∫ (3+5x)2 ______________________

3.14.28 \(\int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx\)

Optimal. Leaf size=98 \[ -\frac {1936}{823543 (3 x+2)}-\frac {484}{117649 (3 x+2)^2}-\frac {484}{50421 (3 x+2)^3}-\frac {121}{4802 (3 x+2)^4}-\frac {121}{1715 (3 x+2)^5}+\frac {34}{1323 (3 x+2)^6}-\frac {1}{441 (3 x+2)^7}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (3 x+2)}{5764801} \]

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Rubi [A] time = 0.04, antiderivative size = 98, 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 {1936}{823543 (3 x+2)}-\frac {484}{117649 (3 x+2)^2}-\frac {484}{50421 (3 x+2)^3}-\frac {121}{4802 (3 x+2)^4}-\frac {121}{1715 (3 x+2)^5}+\frac {34}{1323 (3 x+2)^6}-\frac {1}{441 (3 x+2)^7}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (3 x+2)}{5764801} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

-1/(441*(2 + 3*x)^7) + 34/(1323*(2 + 3*x)^6) - 121/(1715*(2 + 3*x)^5) - 121/(4802*(2 + 3*x)^4) - 484/(50421*(2

+ 3*x)^3) - 484/(117649*(2 + 3*x)^2) - 1936/(823543*(2 + 3*x)) - (3872*Log[1 - 2*x])/5764801 + (3872*Log[2 +

3*x])/5764801

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 {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx &=\int \left (-\frac {7744}{5764801 (-1+2 x)}+\frac {1}{21 (2+3 x)^8}-\frac {68}{147 (2+3 x)^7}+\frac {363}{343 (2+3 x)^6}+\frac {726}{2401 (2+3 x)^5}+\frac {1452}{16807 (2+3 x)^4}+\frac {2904}{117649 (2+3 x)^3}+\frac {5808}{823543 (2+3 x)^2}+\frac {11616}{5764801 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{441 (2+3 x)^7}+\frac {34}{1323 (2+3 x)^6}-\frac {121}{1715 (2+3 x)^5}-\frac {121}{4802 (2+3 x)^4}-\frac {484}{50421 (2+3 x)^3}-\frac {484}{117649 (2+3 x)^2}-\frac {1936}{823543 (2+3 x)}-\frac {3872 \log (1-2 x)}{5764801}+\frac {3872 \log (2+3 x)}{5764801}\\ \end {align*}

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Mathematica [A] time = 0.11, size = 62, normalized size = 0.63 \begin {gather*} \frac {8 \left (-\frac {7 \left (381062880 x^6+1746538200 x^5+3454264440 x^4+3858408675 x^3+2692491516 x^2+1098354408 x+193528666\right )}{16 (3 x+2)^7}-65340 \log (1-2 x)+65340 \log (6 x+4)\right )}{778248135} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

(8*((-7*(193528666 + 1098354408*x + 2692491516*x^2 + 3858408675*x^3 + 3454264440*x^4 + 1746538200*x^5 + 381062

880*x^6))/(16*(2 + 3*x)^7) - 65340*Log[1 - 2*x] + 65340*Log[4 + 6*x]))/778248135

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^2}{(1-2 x) (2+3 x)^8} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^8), x]

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fricas [A] time = 1.25, size = 155, normalized size = 1.58 \begin {gather*} -\frac {2667440160 \, x^{6} + 12225767400 \, x^{5} + 24179851080 \, x^{4} + 27008860725 \, x^{3} + 18847440612 \, x^{2} - 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 1045440 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 7688480856 \, x + 1354700662}{1556496270 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/1556496270*(2667440160*x^6 + 12225767400*x^5 + 24179851080*x^4 + 27008860725*x^3 + 18847440612*x^2 - 104544

0*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 1045440*

(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(2*x - 1) + 7688480856

*x + 1354700662)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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giac [A] time = 0.94, size = 58, normalized size = 0.59 \begin {gather*} -\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (3 \, x + 2\right )}^{7}} + \frac {3872}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3872}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675*x^3 + 2692491516*x^2 + 1098354408*x

