\(\int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx\) [1449]

Optimal result

Integrand size = 20, antiderivative size = 87 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=\frac {1}{126 (2+3 x)^6}-\frac {11}{245 (2+3 x)^5}-\frac {11}{686 (2+3 x)^4}-\frac {44}{7203 (2+3 x)^3}-\frac {44}{16807 (2+3 x)^2}-\frac {176}{117649 (2+3 x)}-\frac {352 \log (1-2 x)}{823543}+\frac {352 \log (2+3 x)}{823543} \]

[Out]

1/126/(2+3*x)^6-11/245/(2+3*x)^5-11/686/(2+3*x)^4-44/7203/(2+3*x)^3-44/16807/(2+3*x)^2-176/117649/(2+3*x)-352/
823543*ln(1-2*x)+352/823543*ln(2+3*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, 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} \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=-\frac {176}{117649 (3 x+2)}-\frac {44}{16807 (3 x+2)^2}-\frac {44}{7203 (3 x+2)^3}-\frac {11}{686 (3 x+2)^4}-\frac {11}{245 (3 x+2)^5}+\frac {1}{126 (3 x+2)^6}-\frac {352 \log (1-2 x)}{823543}+\frac {352 \log (3 x+2)}{823543} \]

[In]

Int[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

1/(126*(2 + 3*x)^6) - 11/(245*(2 + 3*x)^5) - 11/(686*(2 + 3*x)^4) - 44/(7203*(2 + 3*x)^3) - 44/(16807*(2 + 3*x
)^2) - 176/(117649*(2 + 3*x)) - (352*Log[1 - 2*x])/823543 + (352*Log[2 + 3*x])/823543

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 {704}{823543 (-1+2 x)}-\frac {1}{7 (2+3 x)^7}+\frac {33}{49 (2+3 x)^6}+\frac {66}{343 (2+3 x)^5}+\frac {132}{2401 (2+3 x)^4}+\frac {264}{16807 (2+3 x)^3}+\frac {528}{117649 (2+3 x)^2}+\frac {1056}{823543 (2+3 x)}\right ) \, dx \\ & = \frac {1}{126 (2+3 x)^6}-\frac {11}{245 (2+3 x)^5}-\frac {11}{686 (2+3 x)^4}-\frac {44}{7203 (2+3 x)^3}-\frac {44}{16807 (2+3 x)^2}-\frac {176}{117649 (2+3 x)}-\frac {352 \log (1-2 x)}{823543}+\frac {352 \log (2+3 x)}{823543} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.63 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=\frac {-\frac {7 \left (3013741+12254814 x+22413105 x^2+24841080 x^3+15075720 x^4+3849120 x^5\right )}{(2+3 x)^6}-31680 \log (3-6 x)+31680 \log (2+3 x)}{74118870} \]

[In]

Integrate[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

((-7*(3013741 + 12254814*x + 22413105*x^2 + 24841080*x^3 + 15075720*x^4 + 3849120*x^5))/(2 + 3*x)^6 - 31680*Lo
g[3 - 6*x] + 31680*Log[2 + 3*x])/74118870

