∫ (3+5x)2 ________________________

3.1569 \(\int \frac {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac {1936}{823543 (1-2 x)}-\frac {11264}{823543 (3 x+2)}-\frac {2090}{117649 (3 x+2)^2}-\frac {1364}{50421 (3 x+2)^3}-\frac {319}{9604 (3 x+2)^4}+\frac {22}{1715 (3 x+2)^5}-\frac {1}{882 (3 x+2)^6}-\frac {4048 \log (1-2 x)}{823543}+\frac {4048 \log (3 x+2)}{823543} \]

[Out]

1936/823543/(1-2*x)-1/882/(2+3*x)^6+22/1715/(2+3*x)^5-319/9604/(2+3*x)^4-1364/50421/(2+3*x)^3-2090/117649/(2+3

*x)^2-11264/823543/(2+3*x)-4048/823543*ln(1-2*x)+4048/823543*ln(2+3*x)

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Rubi [A] time = 0.05, 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} \[ \frac {1936}{823543 (1-2 x)}-\frac {11264}{823543 (3 x+2)}-\frac {2090}{117649 (3 x+2)^2}-\frac {1364}{50421 (3 x+2)^3}-\frac {319}{9604 (3 x+2)^4}+\frac {22}{1715 (3 x+2)^5}-\frac {1}{882 (3 x+2)^6}-\frac {4048 \log (1-2 x)}{823543}+\frac {4048 \log (3 x+2)}{823543} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1936/(823543*(1 - 2*x)) - 1/(882*(2 + 3*x)^6) + 22/(1715*(2 + 3*x)^5) - 319/(9604*(2 + 3*x)^4) - 1364/(50421*(

2 + 3*x)^3) - 2090/(117649*(2 + 3*x)^2) - 11264/(823543*(2 + 3*x)) - (4048*Log[1 - 2*x])/823543 + (4048*Log[2

+ 3*x])/823543

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 (2+3 x)^7} \, dx &=\int \left (\frac {3872}{823543 (-1+2 x)^2}-\frac {8096}{823543 (-1+2 x)}+\frac {1}{49 (2+3 x)^7}-\frac {66}{343 (2+3 x)^6}+\frac {957}{2401 (2+3 x)^5}+\frac {4092}{16807 (2+3 x)^4}+\frac {12540}{117649 (2+3 x)^3}+\frac {33792}{823543 (2+3 x)^2}+\frac {12144}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {1936}{823543 (1-2 x)}-\frac {1}{882 (2+3 x)^6}+\frac {22}{1715 (2+3 x)^5}-\frac {319}{9604 (2+3 x)^4}-\frac {1364}{50421 (2+3 x)^3}-\frac {2090}{117649 (2+3 x)^2}-\frac {11264}{823543 (2+3 x)}-\frac {4048 \log (1-2 x)}{823543}+\frac {4048 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A] time = 0.09, size = 69, normalized size = 0.70 \[ \frac {4 \left (-\frac {7 \left (177059520 x^6+604953360 x^5+795948120 x^4+459657990 x^3+48220029 x^2-60874336 x-18979078\right )}{16 (2 x-1) (3 x+2)^6}-45540 \log (1-2 x)+45540 \log (6 x+4)\right )}{37059435} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*((-7*(-18979078 - 60874336*x + 48220029*x^2 + 459657990*x^3 + 795948120*x^4 + 604953360*x^5 + 177059520*x^6

))/(16*(-1 + 2*x)*(2 + 3*x)^6) - 45540*Log[1 - 2*x] + 45540*Log[4 + 6*x]))/37059435

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fricas [A] time = 0.82, size = 140, normalized size = 1.43 \[ -\frac {1239416640 \, x^{6} + 4234673520 \, x^{5} + 5571636840 \, x^{4} + 3217605930 \, x^{3} + 337540203 \, x^{2} - 728640 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 728640 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 426120352 \, x - 132853546}{148237740 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/148237740*(1239416640*x^6 + 4234673520*x^5 + 5571636840*x^4 + 3217605930*x^3 + 337540203*x^2 - 728640*(1458

*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 728640*(1458*x^7 + 5103*x^6 + 68

04*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(2*x - 1) - 426120352*x - 132853546)/(1458*x^7 + 5103*x^6 + 6804

*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)

