∫ (2+3x)5 ________________________

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

Optimal. Leaf size=59 \[ \frac {243 x}{500}+\frac {16807}{10648 (1-2 x)}-\frac {169}{831875 (5 x+3)}-\frac {1}{151250 (5 x+3)^2}+\frac {36015 \log (1-2 x)}{29282}+\frac {11562 \log (5 x+3)}{9150625} \]

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Rubi [A] time = 0.03, antiderivative size = 59, 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 {243 x}{500}+\frac {16807}{10648 (1-2 x)}-\frac {169}{831875 (5 x+3)}-\frac {1}{151250 (5 x+3)^2}+\frac {36015 \log (1-2 x)}{29282}+\frac {11562 \log (5 x+3)}{9150625} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

16807/(10648*(1 - 2*x)) + (243*x)/500 - 1/(151250*(3 + 5*x)^2) - 169/(831875*(3 + 5*x)) + (36015*Log[1 - 2*x])

/29282 + (11562*Log[3 + 5*x])/9150625

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)^5}{(1-2 x)^2 (3+5 x)^3} \, dx &=\int \left (\frac {243}{500}+\frac {16807}{5324 (-1+2 x)^2}+\frac {36015}{14641 (-1+2 x)}+\frac {1}{15125 (3+5 x)^3}+\frac {169}{166375 (3+5 x)^2}+\frac {11562}{1830125 (3+5 x)}\right ) \, dx\\ &=\frac {16807}{10648 (1-2 x)}+\frac {243 x}{500}-\frac {1}{151250 (3+5 x)^2}-\frac {169}{831875 (3+5 x)}+\frac {36015 \log (1-2 x)}{29282}+\frac {11562 \log (3+5 x)}{9150625}\\ \end {align*}

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Mathematica [A] time = 0.06, size = 55, normalized size = 0.93 \begin {gather*} \frac {17788815 (2 x-1)+\frac {115548125}{1-2 x}-\frac {14872}{5 x+3}-\frac {484}{(5 x+3)^2}+90037500 \log (1-2 x)+92496 \log (10 x+6)}{73205000} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(115548125/(1 - 2*x) + 17788815*(-1 + 2*x) - 484/(3 + 5*x)^2 - 14872/(3 + 5*x) + 90037500*Log[1 - 2*x] + 92496

*Log[6 + 10*x])/73205000

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

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.36, size = 85, normalized size = 1.44 \begin {gather*} \frac {1778881500 \, x^{4} + 1245217050 \, x^{3} - 3315783405 \, x^{2} + 92496 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 90037500 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 3786658260 \, x - 1039888025}{73205000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/73205000*(1778881500*x^4 + 1245217050*x^3 - 3315783405*x^2 + 92496*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3)

+ 90037500*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) - 3786658260*x - 1039888025)/(50*x^3 + 35*x^2 - 12*x - 9

)

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giac [A] time = 1.20, size = 83, normalized size = 1.41 \begin {gather*} \frac {{\left (2 \, x - 1\right )} {\left (\frac {391367530}{2 \, x - 1} + \frac {430519419}{{\left (2 \, x - 1\right )}^{2}} + 88944075\right )}}{14641000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {16807}{10648 \, {\left (2 \, x - 1\right )}} - \frac {1539}{1250} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {11562}{9150625} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14641000*(2*x - 1)*(391367530/(2*x - 1) + 430519419/(2*x - 1)^2 + 88944075)/(11/(2*x - 1) + 5)^2 - 16807/106

48/(2*x - 1) - 1539/1250*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 11562/9150625*log(abs(-11/(2*x - 1) - 5))

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maple [A] time = 0.01, size = 48, normalized size = 0.81 \begin {gather*} \frac {243 x}{500}+\frac {36015 \ln \left (2 x -1\right )}{29282}+\frac {11562 \ln \left (5 x +3\right )}{9150625}-\frac {1}{151250 \left (5 x +3\right )^{2}}-\frac {169}{831875 \left (5 x +3\right )}-\frac {16807}{10648 \left (2 x -1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/500*x-1/151250/(5*x+3)^2-169/831875/(5*x+3)+11562/9150625*ln(5*x+3)-16807/10648/(2*x-1)+36015/29282*ln(2*x

-1)

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maxima [A] time = 0.46, size = 49, normalized size = 0.83 \begin {gather*} \frac {243}{500} \, x - \frac {52524579 \, x^{2} + 63026538 \, x + 18907055}{1331000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {11562}{9150625} \, \log \left (5 \, x + 3\right ) + \frac {36015}{29282} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/500*x - 1/1331000*(52524579*x^2 + 63026538*x + 18907055)/(50*x^3 + 35*x^2 - 12*x - 9) + 11562/9150625*log(

5*x + 3) + 36015/29282*log(2*x - 1)

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mupad [B] time = 1.07, size = 44, normalized size = 0.75 \begin {gather*} \frac {243\,x}{500}+\frac {36015\,\ln \left (x-\frac {1}{2}\right )}{29282}+\frac {11562\,\ln \left (x+\frac {3}{5}\right )}{9150625}+\frac {\frac {52524579\,x^2}{66550000}+\frac {31513269\,x}{33275000}+\frac {3781411}{13310000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(243*x)/500 + (36015*log(x - 1/2))/29282 + (11562*log(x + 3/5))/9150625 + ((31513269*x)/33275000 + (52524579*x

^2)/66550000 + 3781411/13310000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50)

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sympy [A] time = 0.19, size = 51, normalized size = 0.86 \begin {gather*} \frac {243 x}{500} + \frac {- 52524579 x^{2} - 63026538 x - 18907055}{66550000 x^{3} + 46585000 x^{2} - 15972000 x - 11979000} + \frac {36015 \log {\left (x - \frac {1}{2} \right )}}{29282} + \frac {11562 \log {\left (x + \frac {3}{5} \right )}}{9150625} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243*x/500 + (-52524579*x**2 - 63026538*x - 18907055)/(66550000*x**3 + 46585000*x**2 - 15972000*x - 11979000) +

36015*log(x - 1/2)/29282 + 11562*log(x + 3/5)/9150625

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