∫ (2+3x)7 ________________________

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

Optimal. Leaf size=73 \[ \frac {729 x^3}{500}+\frac {21141 x^2}{2500}+\frac {1467477 x}{50000}+\frac {823543}{42592 (1-2 x)}-\frac {47}{4159375 (5 x+3)}-\frac {1}{3781250 (5 x+3)^2}+\frac {7411887 \log (1-2 x)}{234256}+\frac {4761 \log (5 x+3)}{45753125} \]

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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}{500}+\frac {21141 x^2}{2500}+\frac {1467477 x}{50000}+\frac {823543}{42592 (1-2 x)}-\frac {47}{4159375 (5 x+3)}-\frac {1}{3781250 (5 x+3)^2}+\frac {7411887 \log (1-2 x)}{234256}+\frac {4761 \log (5 x+3)}{45753125} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

823543/(42592*(1 - 2*x)) + (1467477*x)/50000 + (21141*x^2)/2500 + (729*x^3)/500 - 1/(3781250*(3 + 5*x)^2) - 47

/(4159375*(3 + 5*x)) + (7411887*Log[1 - 2*x])/234256 + (4761*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)^2 (3+5 x)^3} \, dx &=\int \left (\frac {1467477}{50000}+\frac {21141 x}{1250}+\frac {2187 x^2}{500}+\frac {823543}{21296 (-1+2 x)^2}+\frac {7411887}{117128 (-1+2 x)}+\frac {1}{378125 (3+5 x)^3}+\frac {47}{831875 (3+5 x)^2}+\frac {4761}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac {823543}{42592 (1-2 x)}+\frac {1467477 x}{50000}+\frac {21141 x^2}{2500}+\frac {729 x^3}{500}-\frac {1}{3781250 (3+5 x)^2}-\frac {47}{4159375 (3+5 x)}+\frac {7411887 \log (1-2 x)}{234256}+\frac {4761 \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 {11 \left (48514950000 x^6+315347175000 x^5+1161933052500 x^4+42644641050 x^3-1002031406415 x^2-426293494632 x-14162188399\right )}{(2 x-1) (5 x+3)^2}+231621468750 \log (1-2 x)+761760 \log (10 x+6)}{7320500000} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((11*(-14162188399 - 426293494632*x - 1002031406415*x^2 + 42644641050*x^3 + 1161933052500*x^4 + 315347175000*x

^5 + 48514950000*x^6))/((-1 + 2*x)*(3 + 5*x)^2) + 231621468750*Log[1 - 2*x] + 761760*Log[6 + 10*x])/7320500000

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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)^2 (3+5 x)^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.57, size = 95, normalized size = 1.30 \begin {gather*} \frac {533664450000 \, x^{6} + 3468818925000 \, x^{5} + 12781263577500 \, x^{4} + 6680945249550 \, x^{3} - 6674047531965 \, x^{2} + 761760 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 231621468750 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 6180073448472 \, x - 1273917828029}{7320500000 \, {\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)^7/(1-2*x)^2/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/7320500000*(533664450000*x^6 + 3468818925000*x^5 + 12781263577500*x^4 + 6680945249550*x^3 - 6674047531965*x^

2 + 761760*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 231621468750*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1)

- 6180073448472*x - 1273917828029)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [A] time = 1.20, size = 103, normalized size = 1.41 \begin {gather*} \frac {{\left (2 \, x - 1\right )}^{3} {\left (\frac {25349061375}{2 \, x - 1} + \frac {234545525775}{{\left (2 \, x - 1\right )}^{2}} + \frac {720756547985}{{\left (2 \, x - 1\right )}^{3}} + \frac {689127341628}{{\left (2 \, x - 1\right )}^{4}} + 1334161125\right )}}{292820000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {823543}{42592 \, {\left (2 \, x - 1\right )}} - \frac {1582011}{50000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {4761}{45753125} \, \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)^7/(1-2*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