+ 193528666)/(3*x + 2)^7 + 3872/5764801*log(abs(3*x + 2)) - 3872/5764801*log(abs(2*x - 1))

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maple [A] time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {3872 \ln \left (2 x -1\right )}{5764801}+\frac {3872 \ln \left (3 x +2\right )}{5764801}-\frac {1}{441 \left (3 x +2\right )^{7}}+\frac {34}{1323 \left (3 x +2\right )^{6}}-\frac {121}{1715 \left (3 x +2\right )^{5}}-\frac {121}{4802 \left (3 x +2\right )^{4}}-\frac {484}{50421 \left (3 x +2\right )^{3}}-\frac {484}{117649 \left (3 x +2\right )^{2}}-\frac {1936}{823543 \left (3 x +2\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^2/(1-2*x)/(3*x+2)^8,x)

[Out]

-1/441/(3*x+2)^7+34/1323/(3*x+2)^6-121/1715/(3*x+2)^5-121/4802/(3*x+2)^4-484/50421/(3*x+2)^3-484/117649/(3*x+2

)^2-1936/823543/(3*x+2)+3872/5764801*ln(3*x+2)-3872/5764801*ln(2*x-1)

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maxima [A] time = 0.65, size = 86, normalized size = 0.88 \begin {gather*} -\frac {381062880 \, x^{6} + 1746538200 \, x^{5} + 3454264440 \, x^{4} + 3858408675 \, x^{3} + 2692491516 \, x^{2} + 1098354408 \, x + 193528666}{222356610 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {3872}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {3872}{5764801} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/222356610*(381062880*x^6 + 1746538200*x^5 + 3454264440*x^4 + 3858408675*x^3 + 2692491516*x^2 + 1098354408*x

+ 193528666)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 3872/5764

801*log(3*x + 2) - 3872/5764801*log(2*x - 1)

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mupad [B] time = 0.05, size = 76, normalized size = 0.78 \begin {gather*} \frac {7744\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {1936\,x^6}{2470629}+\frac {26620\,x^5}{7411887}+\frac {473836\,x^4}{66706983}+\frac {3175645\,x^3}{400241898}+\frac {1846702\,x^2}{333534915}+\frac {183059068\,x}{81048984345}+\frac {96764333}{243146953035}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^2/((2*x - 1)*(3*x + 2)^8),x)

[Out]

(7744*atanh((12*x)/7 + 1/7))/5764801 - ((183059068*x)/81048984345 + (1846702*x^2)/333534915 + (3175645*x^3)/40

0241898 + (473836*x^4)/66706983 + (26620*x^5)/7411887 + (1936*x^6)/2470629 + 96764333/243146953035)/((448*x)/7

29 + (224*x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3 + (14*x^6)/3 + x^7 + 128/2187)

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sympy [A] time = 0.23, size = 85, normalized size = 0.87 \begin {gather*} - \frac {381062880 x^{6} + 1746538200 x^{5} + 3454264440 x^{4} + 3858408675 x^{3} + 2692491516 x^{2} + 1098354408 x + 193528666}{486293906070 x^{7} + 2269371561660 x^{6} + 4538743123320 x^{5} + 5043047914800 x^{4} + 3362031943200 x^{3} + 1344812777280 x^{2} + 298847283840 x + 28461646080} - \frac {3872 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {3872 \log {\left (x + \frac {2}{3} \right )}}{5764801} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2/(1-2*x)/(2+3*x)**8,x)

[Out]

-(381062880*x**6 + 1746538200*x**5 + 3454264440*x**4 + 3858408675*x**3 + 2692491516*x**2 + 1098354408*x + 1935

28666)/(486293906070*x**7 + 2269371561660*x**6 + 4538743123320*x**5 + 5043047914800*x**4 + 3362031943200*x**3

+ 1344812777280*x**2 + 298847283840*x + 28461646080) - 3872*log(x - 1/2)/5764801 + 3872*log(x + 2/3)/5764801

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