\(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx\) [1367]

Optimal result

Integrand size = 22, antiderivative size = 56 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {1780 x}{243}-\frac {100 x^2}{81}-\frac {343}{2187 (2+3 x)^3}+\frac {1862}{729 (2+3 x)^2}-\frac {11599}{729 (2+3 x)}-\frac {8198}{729} \log (2+3 x) \]

[Out]

1780/243*x-100/81*x^2-343/2187/(2+3*x)^3+1862/729/(2+3*x)^2-11599/729/(2+3*x)-8198/729*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {100 x^2}{81}+\frac {1780 x}{243}-\frac {11599}{729 (3 x+2)}+\frac {1862}{729 (3 x+2)^2}-\frac {343}{2187 (3 x+2)^3}-\frac {8198}{729} \log (3 x+2) \]

[In]

Int[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(1780*x)/243 - (100*x^2)/81 - 343/(2187*(2 + 3*x)^3) + 1862/(729*(2 + 3*x)^2) - 11599/(729*(2 + 3*x)) - (8198*
Log[2 + 3*x])/729

Rule 90

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*} \text {integral}& = \int \left (\frac {1780}{243}-\frac {200 x}{81}+\frac {343}{243 (2+3 x)^4}-\frac {3724}{243 (2+3 x)^3}+\frac {11599}{243 (2+3 x)^2}-\frac {8198}{243 (2+3 x)}\right ) \, dx \\ & = \frac {1780 x}{243}-\frac {100 x^2}{81}-\frac {343}{2187 (2+3 x)^3}+\frac {1862}{729 (2+3 x)^2}-\frac {11599}{729 (2+3 x)}-\frac {8198}{729} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {-33319+155034 x+883467 x^2+1088640 x^3+286740 x^4-72900 x^5-24594 (2+3 x)^3 \log (20+30 x)}{2187 (2+3 x)^3} \]

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(-33319 + 155034*x + 883467*x^2 + 1088640*x^3 + 286740*x^4 - 72900*x^5 - 24594*(2 + 3*x)^3*Log[20 + 30*x])/(21
87*(2 + 3*x)^3)

Maple [A] (verified)

Time = 2.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {100 x^{2}}{81}+\frac {1780 x}{243}+\frac {-\frac {11599}{81} x^{2}-\frac {44534}{243} x -\frac {128359}{2187}}{\left (2+3 x \right )^{3}}-\frac {8198 \ln \left (2+3 x \right )}{729}\)	\(37\)
norman	\(\frac {\frac {67771}{486} x +\frac {164203}{324} x^{2}+\frac {355879}{648} x^{3}+\frac {1180}{9} x^{4}-\frac {100}{3} x^{5}}{\left (2+3 x \right )^{3}}-\frac {8198 \ln \left (2+3 x \right )}{729}\)	\(42\)
default	\(\frac {1780 x}{243}-\frac {100 x^{2}}{81}-\frac {343}{2187 \left (2+3 x \right )^{3}}+\frac {1862}{729 \left (2+3 x \right )^{2}}-\frac {11599}{729 \left (2+3 x \right )}-\frac {8198 \ln \left (2+3 x \right )}{729}\)	\(45\)
parallelrisch	\(-\frac {194400 x^{5}+1770768 \ln \left (\frac {2}{3}+x \right ) x^{3}-764640 x^{4}+3541536 \ln \left (\frac {2}{3}+x \right ) x^{2}-3202911 x^{3}+2361024 \ln \left (\frac {2}{3}+x \right ) x -2955654 x^{2}+524672 \ln \left (\frac {2}{3}+x \right )-813252 x}{5832 \left (2+3 x \right )^{3}}\)	\(65\)
meijerg	\(\frac {3 x \left (\frac {9}{4} x^{2}+\frac {9}{2} x +3\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {x^{2} \left (3+\frac {3 x}{2}\right )}{4 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {47 x^{3}}{48 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {23 x \left (\frac {99}{2} x^{2}+45 x +12\right )}{108 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {8198 \ln \left (1+\frac {3 x}{2}\right )}{729}+\frac {4 x \left (\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{81 \left (1+\frac {3 x}{2}\right )^{3}}+\frac {200 x \left (-\frac {243}{16} x^{4}+\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{729 \left (1+\frac {3 x}{2}\right )^{3}}\)	\(134\)

[In]

int((1-2*x)^3*(3+5*x)^2/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

-100/81*x^2+1780/243*x+27*(-11599/2187*x^2-44534/6561*x-128359/59049)/(2+3*x)^3-8198/729*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.20 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {72900 \, x^{5} - 286740 \, x^{4} - 767880 \, x^{3} - 241947 \, x^{2} + 24594 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 272646 \, x + 128359}{2187 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/2187*(72900*x^5 - 286740*x^4 - 767880*x^3 - 241947*x^2 + 24594*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) +
272646*x + 128359)/(27*x^3 + 54*x^2 + 36*x + 8)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=- \frac {100 x^{2}}{81} + \frac {1780 x}{243} - \frac {313173 x^{2} + 400806 x + 128359}{59049 x^{3} + 118098 x^{2} + 78732 x + 17496} - \frac {8198 \log {\left (3 x + 2 \right )}}{729} \]

[In]

integrate((1-2*x)**3*(3+5*x)**2/(2+3*x)**4,x)

[Out]

-100*x**2/81 + 1780*x/243 - (313173*x**2 + 400806*x + 128359)/(59049*x**3 + 118098*x**2 + 78732*x + 17496) - 8
198*log(3*x + 2)/729

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {100}{81} \, x^{2} + \frac {1780}{243} \, x - \frac {7 \, {\left (44739 \, x^{2} + 57258 \, x + 18337\right )}}{2187 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {8198}{729} \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^4,x, algorithm="maxima")

[Out]

-100/81*x^2 + 1780/243*x - 7/2187*(44739*x^2 + 57258*x + 18337)/(27*x^3 + 54*x^2 + 36*x + 8) - 8198/729*log(3*
x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=-\frac {100}{81} \, x^{2} + \frac {1780}{243} \, x - \frac {7 \, {\left (44739 \, x^{2} + 57258 \, x + 18337\right )}}{2187 \, {\left (3 \, x + 2\right )}^{3}} - \frac {8198}{729} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)^3*(3+5*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

-100/81*x^2 + 1780/243*x - 7/2187*(44739*x^2 + 57258*x + 18337)/(3*x + 2)^3 - 8198/729*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {1780\,x}{243}-\frac {8198\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {\frac {11599\,x^2}{2187}+\frac {44534\,x}{6561}+\frac {128359}{59049}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}-\frac {100\,x^2}{81} \]

[In]

int(-((2*x - 1)^3*(5*x + 3)^2)/(3*x + 2)^4,x)

[Out]

(1780*x)/243 - (8198*log(x + 2/3))/729 - ((44534*x)/6561 + (11599*x^2)/2187 + 128359/59049)/((4*x)/3 + 2*x^2 +
x^3 + 8/27) - (100*x^2)/81