∫ (2+3x)7 ________________________

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

Optimal. Leaf size=73 \[ -\frac {729 x^3}{200}-\frac {108621 x^2}{4000}-\frac {1258983 x}{10000}-\frac {18941489}{85184 (1-2 x)}-\frac {1}{4159375 (5 x+3)}+\frac {823543}{15488 (1-2 x)^2}-\frac {87177909 \log (1-2 x)}{468512}+\frac {237 \log (5 x+3)}{45753125} \]

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Rubi [A] time = 0.04, antiderivative size = 73, 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^3}{200}-\frac {108621 x^2}{4000}-\frac {1258983 x}{10000}-\frac {18941489}{85184 (1-2 x)}-\frac {1}{4159375 (5 x+3)}+\frac {823543}{15488 (1-2 x)^2}-\frac {87177909 \log (1-2 x)}{468512}+\frac {237 \log (5 x+3)}{45753125} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

823543/(15488*(1 - 2*x)^2) - 18941489/(85184*(1 - 2*x)) - (1258983*x)/10000 - (108621*x^2)/4000 - (729*x^3)/20

0 - 1/(4159375*(3 + 5*x)) - (87177909*Log[1 - 2*x])/468512 + (237*Log[3 + 5*x])/45753125

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)^7}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac {1258983}{10000}-\frac {108621 x}{2000}-\frac {2187 x^2}{200}-\frac {823543}{3872 (-1+2 x)^3}-\frac {18941489}{42592 (-1+2 x)^2}-\frac {87177909}{234256 (-1+2 x)}+\frac {1}{831875 (3+5 x)^2}+\frac {237}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac {823543}{15488 (1-2 x)^2}-\frac {18941489}{85184 (1-2 x)}-\frac {1258983 x}{10000}-\frac {108621 x^2}{4000}-\frac {729 x^3}{200}-\frac {1}{4159375 (3+5 x)}-\frac {87177909 \log (1-2 x)}{468512}+\frac {237 \log (3+5 x)}{45753125}\\ \end {align*}

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Mathematica [A] time = 0.06, size = 67, normalized size = 0.92 \begin {gather*} \frac {-\frac {22 \left (4851495000 x^6+34203039750 x^5+151415158950 x^4-172378468845 x^3-163837494156 x^2+25343933346 x+19763981131\right )}{(1-2 x)^2 (5 x+3)}-272430965625 \log (1-2 x)+7584 \log (10 x+6)}{1464100000} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-22*(19763981131 + 25343933346*x - 163837494156*x^2 - 172378468845*x^3 + 151415158950*x^4 + 34203039750*x^5

+ 4851495000*x^6))/((1 - 2*x)^2*(3 + 5*x)) - 272430965625*Log[1 - 2*x] + 7584*Log[6 + 10*x])/1464100000

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.57, size = 95, normalized size = 1.30 \begin {gather*} -\frac {426931560000 \, x^{6} + 3009867498000 \, x^{5} + 13324533987600 \, x^{4} - 6947670741660 \, x^{3} - 17706353292408 \, x^{2} - 30336 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1089723862500 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 647305946397 \, x + 2972475517033}{5856400000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5856400000*(426931560000*x^6 + 3009867498000*x^5 + 13324533987600*x^4 - 6947670741660*x^3 - 17706353292408*

x^2 - 30336*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 1089723862500*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) -

647305946397*x + 2972475517033)/(20*x^3 - 8*x^2 - 7*x + 3)

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giac [A] time = 1.16, size = 103, normalized size = 1.41 \begin {gather*} -\frac {{\left (5 \, x + 3\right )}^{3} {\left (\frac {1472913882}{5 \, x + 3} + \frac {33001809588}{{\left (5 \, x + 3\right )}^{2}} - \frac {817302548083}{{\left (5 \, x + 3\right )}^{3}} + \frac {2996736348771}{{\left (5 \, x + 3\right )}^{4}} + 85386312\right )}}{732050000 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {1}{4159375 \, {\left (5 \, x + 3\right )}} + \frac {18607401}{100000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {87177909}{468512} \, \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)^7/(1-2*x)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/732050000*(5*x + 3)^3*(1472913882/(5*x + 3) + 33001809588/(5*x + 3)^2 - 817302548083/(5*x + 3)^3 + 29967363