1/292820000*(2*x - 1)^3*(25349061375/(2*x - 1) + 234545525775/(2*x - 1)^2 + 720756547985/(2*x - 1)^3 + 6891273

41628/(2*x - 1)^4 + 1334161125)/(11/(2*x - 1) + 5)^2 - 823543/42592/(2*x - 1) - 1582011/50000*log(1/2*abs(2*x

- 1)/(2*x - 1)^2) + 4761/45753125*log(abs(-11/(2*x - 1) - 5))

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maple [A] time = 0.01, size = 58, normalized size = 0.79 \begin {gather*} \frac {729 x^{3}}{500}+\frac {21141 x^{2}}{2500}+\frac {1467477 x}{50000}+\frac {7411887 \ln \left (2 x -1\right )}{234256}+\frac {4761 \ln \left (5 x +3\right )}{45753125}-\frac {1}{3781250 \left (5 x +3\right )^{2}}-\frac {47}{4159375 \left (5 x +3\right )}-\frac {823543}{42592 \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)^2/(5*x+3)^3,x)

[Out]

729/500*x^3+21141/2500*x^2+1467477/50000*x-1/3781250/(5*x+3)^2-47/4159375/(5*x+3)+4761/45753125*ln(5*x+3)-8235

43/42592/(2*x-1)+7411887/234256*ln(2*x-1)

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maxima [A] time = 0.55, size = 59, normalized size = 0.81 \begin {gather*} \frac {729}{500} \, x^{3} + \frac {21141}{2500} \, x^{2} + \frac {1467477}{50000} \, x - \frac {321696559575 \, x^{2} + 386035789122 \, x + 115810711639}{665500000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {4761}{45753125} \, \log \left (5 \, x + 3\right ) + \frac {7411887}{234256} \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

729/500*x^3 + 21141/2500*x^2 + 1467477/50000*x - 1/665500000*(321696559575*x^2 + 386035789122*x + 115810711639

)/(50*x^3 + 35*x^2 - 12*x - 9) + 4761/45753125*log(5*x + 3) + 7411887/234256*log(2*x - 1)

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mupad [B] time = 0.04, size = 54, normalized size = 0.74 \begin {gather*} \frac {1467477\,x}{50000}+\frac {7411887\,\ln \left (x-\frac {1}{2}\right )}{234256}+\frac {4761\,\ln \left (x+\frac {3}{5}\right )}{45753125}+\frac {\frac {12867862383\,x^2}{1331000000}+\frac {193017894561\,x}{16637500000}+\frac {115810711639}{33275000000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}}+\frac {21141\,x^2}{2500}+\frac {729\,x^3}{500} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1467477*x)/50000 + (7411887*log(x - 1/2))/234256 + (4761*log(x + 3/5))/45753125 + ((193017894561*x)/166375000

00 + (12867862383*x^2)/1331000000 + 115810711639/33275000000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50) + (21141*x^

2)/2500 + (729*x^3)/500

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sympy [A] time = 0.20, size = 65, normalized size = 0.89 \begin {gather*} \frac {729 x^{3}}{500} + \frac {21141 x^{2}}{2500} + \frac {1467477 x}{50000} + \frac {- 321696559575 x^{2} - 386035789122 x - 115810711639}{33275000000 x^{3} + 23292500000 x^{2} - 7986000000 x - 5989500000} + \frac {7411887 \log {\left (x - \frac {1}{2} \right )}}{234256} + \frac {4761 \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)**2/(3+5*x)**3,x)

[Out]

729*x**3/500 + 21141*x**2/2500 + 1467477*x/50000 + (-321696559575*x**2 - 386035789122*x - 115810711639)/(33275

000000*x**3 + 23292500000*x**2 - 7986000000*x - 5989500000) + 7411887*log(x - 1/2)/234256 + 4761*log(x + 3/5)/

45753125

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