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

Optimal. Leaf size=77 \[ -\frac {3700}{729 (3 x+2)}+\frac {7195}{729 (3 x+2)^2}-\frac {66193}{6561 (3 x+2)^3}+\frac {10073}{2916 (3 x+2)^4}-\frac {1813}{3645 (3 x+2)^5}+\frac {343}{13122 (3 x+2)^6}-\frac {1000 \log (3 x+2)}{2187} \]

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Rubi [A]  time = 0.03, antiderivative size = 77, 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 {3700}{729 (3 x+2)}+\frac {7195}{729 (3 x+2)^2}-\frac {66193}{6561 (3 x+2)^3}+\frac {10073}{2916 (3 x+2)^4}-\frac {1813}{3645 (3 x+2)^5}+\frac {343}{13122 (3 x+2)^6}-\frac {1000 \log (3 x+2)}{2187} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

343/(13122*(2 + 3*x)^6) - 1813/(3645*(2 + 3*x)^5) + 10073/(2916*(2 + 3*x)^4) - 66193/(6561*(2 + 3*x)^3) + 7195
/(729*(2 + 3*x)^2) - 3700/(729*(2 + 3*x)) - (1000*Log[2 + 3*x])/2187

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 {(1-2 x)^3 (3+5 x)^3}{(2+3 x)^7} \, dx &=\int \left (-\frac {343}{729 (2+3 x)^7}+\frac {1813}{243 (2+3 x)^6}-\frac {10073}{243 (2+3 x)^5}+\frac {66193}{729 (2+3 x)^4}-\frac {14390}{243 (2+3 x)^3}+\frac {3700}{243 (2+3 x)^2}-\frac {1000}{729 (2+3 x)}\right ) \, dx\\ &=\frac {343}{13122 (2+3 x)^6}-\frac {1813}{3645 (2+3 x)^5}+\frac {10073}{2916 (2+3 x)^4}-\frac {66193}{6561 (2+3 x)^3}+\frac {7195}{729 (2+3 x)^2}-\frac {3700}{729 (2+3 x)}-\frac {1000 \log (2+3 x)}{2187}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 51, normalized size = 0.66 \begin {gather*} -\frac {53946000 x^5+144852300 x^4+158427540 x^3+89062425 x^2+25975248 x+20000 (3 x+2)^6 \log (3 x+2)+3165082}{43740 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/43740*(3165082 + 25975248*x + 89062425*x^2 + 158427540*x^3 + 144852300*x^4 + 53946000*x^5 + 20000*(2 + 3*x)
^6*Log[2 + 3*x])/(2 + 3*x)^6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.42, size = 97, normalized size = 1.26 \begin {gather*} -\frac {53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 20000 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 25975248 \, x + 3165082}{43740 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 20000*(729*x^6 + 2916*x^5 + 4860*x^4 +
4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 25975248*x + 3165082)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64)

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giac [A]  time = 0.83, size = 44, normalized size = 0.57 \begin {gather*} -\frac {53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 25975248 \, x + 3165082}{43740 \, {\left (3 \, x + 2\right )}^{6}} - \frac {1000}{2187} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 25975248*x + 3165082)/(3*x + 2)^6 - 10
00/2187*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 64, normalized size = 0.83 \begin {gather*} -\frac {1000 \ln \left (3 x +2\right )}{2187}+\frac {343}{13122 \left (3 x +2\right )^{6}}-\frac {1813}{3645 \left (3 x +2\right )^{5}}+\frac {10073}{2916 \left (3 x +2\right )^{4}}-\frac {66193}{6561 \left (3 x +2\right )^{3}}+\frac {7195}{729 \left (3 x +2\right )^{2}}-\frac {3700}{729 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/13122/(3*x+2)^6-1813/3645/(3*x+2)^5+10073/2916/(3*x+2)^4-66193/6561/(3*x+2)^3+7195/729/(3*x+2)^2-3700/729/
(3*x+2)-1000/2187*ln(3*x+2)

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maxima [A]  time = 0.59, size = 68, normalized size = 0.88 \begin {gather*} -\frac {53946000 \, x^{5} + 144852300 \, x^{4} + 158427540 \, x^{3} + 89062425 \, x^{2} + 25975248 \, x + 3165082}{43740 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} - \frac {1000}{2187} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43740*(53946000*x^5 + 144852300*x^4 + 158427540*x^3 + 89062425*x^2 + 25975248*x + 3165082)/(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) - 1000/2187*log(3*x + 2)

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mupad [B]  time = 0.05, size = 64, normalized size = 0.83 \begin {gather*} -\frac {1000\,\ln \left (x+\frac {2}{3}\right )}{2187}-\frac {\frac {3700\,x^5}{2187}+\frac {9935\,x^4}{2187}+\frac {880153\,x^3}{177147}+\frac {1979165\,x^2}{708588}+\frac {2164604\,x}{2657205}+\frac {1582541}{15943230}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (1000*log(x + 2/3))/2187 - ((2164604*x)/2657205 + (1979165*x^2)/708588 + (880153*x^3)/177147 + (9935*x^4)/21
87 + (3700*x^5)/2187 + 1582541/15943230)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x^5 + x^6 +
64/729)

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sympy [A]  time = 0.19, size = 66, normalized size = 0.86 \begin {gather*} - \frac {53946000 x^{5} + 144852300 x^{4} + 158427540 x^{3} + 89062425 x^{2} + 25975248 x + 3165082}{31886460 x^{6} + 127545840 x^{5} + 212576400 x^{4} + 188956800 x^{3} + 94478400 x^{2} + 25194240 x + 2799360} - \frac {1000 \log {\left (3 x + 2 \right )}}{2187} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(53946000*x**5 + 144852300*x**4 + 158427540*x**3 + 89062425*x**2 + 25975248*x + 3165082)/(31886460*x**6 + 127
545840*x**5 + 212576400*x**4 + 188956800*x**3 + 94478400*x**2 + 25194240*x + 2799360) - 1000*log(3*x + 2)/2187

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