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

Optimal. Leaf size=59 \[ -\frac {250 x^4}{27}+\frac {1700 x^3}{81}-\frac {1795 x^2}{81}+\frac {16253 x}{729}-\frac {1813}{729 (3 x+2)}+\frac {343}{4374 (3 x+2)^2}-\frac {10073}{729} \log (3 x+2) \]

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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 {250 x^4}{27}+\frac {1700 x^3}{81}-\frac {1795 x^2}{81}+\frac {16253 x}{729}-\frac {1813}{729 (3 x+2)}+\frac {343}{4374 (3 x+2)^2}-\frac {10073}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16253*x)/729 - (1795*x^2)/81 + (1700*x^3)/81 - (250*x^4)/27 + 343/(4374*(2 + 3*x)^2) - 1813/(729*(2 + 3*x)) -
(10073*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)^3} \, dx &=\int \left (\frac {16253}{729}-\frac {3590 x}{81}+\frac {1700 x^2}{27}-\frac {1000 x^3}{27}-\frac {343}{729 (2+3 x)^3}+\frac {1813}{243 (2+3 x)^2}-\frac {10073}{243 (2+3 x)}\right ) \, dx\\ &=\frac {16253 x}{729}-\frac {1795 x^2}{81}+\frac {1700 x^3}{81}-\frac {250 x^4}{27}+\frac {343}{4374 (2+3 x)^2}-\frac {1813}{729 (2+3 x)}-\frac {10073}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 56, normalized size = 0.95 \begin {gather*} \frac {-364500 x^6+340200 x^5+67230 x^4+81702 x^3+2072124 x^2+2076942 x-60438 (3 x+2)^2 \log (3 x+2)+551755}{4374 (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(551755 + 2076942*x + 2072124*x^2 + 81702*x^3 + 67230*x^4 + 340200*x^5 - 364500*x^6 - 60438*(2 + 3*x)^2*Log[2
+ 3*x])/(4374*(2 + 3*x)^2)

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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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 62, normalized size = 1.05 \begin {gather*} -\frac {364500 \, x^{6} - 340200 \, x^{5} - 67230 \, x^{4} - 81702 \, x^{3} - 782496 \, x^{2} + 60438 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 357438 \, x + 21413}{4374 \, {\left (9 \, x^{2} + 12 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

-1/4374*(364500*x^6 - 340200*x^5 - 67230*x^4 - 81702*x^3 - 782496*x^2 + 60438*(9*x^2 + 12*x + 4)*log(3*x + 2)
- 357438*x + 21413)/(9*x^2 + 12*x + 4)

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giac [A]  time = 0.96, size = 42, normalized size = 0.71 \begin {gather*} -\frac {250}{27} \, x^{4} + \frac {1700}{81} \, x^{3} - \frac {1795}{81} \, x^{2} + \frac {16253}{729} \, x - \frac {49 \, {\left (666 \, x + 437\right )}}{4374 \, {\left (3 \, x + 2\right )}^{2}} - \frac {10073}{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)^3,x, algorithm="giac")

[Out]

-250/27*x^4 + 1700/81*x^3 - 1795/81*x^2 + 16253/729*x - 49/4374*(666*x + 437)/(3*x + 2)^2 - 10073/729*log(abs(
3*x + 2))

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maple [A]  time = 0.01, size = 46, normalized size = 0.78 \begin {gather*} -\frac {250 x^{4}}{27}+\frac {1700 x^{3}}{81}-\frac {1795 x^{2}}{81}+\frac {16253 x}{729}-\frac {10073 \ln \left (3 x +2\right )}{729}+\frac {343}{4374 \left (3 x +2\right )^{2}}-\frac {1813}{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)^3,x)

[Out]

16253/729*x-1795/81*x^2+1700/81*x^3-250/27*x^4+343/4374/(3*x+2)^2-1813/729/(3*x+2)-10073/729*ln(3*x+2)

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maxima [A]  time = 0.53, size = 46, normalized size = 0.78 \begin {gather*} -\frac {250}{27} \, x^{4} + \frac {1700}{81} \, x^{3} - \frac {1795}{81} \, x^{2} + \frac {16253}{729} \, x - \frac {49 \, {\left (666 \, x + 437\right )}}{4374 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {10073}{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)^3,x, algorithm="maxima")

[Out]

-250/27*x^4 + 1700/81*x^3 - 1795/81*x^2 + 16253/729*x - 49/4374*(666*x + 437)/(9*x^2 + 12*x + 4) - 10073/729*l
og(3*x + 2)

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mupad [B]  time = 0.03, size = 42, normalized size = 0.71 \begin {gather*} \frac {16253\,x}{729}-\frac {10073\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {\frac {1813\,x}{2187}+\frac {21413}{39366}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}-\frac {1795\,x^2}{81}+\frac {1700\,x^3}{81}-\frac {250\,x^4}{27} \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)^3,x)

[Out]

(16253*x)/729 - (10073*log(x + 2/3))/729 - ((1813*x)/2187 + 21413/39366)/((4*x)/3 + x^2 + 4/9) - (1795*x^2)/81
+ (1700*x^3)/81 - (250*x^4)/27

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sympy [A]  time = 0.14, size = 49, normalized size = 0.83 \begin {gather*} - \frac {250 x^{4}}{27} + \frac {1700 x^{3}}{81} - \frac {1795 x^{2}}{81} + \frac {16253 x}{729} - \frac {32634 x + 21413}{39366 x^{2} + 52488 x + 17496} - \frac {10073 \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)**3,x)

[Out]

-250*x**4/27 + 1700*x**3/81 - 1795*x**2/81 + 16253*x/729 - (32634*x + 21413)/(39366*x**2 + 52488*x + 17496) -
10073*log(3*x + 2)/729

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