∫ (3+5x)3 ________________________

3.15.46 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx\)

Optimal. Leaf size=87 \[ \frac {5324}{117649 (1-2 x)}-\frac {14520}{117649 (3 x+2)}-\frac {3267}{33614 (3 x+2)^2}+\frac {121}{2401 (3 x+2)^3}-\frac {101}{12348 (3 x+2)^4}+\frac {1}{2205 (3 x+2)^5}-\frac {45012 \log (1-2 x)}{823543}+\frac {45012 \log (3 x+2)}{823543} \]

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Rubi [A] time = 0.04, antiderivative size = 87, 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 {5324}{117649 (1-2 x)}-\frac {14520}{117649 (3 x+2)}-\frac {3267}{33614 (3 x+2)^2}+\frac {121}{2401 (3 x+2)^3}-\frac {101}{12348 (3 x+2)^4}+\frac {1}{2205 (3 x+2)^5}-\frac {45012 \log (1-2 x)}{823543}+\frac {45012 \log (3 x+2)}{823543} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6),x]

[Out]

5324/(117649*(1 - 2*x)) + 1/(2205*(2 + 3*x)^5) - 101/(12348*(2 + 3*x)^4) + 121/(2401*(2 + 3*x)^3) - 3267/(3361

4*(2 + 3*x)^2) - 14520/(117649*(2 + 3*x)) - (45012*Log[1 - 2*x])/823543 + (45012*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)^3}{(1-2 x)^2 (2+3 x)^6} \, dx &=\int \left (\frac {10648}{117649 (-1+2 x)^2}-\frac {90024}{823543 (-1+2 x)}-\frac {1}{147 (2+3 x)^6}+\frac {101}{1029 (2+3 x)^5}-\frac {1089}{2401 (2+3 x)^4}+\frac {9801}{16807 (2+3 x)^3}+\frac {43560}{117649 (2+3 x)^2}+\frac {135036}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {5324}{117649 (1-2 x)}+\frac {1}{2205 (2+3 x)^5}-\frac {101}{12348 (2+3 x)^4}+\frac {121}{2401 (2+3 x)^3}-\frac {3267}{33614 (2+3 x)^2}-\frac {14520}{117649 (2+3 x)}-\frac {45012 \log (1-2 x)}{823543}+\frac {45012 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A] time = 0.11, size = 64, normalized size = 0.74 \begin {gather*} \frac {2 \left (-\frac {7 \left (656274960 x^5+1804756140 x^4+1747028250 x^3+649342770 x^2+25985087 x-23684986\right )}{8 (2 x-1) (3 x+2)^5}-1012770 \log (1-2 x)+1012770 \log (6 x+4)\right )}{37059435} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6),x]

[Out]

(2*((-7*(-23684986 + 25985087*x + 649342770*x^2 + 1747028250*x^3 + 1804756140*x^4 + 656274960*x^5))/(8*(-1 + 2

*x)*(2 + 3*x)^5) - 1012770*Log[1 - 2*x] + 1012770*Log[4 + 6*x]))/37059435

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^6} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^6), x]

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fricas [A] time = 1.71, size = 135, normalized size = 1.55 \begin {gather*} -\frac {4593924720 \, x^{5} + 12633292980 \, x^{4} + 12229197750 \, x^{3} + 4545399390 \, x^{2} - 8102160 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 8102160 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) + 181895609 \, x - 165794902}{148237740 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/148237740*(4593924720*x^5 + 12633292980*x^4 + 12229197750*x^3 + 4545399390*x^2 - 8102160*(486*x^6 + 1377*x^

5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) + 8102160*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3

- 240*x^2 - 176*x - 32)*log(2*x - 1) + 181895609*x - 165794902)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 24

0*x^2 - 176*x - 32)

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giac [A] time = 0.96, size = 78, normalized size = 0.90 \begin {gather*} -\frac {5324}{117649 \, {\left (2 \, x - 1\right )}} + \frac {2 \, {\left (\frac {204418935}{2 \, x - 1} + \frac {740244225}{{\left (2 \, x - 1\right )}^{2}} + \frac {1185622375}{{\left (2 \, x - 1\right )}^{3}} + \frac {709135350}{{\left (2 \, x - 1\right )}^{4}} + 21049983\right )}}{4117715 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{5}} + \frac {45012}{823543} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="giac")

