])/(1 - 2*x)^
2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^7 (3+5 x)^2}{(1-2 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((2 + 3*x)^7*(3 + 5*x)^2)/(1 - 2*x)^3,x]

[Out]

IntegrateAlgebraic[((2 + 3*x)^7*(3 + 5*x)^2)/(1 - 2*x)^3, x]

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fricas [A]  time = 1.51, size = 77, normalized size = 0.96 \begin {gather*} -\frac {279936000 \, x^{9} + 2125958400 \, x^{8} + 7694383104 \, x^{7} + 18122662656 \, x^{6} + 32961712896 \, x^{5} + 55697987520 \, x^{4} + 123980115840 \, x^{3} - 189590514540 \, x^{2} + 47370193360 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 461405140 \, x + 23779804125}{71680 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)^2/(1-2*x)^3,x, algorithm="fricas")

[Out]

-1/71680*(279936000*x^9 + 2125958400*x^8 + 7694383104*x^7 + 18122662656*x^6 + 32961712896*x^5 + 55697987520*x^
4 + 123980115840*x^3 - 189590514540*x^2 + 47370193360*(4*x^2 - 4*x + 1)*log(2*x - 1) - 461405140*x + 237798041
25)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.22, size = 57, normalized size = 0.71 \begin {gather*} -\frac {54675}{56} \, x^{7} - \frac {268515}{32} \, x^{6} - \frac {2798631}{80} \, x^{5} - \frac {12299769}{128} \, x^{4} - \frac {25895367}{128} \, x^{3} - \frac {190742391}{512} \, x^{2} - \frac {48280011}{64} \, x + \frac {9058973 \, {\left (172 \, x - 75\right )}}{2048 \, {\left (2 \, x - 1\right )}^{2}} - \frac {84589631}{128} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)^2/(1-2*x)^3,x, algorithm="giac")

[Out]

-54675/56*x^7 - 268515/32*x^6 - 2798631/80*x^5 - 12299769/128*x^4 - 25895367/128*x^3 - 190742391/512*x^2 - 482
80011/64*x + 9058973/2048*(172*x - 75)/(2*x - 1)^2 - 84589631/128*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 61, normalized size = 0.76 \begin {gather*} -\frac {54675 x^{7}}{56}-\frac {268515 x^{6}}{32}-\frac {2798631 x^{5}}{80}-\frac {12299769 x^{4}}{128}-\frac {25895367 x^{3}}{128}-\frac {190742391 x^{2}}{512}-\frac {48280011 x}{64}-\frac {84589631 \ln \left (2 x -1\right )}{128}+\frac {99648703}{2048 \left (2 x -1\right )^{2}}+\frac {389535839}{1024 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^7*(5*x+3)^2/(1-2*x)^3,x)

[Out]

-54675/56*x^7-268515/32*x^6-2798631/80*x^5-12299769/128*x^4-25895367/128*x^3-190742391/512*x^2-48280011/64*x+9
9648703/2048/(2*x-1)^2+389535839/1024/(2*x-1)-84589631/128*ln(2*x-1)

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maxima [A]  time = 0.46, size = 61, normalized size = 0.76 \begin {gather*} -\frac {54675}{56} \, x^{7} - \frac {268515}{32} \, x^{6} - \frac {2798631}{80} \, x^{5} - \frac {12299769}{128} \, x^{4} - \frac {25895367}{128} \, x^{3} - \frac {190742391}{512} \, x^{2} - \frac {48280011}{64} \, x + \frac {9058973 \, {\left (172 \, x - 75\right )}}{2048 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {84589631}{128} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7*(3+5*x)^2/(1-2*x)^3,x, algorithm="maxima")

[Out]

-54675/56*x^7 - 268515/32*x^6 - 2798631/80*x^5 - 12299769/128*x^4 - 25895367/128*x^3 - 190742391/512*x^2 - 482
80011/64*x + 9058973/2048*(172*x - 75)/(4*x^2 - 4*x + 1) - 84589631/128*log(2*x - 1)

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mupad [B]  time = 1.06, size = 56, normalized size = 0.70 \begin {gather*} \frac {\frac {389535839\,x}{2048}-\frac {679422975}{8192}}{x^2-x+\frac {1}{4}}-\frac {84589631\,\ln \left (x-\frac {1}{2}\right )}{128}-\frac {48280011\,x}{64}-\frac {190742391\,x^2}{512}-\frac {25895367\,x^3}{128}-\frac {12299769\,x^4}{128}-\frac {2798631\,x^5}{80}-\frac {268515\,x^6}{32}-\frac {54675\,x^7}{56} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x + 2)^7*(5*x + 3)^2)/(2*x - 1)^3,x)

[Out]

((389535839*x)/2048 - 679422975/8192)/(x^2 - x + 1/4) - (84589631*log(x - 1/2))/128 - (48280011*x)/64 - (19074
2391*x^2)/512 - (25895367*x^3)/128 - (12299769*x^4)/128 - (2798631*x^5)/80 - (268515*x^6)/32 - (54675*x^7)/56

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sympy [A]  time = 0.16, size = 71, normalized size = 0.89 \begin {gather*} - \frac {54675 x^{7}}{56} - \frac {268515 x^{6}}{32} - \frac {2798631 x^{5}}{80} - \frac {12299769 x^{4}}{128} - \frac {25895367 x^{3}}{128} - \frac {190742391 x^{2}}{512} - \frac {48280011 x}{64} - \frac {679422975 - 1558143356 x}{8192 x^{2} - 8192 x + 2048} - \frac {84589631 \log {\left (2 x - 1 \right )}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**7*(3+5*x)**2/(1-2*x)**3,x)

[Out]

-54675*x**7/56 - 268515*x**6/32 - 2798631*x**5/80 - 12299769*x**4/128 - 25895367*x**3/128 - 190742391*x**2/512
- 48280011*x/64 - (679422975 - 1558143356*x)/(8192*x**2 - 8192*x + 2048) - 84589631*log(2*x - 1)/128

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