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

Optimal. Leaf size=71 \[ -\frac {1000 x}{729}+\frac {14390}{729 (3 x+2)}-\frac {66193}{4374 (3 x+2)^2}+\frac {10073}{2187 (3 x+2)^3}-\frac {1813}{2916 (3 x+2)^4}+\frac {343}{10935 (3 x+2)^5}+\frac {3700}{729} \log (3 x+2) \]

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Rubi [A]  time = 0.03, antiderivative size = 71, 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 {1000 x}{729}+\frac {14390}{729 (3 x+2)}-\frac {66193}{4374 (3 x+2)^2}+\frac {10073}{2187 (3 x+2)^3}-\frac {1813}{2916 (3 x+2)^4}+\frac {343}{10935 (3 x+2)^5}+\frac {3700}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1000*x)/729 + 343/(10935*(2 + 3*x)^5) - 1813/(2916*(2 + 3*x)^4) + 10073/(2187*(2 + 3*x)^3) - 66193/(4374*(2
+ 3*x)^2) + 14390/(729*(2 + 3*x)) + (3700*Log[2 + 3*x])/729

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)^6} \, dx &=\int \left (-\frac {1000}{729}-\frac {343}{729 (2+3 x)^6}+\frac {1813}{243 (2+3 x)^5}-\frac {10073}{243 (2+3 x)^4}+\frac {66193}{729 (2+3 x)^3}-\frac {14390}{243 (2+3 x)^2}+\frac {3700}{243 (2+3 x)}\right ) \, dx\\ &=-\frac {1000 x}{729}+\frac {343}{10935 (2+3 x)^5}-\frac {1813}{2916 (2+3 x)^4}+\frac {10073}{2187 (2+3 x)^3}-\frac {66193}{4374 (2+3 x)^2}+\frac {14390}{729 (2+3 x)}+\frac {3700}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 56, normalized size = 0.79 \begin {gather*} \frac {-14580000 x^6-58320000 x^5-27264600 x^4+82222290 x^3+109363320 x^2+49872855 x+222000 (3 x+2)^5 \log (3 x+2)+7991782}{43740 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7991782 + 49872855*x + 109363320*x^2 + 82222290*x^3 - 27264600*x^4 - 58320000*x^5 - 14580000*x^6 + 222000*(2
+ 3*x)^5*Log[2 + 3*x])/(43740*(2 + 3*x)^5)

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 92, normalized size = 1.30 \begin {gather*} -\frac {4860000 \, x^{6} + 16200000 \, x^{5} - 1711800 \, x^{4} - 41807430 \, x^{3} - 46054440 \, x^{2} - 74000 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) - 19824285 \, x - 3090594}{14580 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/14580*(4860000*x^6 + 16200000*x^5 - 1711800*x^4 - 41807430*x^3 - 46054440*x^2 - 74000*(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) - 19824285*x - 3090594)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2
+ 240*x + 32)

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giac [A]  time = 0.97, size = 42, normalized size = 0.59 \begin {gather*} -\frac {1000}{729} \, x + \frac {23311800 \, x^{4} + 56207430 \, x^{3} + 50854440 \, x^{2} + 20464285 \, x + 3090594}{14580 \, {\left (3 \, x + 2\right )}^{5}} + \frac {3700}{729} \, \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)^6,x, algorithm="giac")

[Out]

-1000/729*x + 1/14580*(23311800*x^4 + 56207430*x^3 + 50854440*x^2 + 20464285*x + 3090594)/(3*x + 2)^5 + 3700/7
29*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 58, normalized size = 0.82 \begin {gather*} -\frac {1000 x}{729}+\frac {3700 \ln \left (3 x +2\right )}{729}+\frac {343}{10935 \left (3 x +2\right )^{5}}-\frac {1813}{2916 \left (3 x +2\right )^{4}}+\frac {10073}{2187 \left (3 x +2\right )^{3}}-\frac {66193}{4374 \left (3 x +2\right )^{2}}+\frac {14390}{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)^6,x)

[Out]

-1000/729*x+343/10935/(3*x+2)^5-1813/2916/(3*x+2)^4+10073/2187/(3*x+2)^3-66193/4374/(3*x+2)^2+14390/729/(3*x+2
)+3700/729*ln(3*x+2)

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maxima [A]  time = 0.59, size = 61, normalized size = 0.86 \begin {gather*} -\frac {1000}{729} \, x + \frac {23311800 \, x^{4} + 56207430 \, x^{3} + 50854440 \, x^{2} + 20464285 \, x + 3090594}{14580 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {3700}{729} \, \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)^6,x, algorithm="maxima")

[Out]

-1000/729*x + 1/14580*(23311800*x^4 + 56207430*x^3 + 50854440*x^2 + 20464285*x + 3090594)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32) + 3700/729*log(3*x + 2)

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mupad [B]  time = 1.13, size = 56, normalized size = 0.79 \begin {gather*} \frac {3700\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {1000\,x}{729}+\frac {\frac {14390\,x^4}{2187}+\frac {624527\,x^3}{39366}+\frac {847574\,x^2}{59049}+\frac {4092857\,x}{708588}+\frac {515099}{590490}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \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)^6,x)

[Out]

(3700*log(x + 2/3))/729 - (1000*x)/729 + ((4092857*x)/708588 + (847574*x^2)/59049 + (624527*x^3)/39366 + (1439
0*x^4)/2187 + 515099/590490)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)

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sympy [A]  time = 0.17, size = 61, normalized size = 0.86 \begin {gather*} - \frac {1000 x}{729} - \frac {- 23311800 x^{4} - 56207430 x^{3} - 50854440 x^{2} - 20464285 x - 3090594}{3542940 x^{5} + 11809800 x^{4} + 15746400 x^{3} + 10497600 x^{2} + 3499200 x + 466560} + \frac {3700 \log {\left (3 x + 2 \right )}}{729} \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)**6,x)

[Out]

-1000*x/729 - (-23311800*x**4 - 56207430*x**3 - 50854440*x**2 - 20464285*x - 3090594)/(3542940*x**5 + 11809800
*x**4 + 15746400*x**3 + 10497600*x**2 + 3499200*x + 466560) + 3700*log(3*x + 2)/729

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