∫ (2+3x)6 ______________________

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

Optimal. Leaf size=58 \[ -\frac {729 x^4}{200}-\frac {8019 x^3}{500}-\frac {335097 x^2}{10000}-\frac {2682909 x}{50000}-\frac {1}{171875 (5 x+3)}-\frac {117649 \log (1-2 x)}{3872}+\frac {8 \log (5 x+3)}{75625} \]

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Rubi [A] time = 0.03, antiderivative size = 58, 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 {729 x^4}{200}-\frac {8019 x^3}{500}-\frac {335097 x^2}{10000}-\frac {2682909 x}{50000}-\frac {1}{171875 (5 x+3)}-\frac {117649 \log (1-2 x)}{3872}+\frac {8 \log (5 x+3)}{75625} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2682909*x)/50000 - (335097*x^2)/10000 - (8019*x^3)/500 - (729*x^4)/200 - 1/(171875*(3 + 5*x)) - (117649*Log[

1 - 2*x])/3872 + (8*Log[3 + 5*x])/75625

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)^6}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac {2682909}{50000}-\frac {335097 x}{5000}-\frac {24057 x^2}{500}-\frac {729 x^3}{50}-\frac {117649}{1936 (-1+2 x)}+\frac {1}{34375 (3+5 x)^2}+\frac {8}{15125 (3+5 x)}\right ) \, dx\\ &=-\frac {2682909 x}{50000}-\frac {335097 x^2}{10000}-\frac {8019 x^3}{500}-\frac {729 x^4}{200}-\frac {1}{171875 (3+5 x)}-\frac {117649 \log (1-2 x)}{3872}+\frac {8 \log (3+5 x)}{75625}\\ \end {align*}

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Mathematica [A] time = 0.05, size = 54, normalized size = 0.93 \begin {gather*} \frac {-80190000 x^4-352836000 x^3-737213400 x^2-1180479960 x-\frac {128}{5 x+3}+823659705}{22000000}-\frac {117649 \log (1-2 x)}{3872}+\frac {8 \log (10 x+6)}{75625} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(823659705 - 1180479960*x - 737213400*x^2 - 352836000*x^3 - 80190000*x^4 - 128/(3 + 5*x))/22000000 - (117649*L

og[1 - 2*x])/3872 + (8*Log[6 + 10*x])/75625

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

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.36, size = 60, normalized size = 1.03 \begin {gather*} -\frac {1102612500 \, x^{5} + 5513062500 \, x^{4} + 13047581250 \, x^{3} + 22313610000 \, x^{2} - 6400 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1838265625 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 9738959670 \, x + 352}{60500000 \, {\left (5 \, x + 3\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60500000*(1102612500*x^5 + 5513062500*x^4 + 13047581250*x^3 + 22313610000*x^2 - 6400*(5*x + 3)*log(5*x + 3)

+ 1838265625*(5*x + 3)*log(2*x - 1) + 9738959670*x + 352)/(5*x + 3)

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giac [A] time = 1.01, size = 81, normalized size = 1.40 \begin {gather*} -\frac {27}{250000} \, {\left (5 \, x + 3\right )}^{4} {\left (\frac {540}{5 \, x + 3} + \frac {4635}{{\left (5 \, x + 3\right )}^{2}} + \frac {51145}{{\left (5 \, x + 3\right )}^{3}} + 54\right )} - \frac {1}{171875 \, {\left (5 \, x + 3\right )}} + \frac {607689}{20000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {117649}{3872} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/250000*(5*x + 3)^4*(540/(5*x + 3) + 4635/(5*x + 3)^2 + 51145/(5*x + 3)^3 + 54) - 1/171875/(5*x + 3) + 6076

89/20000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 117649/3872*log(abs(-11/(5*x + 3) + 2))

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maple [A] time = 0.01, size = 45, normalized size = 0.78 \begin {gather*} -\frac {729 x^{4}}{200}-\frac {8019 x^{3}}{500}-\frac {335097 x^{2}}{10000}-\frac {2682909 x}{50000}-\frac {117649 \ln \left (2 x -1\right )}{3872}+\frac {8 \ln \left (5 x +3\right )}{75625}-\frac {1}{171875 \left (5 x +3\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/200*x^4-8019/500*x^3-335097/10000*x^2-2682909/50000*x-1/171875/(5*x+3)+8/75625*ln(5*x+3)-117649/3872*ln(2

*x-1)

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maxima [A] time = 0.67, size = 44, normalized size = 0.76 \begin {gather*} -\frac {729}{200} \, x^{4} - \frac {8019}{500} \, x^{3} - \frac {335097}{10000} \, x^{2} - \frac {2682909}{50000} \, x - \frac {1}{171875 \, {\left (5 \, x + 3\right )}} + \frac {8}{75625} \, \log \left (5 \, x + 3\right ) - \frac {117649}{3872} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/200*x^4 - 8019/500*x^3 - 335097/10000*x^2 - 2682909/50000*x - 1/171875/(5*x + 3) + 8/75625*log(5*x + 3) -

117649/3872*log(2*x - 1)

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mupad [B] time = 1.10, size = 40, normalized size = 0.69 \begin {gather*} \frac {8\,\ln \left (x+\frac {3}{5}\right )}{75625}-\frac {117649\,\ln \left (x-\frac {1}{2}\right )}{3872}-\frac {2682909\,x}{50000}-\frac {1}{859375\,\left (x+\frac {3}{5}\right )}-\frac {335097\,x^2}{10000}-\frac {8019\,x^3}{500}-\frac {729\,x^4}{200} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*log(x + 3/5))/75625 - (117649*log(x - 1/2))/3872 - (2682909*x)/50000 - 1/(859375*(x + 3/5)) - (335097*x^2)/

10000 - (8019*x^3)/500 - (729*x^4)/200

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sympy [A] time = 0.16, size = 51, normalized size = 0.88 \begin {gather*} - \frac {729 x^{4}}{200} - \frac {8019 x^{3}}{500} - \frac {335097 x^{2}}{10000} - \frac {2682909 x}{50000} - \frac {117649 \log {\left (x - \frac {1}{2} \right )}}{3872} + \frac {8 \log {\left (x + \frac {3}{5} \right )}}{75625} - \frac {1}{859375 x + 515625} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729*x**4/200 - 8019*x**3/500 - 335097*x**2/10000 - 2682909*x/50000 - 117649*log(x - 1/2)/3872 + 8*log(x + 3/5

)/75625 - 1/(859375*x + 515625)

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