\(\int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx\) [1264]

Optimal result

Integrand size = 22, antiderivative size = 45 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {20 x}{3}+\frac {50 x^2}{27}-\frac {49}{486 (2+3 x)^2}+\frac {518}{243 (2+3 x)}+\frac {503}{81} \log (2+3 x) \]

[Out]

-20/3*x+50/27*x^2-49/486/(2+3*x)^2+518/243/(2+3*x)+503/81*ln(2+3*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, 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} \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50 x^2}{27}-\frac {20 x}{3}+\frac {518}{243 (3 x+2)}-\frac {49}{486 (3 x+2)^2}+\frac {503}{81} \log (3 x+2) \]

[In]

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

[Out]

(-20*x)/3 + (50*x^2)/27 - 49/(486*(2 + 3*x)^2) + 518/(243*(2 + 3*x)) + (503*Log[2 + 3*x])/81

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 {20}{3}+\frac {100 x}{27}+\frac {49}{81 (2+3 x)^3}-\frac {518}{81 (2+3 x)^2}+\frac {503}{27 (2+3 x)}\right ) \, dx \\ & = -\frac {20 x}{3}+\frac {50 x^2}{27}-\frac {49}{486 (2+3 x)^2}+\frac {518}{243 (2+3 x)}+\frac {503}{81} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=-\frac {913+4508 x+6480 x^2+2040 x^3-900 x^4}{54 (2+3 x)^2}+\frac {503}{81} \log (2+3 x) \]

[In]

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

[Out]

-1/54*(913 + 4508*x + 6480*x^2 + 2040*x^3 - 900*x^4)/(2 + 3*x)^2 + (503*Log[2 + 3*x])/81

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {50 x^{2}}{27}-\frac {20 x}{3}+\frac {\frac {518 x}{81}+\frac {2023}{486}}{\left (2+3 x \right )^{2}}+\frac {503 \ln \left (2+3 x \right )}{81}\)	\(32\)
default	\(-\frac {20 x}{3}+\frac {50 x^{2}}{27}-\frac {49}{486 \left (2+3 x \right )^{2}}+\frac {518}{243 \left (2+3 x \right )}+\frac {503 \ln \left (2+3 x \right )}{81}\)	\(36\)
norman	\(\frac {-\frac {1769}{54} x -\frac {1967}{24} x^{2}-\frac {340}{9} x^{3}+\frac {50}{3} x^{4}}{\left (2+3 x \right )^{2}}+\frac {503 \ln \left (2+3 x \right )}{81}\)	\(37\)
parallelrisch	\(\frac {10800 x^{4}+36216 \ln \left (\frac {2}{3}+x \right ) x^{2}-24480 x^{3}+48288 \ln \left (\frac {2}{3}+x \right ) x -53109 x^{2}+16096 \ln \left (\frac {2}{3}+x \right )-21228 x}{648 \left (2+3 x \right )^{2}}\)	\(51\)
meijerg	\(\frac {9 x \left (\frac {3 x}{2}+2\right )}{16 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {3 x^{2}}{8 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {59 x \left (\frac {27 x}{2}+6\right )}{108 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {503 \ln \left (1+\frac {3 x}{2}\right )}{81}+\frac {5 x \left (9 x^{2}+27 x +12\right )}{27 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {20 x \left (-\frac {135}{8} x^{3}+45 x^{2}+135 x +60\right )}{81 \left (1+\frac {3 x}{2}\right )^{2}}\)	\(97\)

[In]

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

[Out]

50/27*x^2-20/3*x+9*(518/729*x+2023/4374)/(2+3*x)^2+503/81*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {8100 \, x^{4} - 18360 \, x^{3} - 35280 \, x^{2} + 3018 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 9852 \, x + 2023}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

[In]

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

[Out]

1/486*(8100*x^4 - 18360*x^3 - 35280*x^2 + 3018*(9*x^2 + 12*x + 4)*log(3*x + 2) - 9852*x + 2023)/(9*x^2 + 12*x
+ 4)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50 x^{2}}{27} - \frac {20 x}{3} + \frac {3108 x + 2023}{4374 x^{2} + 5832 x + 1944} + \frac {503 \log {\left (3 x + 2 \right )}}{81} \]

[In]

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

[Out]

50*x**2/27 - 20*x/3 + (3108*x + 2023)/(4374*x**2 + 5832*x + 1944) + 503*log(3*x + 2)/81

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50}{27} \, x^{2} - \frac {20}{3} \, x + \frac {7 \, {\left (444 \, x + 289\right )}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {503}{81} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

50/27*x^2 - 20/3*x + 7/486*(444*x + 289)/(9*x^2 + 12*x + 4) + 503/81*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {50}{27} \, x^{2} - \frac {20}{3} \, x + \frac {7 \, {\left (444 \, x + 289\right )}}{486 \, {\left (3 \, x + 2\right )}^{2}} + \frac {503}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

50/27*x^2 - 20/3*x + 7/486*(444*x + 289)/(3*x + 2)^2 + 503/81*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^3} \, dx=\frac {503\,\ln \left (x+\frac {2}{3}\right )}{81}-\frac {20\,x}{3}+\frac {\frac {518\,x}{729}+\frac {2023}{4374}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}+\frac {50\,x^2}{27} \]

[In]

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

[Out]

(503*log(x + 2/3))/81 - (20*x)/3 + ((518*x)/729 + 2023/4374)/((4*x)/3 + x^2 + 4/9) + (50*x^2)/27