[Out]

-5324/117649/(2*x - 1) + 2/4117715*(204418935/(2*x - 1) + 740244225/(2*x - 1)^2 + 1185622375/(2*x - 1)^3 + 709

135350/(2*x - 1)^4 + 21049983)/(7/(2*x - 1) + 3)^5 + 45012/823543*log(abs(-7/(2*x - 1) - 3))

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maple [A] time = 0.01, size = 72, normalized size = 0.83 \begin {gather*} -\frac {45012 \ln \left (2 x -1\right )}{823543}+\frac {45012 \ln \left (3 x +2\right )}{823543}+\frac {1}{2205 \left (3 x +2\right )^{5}}-\frac {101}{12348 \left (3 x +2\right )^{4}}+\frac {121}{2401 \left (3 x +2\right )^{3}}-\frac {3267}{33614 \left (3 x +2\right )^{2}}-\frac {14520}{117649 \left (3 x +2\right )}-\frac {5324}{117649 \left (2 x -1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)^2/(3*x+2)^6,x)

[Out]

1/2205/(3*x+2)^5-101/12348/(3*x+2)^4+121/2401/(3*x+2)^3-3267/33614/(3*x+2)^2-14520/117649/(3*x+2)+45012/823543

*ln(3*x+2)-5324/117649/(2*x-1)-45012/823543*ln(2*x-1)

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maxima [A] time = 0.55, size = 76, normalized size = 0.87 \begin {gather*} -\frac {656274960 \, x^{5} + 1804756140 \, x^{4} + 1747028250 \, x^{3} + 649342770 \, x^{2} + 25985087 \, x - 23684986}{21176820 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac {45012}{823543} \, \log \left (3 \, x + 2\right ) - \frac {45012}{823543} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^6,x, algorithm="maxima")

[Out]

-1/21176820*(656274960*x^5 + 1804756140*x^4 + 1747028250*x^3 + 649342770*x^2 + 25985087*x - 23684986)/(486*x^6

+ 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 45012/823543*log(3*x + 2) - 45012/823543*log(2*x -

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mupad [B] time = 0.04, size = 66, normalized size = 0.76 \begin {gather*} \frac {90024\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {7502\,x^5}{117649}+\frac {41261\,x^4}{235298}+\frac {2156825\,x^3}{12706092}+\frac {21644759\,x^2}{343064484}+\frac {25985087\,x}{10291934520}-\frac {11842493}{5145967260}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/((2*x - 1)^2*(3*x + 2)^6),x)

[Out]

(90024*atanh((12*x)/7 + 1/7))/823543 - ((25985087*x)/10291934520 + (21644759*x^2)/343064484 + (2156825*x^3)/12

706092 + (41261*x^4)/235298 + (7502*x^5)/117649 - 11842493/5145967260)/((20*x^3)/27 - (40*x^2)/81 - (88*x)/243

+ (25*x^4)/9 + (17*x^5)/6 + x^6 - 16/243)

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sympy [A] time = 0.20, size = 75, normalized size = 0.86 \begin {gather*} \frac {- 656274960 x^{5} - 1804756140 x^{4} - 1747028250 x^{3} - 649342770 x^{2} - 25985087 x + 23684986}{10291934520 x^{6} + 29160481140 x^{5} + 28588707000 x^{4} + 7623655200 x^{3} - 5082436800 x^{2} - 3727120320 x - 677658240} - \frac {45012 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {45012 \log {\left (x + \frac {2}{3} \right )}}{823543} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)**2/(2+3*x)**6,x)

[Out]

(-656274960*x**5 - 1804756140*x**4 - 1747028250*x**3 - 649342770*x**2 - 25985087*x + 23684986)/(10291934520*x*

*6 + 29160481140*x**5 + 28588707000*x**4 + 7623655200*x**3 - 5082436800*x**2 - 3727120320*x - 677658240) - 450

12*log(x - 1/2)/823543 + 45012*log(x + 2/3)/823543

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