\(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx\) [1188]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {7}{972 (2+3 x)^4}-\frac {107}{729 (2+3 x)^3}+\frac {185}{162 (2+3 x)^2}-\frac {1025}{243 (2+3 x)}-\frac {250}{243} \log (2+3 x) \]

[Out]

7/972/(2+3*x)^4-107/729/(2+3*x)^3+185/162/(2+3*x)^2-1025/243/(2+3*x)-250/243*ln(2+3*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {1025}{243 (3 x+2)}+\frac {185}{162 (3 x+2)^2}-\frac {107}{729 (3 x+2)^3}+\frac {7}{972 (3 x+2)^4}-\frac {250}{243} \log (3 x+2) \]

[In]

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

[Out]

7/(972*(2 + 3*x)^4) - 107/(729*(2 + 3*x)^3) + 185/(162*(2 + 3*x)^2) - 1025/(243*(2 + 3*x)) - (250*Log[2 + 3*x]
)/243

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {7}{81 (2+3 x)^5}+\frac {107}{81 (2+3 x)^4}-\frac {185}{27 (2+3 x)^3}+\frac {1025}{81 (2+3 x)^2}-\frac {250}{81 (2+3 x)}\right ) \, dx \\ & = \frac {7}{972 (2+3 x)^4}-\frac {107}{729 (2+3 x)^3}+\frac {185}{162 (2+3 x)^2}-\frac {1025}{243 (2+3 x)}-\frac {250}{243} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {85915+404124 x+634230 x^2+332100 x^3+3000 (2+3 x)^4 \log (2+3 x)}{2916 (2+3 x)^4} \]

[In]

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

[Out]

-1/2916*(85915 + 404124*x + 634230*x^2 + 332100*x^3 + 3000*(2 + 3*x)^4*Log[2 + 3*x])/(2 + 3*x)^4

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {-\frac {1025}{9} x^{3}-\frac {435}{2} x^{2}-\frac {33677}{243} x -\frac {85915}{2916}}{\left (2+3 x \right )^{4}}-\frac {250 \ln \left (2+3 x \right )}{243}\)	\(34\)
norman	\(\frac {\frac {38935}{216} x^{2}+\frac {61315}{216} x^{3}+\frac {6187}{162} x +\frac {85915}{576} x^{4}}{\left (2+3 x \right )^{4}}-\frac {250 \ln \left (2+3 x \right )}{243}\)	\(37\)
default	\(\frac {7}{972 \left (2+3 x \right )^{4}}-\frac {107}{729 \left (2+3 x \right )^{3}}+\frac {185}{162 \left (2+3 x \right )^{2}}-\frac {1025}{243 \left (2+3 x \right )}-\frac {250 \ln \left (2+3 x \right )}{243}\)	\(46\)
parallelrisch	\(-\frac {1296000 \ln \left (\frac {2}{3}+x \right ) x^{4}+3456000 \ln \left (\frac {2}{3}+x \right ) x^{3}-2319705 x^{4}+3456000 \ln \left (\frac {2}{3}+x \right ) x^{2}-4414680 x^{3}+1536000 \ln \left (\frac {2}{3}+x \right ) x -2803320 x^{2}+256000 \ln \left (\frac {2}{3}+x \right )-593952 x}{15552 \left (2+3 x \right )^{4}}\)	\(69\)
meijerg	\(\frac {27 x \left (\frac {27}{8} x^{3}+9 x^{2}+9 x +4\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {27 x^{2} \left (\frac {9}{4} x^{2}+6 x +6\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {15 x^{3} \left (\frac {3 x}{2}+4\right )}{128 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {325 x^{4}}{128 \left (1+\frac {3 x}{2}\right )^{4}}+\frac {25 x \left (\frac {3375}{8} x^{3}+585 x^{2}+315 x +60\right )}{972 \left (1+\frac {3 x}{2}\right )^{4}}-\frac {250 \ln \left (1+\frac {3 x}{2}\right )}{243}\)	\(111\)

[In]

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

[Out]

81*(-1025/729*x^3-145/54*x^2-33677/19683*x-85915/236196)/(2+3*x)^4-250/243*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.22 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {332100 \, x^{3} + 634230 \, x^{2} + 3000 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 404124 \, x + 85915}{2916 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

[In]

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

[Out]

-1/2916*(332100*x^3 + 634230*x^2 + 3000*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 404124*x + 859
15)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=- \frac {332100 x^{3} + 634230 x^{2} + 404124 x + 85915}{236196 x^{4} + 629856 x^{3} + 629856 x^{2} + 279936 x + 46656} - \frac {250 \log {\left (3 x + 2 \right )}}{243} \]

[In]

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

[Out]

-(332100*x**3 + 634230*x**2 + 404124*x + 85915)/(236196*x**4 + 629856*x**3 + 629856*x**2 + 279936*x + 46656) -
250*log(3*x + 2)/243

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {332100 \, x^{3} + 634230 \, x^{2} + 404124 \, x + 85915}{2916 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {250}{243} \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/2916*(332100*x^3 + 634230*x^2 + 404124*x + 85915)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 250/243*log(3*
x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {1025}{243 \, {\left (3 \, x + 2\right )}} + \frac {185}{162 \, {\left (3 \, x + 2\right )}^{2}} - \frac {107}{729 \, {\left (3 \, x + 2\right )}^{3}} + \frac {7}{972 \, {\left (3 \, x + 2\right )}^{4}} + \frac {250}{243} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) \]

[In]

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

[Out]

-1025/243/(3*x + 2) + 185/162/(3*x + 2)^2 - 107/729/(3*x + 2)^3 + 7/972/(3*x + 2)^4 + 250/243*log(1/3*abs(3*x
+ 2)/(3*x + 2)^2)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {250\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {\frac {1025\,x^3}{9}+\frac {435\,x^2}{2}+\frac {33677\,x}{243}+\frac {85915}{2916}}{{\left (3\,x+2\right )}^4} \]

[In]

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

[Out]

- (250*log(x + 2/3))/243 - ((33677*x)/243 + (435*x^2)/2 + (1025*x^3)/9 + 85915/2916)/(3*x + 2)^4