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giac [A] time = 0.86, size = 87, normalized size = 0.89 \[ -\frac {1936}{823543 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {407084454}{2 \, x - 1} + \frac {2053765665}{{\left (2 \, x - 1\right )}^{2}} + \frac {5220014100}{{\left (2 \, x - 1\right )}^{3}} + \frac {6680782500}{{\left (2 \, x - 1\right )}^{4}} + \frac {3440056760}{{\left (2 \, x - 1\right )}^{5}} + 32498901\right )}}{28824005 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{6}} + \frac {4048}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1936/823543/(2*x - 1) + 4/28824005*(407084454/(2*x - 1) + 2053765665/(2*x - 1)^2 + 5220014100/(2*x - 1)^3 + 6

680782500/(2*x - 1)^4 + 3440056760/(2*x - 1)^5 + 32498901)/(7/(2*x - 1) + 3)^6 + 4048/823543*log(abs(-7/(2*x -

1) - 3))

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maple [A] time = 0.01, size = 81, normalized size = 0.83 \[ -\frac {4048 \ln \left (2 x -1\right )}{823543}+\frac {4048 \ln \left (3 x +2\right )}{823543}-\frac {1}{882 \left (3 x +2\right )^{6}}+\frac {22}{1715 \left (3 x +2\right )^{5}}-\frac {319}{9604 \left (3 x +2\right )^{4}}-\frac {1364}{50421 \left (3 x +2\right )^{3}}-\frac {2090}{117649 \left (3 x +2\right )^{2}}-\frac {11264}{823543 \left (3 x +2\right )}-\frac {1936}{823543 \left (2 x -1\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/882/(3*x+2)^6+22/1715/(3*x+2)^5-319/9604/(3*x+2)^4-1364/50421/(3*x+2)^3-2090/117649/(3*x+2)^2-11264/823543/

(3*x+2)+4048/823543*ln(3*x+2)-1936/823543/(2*x-1)-4048/823543*ln(2*x-1)

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maxima [A] time = 0.49, size = 81, normalized size = 0.83 \[ -\frac {177059520 \, x^{6} + 604953360 \, x^{5} + 795948120 \, x^{4} + 459657990 \, x^{3} + 48220029 \, x^{2} - 60874336 \, x - 18979078}{21176820 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac {4048}{823543} \, \log \left (3 \, x + 2\right ) - \frac {4048}{823543} \, \log \left (2 \, x - 1\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21176820*(177059520*x^6 + 604953360*x^5 + 795948120*x^4 + 459657990*x^3 + 48220029*x^2 - 60874336*x - 18979

078)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 4048/823543*log(3*x + 2) - 4048/823

543*log(2*x - 1)

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mupad [B] time = 1.09, size = 71, normalized size = 0.72 \[ \frac {8096\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {2024\,x^6}{352947}+\frac {20746\,x^5}{1058841}+\frac {245663\,x^4}{9529569}+\frac {567479\,x^3}{38118276}+\frac {595309\,x^2}{381182760}-\frac {7609292\,x}{3859475445}-\frac {9489539}{15437901780}}{x^7+\frac {7\,x^6}{2}+\frac {14\,x^5}{3}+\frac {70\,x^4}{27}-\frac {56\,x^2}{81}-\frac {224\,x}{729}-\frac {32}{729}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8096*atanh((12*x)/7 + 1/7))/823543 - ((595309*x^2)/381182760 - (7609292*x)/3859475445 + (567479*x^3)/38118276

+ (245663*x^4)/9529569 + (20746*x^5)/1058841 + (2024*x^6)/352947 - 9489539/15437901780)/((70*x^4)/27 - (56*x^

2)/81 - (224*x)/729 + (14*x^5)/3 + (7*x^6)/2 + x^7 - 32/729)

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sympy [A] time = 0.21, size = 80, normalized size = 0.82 \[ \frac {- 177059520 x^{6} - 604953360 x^{5} - 795948120 x^{4} - 459657990 x^{3} - 48220029 x^{2} + 60874336 x + 18979078}{30875803560 x^{7} + 108065312460 x^{6} + 144087083280 x^{5} + 80048379600 x^{4} - 21346234560 x^{2} - 9487215360 x - 1355316480} - \frac {4048 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {4048 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-177059520*x**6 - 604953360*x**5 - 795948120*x**4 - 459657990*x**3 - 48220029*x**2 + 60874336*x + 18979078)/(

30875803560*x**7 + 108065312460*x**6 + 144087083280*x**5 + 80048379600*x**4 - 21346234560*x**2 - 9487215360*x

- 1355316480) - 4048*log(x - 1/2)/823543 + 4048*log(x + 2/3)/823543

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