48771/(5*x + 3)^4 + 85386312)/(11/(5*x + 3) - 2)^2 - 1/4159375/(5*x + 3) + 18607401/100000*log(1/5*abs(5*x + 3

)/(5*x + 3)^2) - 87177909/468512*log(abs(-11/(5*x + 3) + 2))

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maple [A] time = 0.01, size = 58, normalized size = 0.79 \begin {gather*} -\frac {729 x^{3}}{200}-\frac {108621 x^{2}}{4000}-\frac {1258983 x}{10000}-\frac {87177909 \ln \left (2 x -1\right )}{468512}+\frac {237 \ln \left (5 x +3\right )}{45753125}-\frac {1}{4159375 \left (5 x +3\right )}+\frac {823543}{15488 \left (2 x -1\right )^{2}}+\frac {18941489}{85184 \left (2 x -1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/200*x^3-108621/4000*x^2-1258983/10000*x-1/4159375/(5*x+3)+237/45753125*ln(5*x+3)+823543/15488/(2*x-1)^2+1

8941489/85184/(2*x-1)-87177909/468512*ln(2*x-1)

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maxima [A] time = 0.49, size = 59, normalized size = 0.81 \begin {gather*} -\frac {729}{200} \, x^{3} - \frac {108621}{4000} \, x^{2} - \frac {1258983}{10000} \, x + \frac {1183843061988 \, x^{2} + 259930759887 \, x - 270225047003}{532400000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {237}{45753125} \, \log \left (5 \, x + 3\right ) - \frac {87177909}{468512} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/200*x^3 - 108621/4000*x^2 - 1258983/10000*x + 1/532400000*(1183843061988*x^2 + 259930759887*x - 270225047

003)/(20*x^3 - 8*x^2 - 7*x + 3) + 237/45753125*log(5*x + 3) - 87177909/468512*log(2*x - 1)

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mupad [B] time = 0.04, size = 55, normalized size = 0.75 \begin {gather*} \frac {237\,\ln \left (x+\frac {3}{5}\right )}{45753125}-\frac {87177909\,\ln \left (x-\frac {1}{2}\right )}{468512}-\frac {1258983\,x}{10000}-\frac {\frac {295960765497\,x^2}{2662000000}+\frac {259930759887\,x}{10648000000}-\frac {270225047003}{10648000000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}}-\frac {108621\,x^2}{4000}-\frac {729\,x^3}{200} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(237*log(x + 3/5))/45753125 - (87177909*log(x - 1/2))/468512 - (1258983*x)/10000 - ((259930759887*x)/106480000

00 + (295960765497*x^2)/2662000000 - 270225047003/10648000000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20) - (108621*x

^2)/4000 - (729*x^3)/200

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sympy [A] time = 0.21, size = 63, normalized size = 0.86 \begin {gather*} - \frac {729 x^{3}}{200} - \frac {108621 x^{2}}{4000} - \frac {1258983 x}{10000} - \frac {- 1183843061988 x^{2} - 259930759887 x + 270225047003}{10648000000 x^{3} - 4259200000 x^{2} - 3726800000 x + 1597200000} - \frac {87177909 \log {\left (x - \frac {1}{2} \right )}}{468512} + \frac {237 \log {\left (x + \frac {3}{5} \right )}}{45753125} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729*x**3/200 - 108621*x**2/4000 - 1258983*x/10000 - (-1183843061988*x**2 - 259930759887*x + 270225047003)/(10

648000000*x**3 - 4259200000*x**2 - 3726800000*x + 1597200000) - 87177909*log(x - 1/2)/468512 + 237*log(x + 3/5

)/45753125

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