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59

method	result	size
norman	\(\frac {-\frac {680823}{588245} x -\frac {498069}{235298} x^{2}-\frac {276012}{117649} x^{3}-\frac {167508}{117649} x^{4}-\frac {42768}{117649} x^{5}-\frac {3013741}{10588410}}{\left (2+3 x \right )^{6}}-\frac {352 \ln \left (-1+2 x \right )}{823543}+\frac {352 \ln \left (2+3 x \right )}{823543}\)	\(51\)
risch	\(\frac {-\frac {680823}{588245} x -\frac {498069}{235298} x^{2}-\frac {276012}{117649} x^{3}-\frac {167508}{117649} x^{4}-\frac {42768}{117649} x^{5}-\frac {3013741}{10588410}}{\left (2+3 x \right )^{6}}-\frac {352 \ln \left (-1+2 x \right )}{823543}+\frac {352 \ln \left (2+3 x \right )}{823543}\)	\(52\)
default	\(-\frac {352 \ln \left (-1+2 x \right )}{823543}+\frac {1}{126 \left (2+3 x \right )^{6}}-\frac {11}{245 \left (2+3 x \right )^{5}}-\frac {11}{686 \left (2+3 x \right )^{4}}-\frac {44}{7203 \left (2+3 x \right )^{3}}-\frac {44}{16807 \left (2+3 x \right )^{2}}-\frac {176}{117649 \left (2+3 x \right )}+\frac {352 \ln \left (2+3 x \right )}{823543}\)	\(72\)
parallelrisch	\(\frac {740138560 x [phone] \ln \left (\frac {2}{3}+x \right ) x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}-1094860800 \ln \left (x -\frac {1}{2}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-973209600 \ln \left (x -\frac {1}{2}\right ) x^{3}-486604800 \ln \left (x -\frac {1}{2}\right ) x^{2}-129761280 \ln \left (x -\frac {1}{2}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-14417920 \ln \left (x -\frac {1}{2}\right )-164229120 \ln \left (x -\frac {1}{2}\right ) x^{6}-656916480 \ln \left (x -\frac {1}{2}\right ) x^{5}}{527067520 \left (2+3 x \right )^{6}}\)	\(155\)

[In]

int((3+5*x)/(1-2*x)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

(-680823/588245*x-498069/235298*x^2-276012/117649*x^3-167508/117649*x^4-42768/117649*x^5-3013741/10588410)/(2+
3*x)^6-352/823543*ln(-1+2*x)+352/823543*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.55 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=-\frac {26943840 \, x^{5} + 105530040 \, x^{4} + 173887560 \, x^{3} + 156891735 \, x^{2} - 31680 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 31680 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 85783698 \, x + 21096187}{74118870 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/74118870*(26943840*x^5 + 105530040*x^4 + 173887560*x^3 + 156891735*x^2 - 31680*(729*x^6 + 2916*x^5 + 4860*x
^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 31680*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x
^2 + 576*x + 64)*log(2*x - 1) + 85783698*x + 21096187)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 +
576*x + 64)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=- \frac {3849120 x^{5} + 15075720 x^{4} + 24841080 x^{3} + 22413105 x^{2} + 12254814 x + 3013741}{7718950890 x^{6} + 30875803560 x^{5} + 51459672600 x^{4} + 45741931200 x^{3} + 22870965600 x^{2} + 6098924160 x + 677658240} - \frac {352 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {352 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)**7,x)

[Out]

-(3849120*x**5 + 15075720*x**4 + 24841080*x**3 + 22413105*x**2 + 12254814*x + 3013741)/(7718950890*x**6 + 3087
5803560*x**5 + 51459672600*x**4 + 45741931200*x**3 + 22870965600*x**2 + 6098924160*x + 677658240) - 352*log(x
- 1/2)/823543 + 352*log(x + 2/3)/823543

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=-\frac {3849120 \, x^{5} + 15075720 \, x^{4} + 24841080 \, x^{3} + 22413105 \, x^{2} + 12254814 \, x + 3013741}{10588410 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {352}{823543} \, \log \left (3 \, x + 2\right ) - \frac {352}{823543} \, \log \left (2 \, x - 1\right ) \]

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/10588410*(3849120*x^5 + 15075720*x^4 + 24841080*x^3 + 22413105*x^2 + 12254814*x + 3013741)/(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 352/823543*log(3*x + 2) - 352/823543*log(2*x - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=-\frac {3849120 \, x^{5} + 15075720 \, x^{4} + 24841080 \, x^{3} + 22413105 \, x^{2} + 12254814 \, x + 3013741}{10588410 \, {\left (3 \, x + 2\right )}^{6}} + \frac {352}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {352}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/10588410*(3849120*x^5 + 15075720*x^4 + 24841080*x^3 + 22413105*x^2 + 12254814*x + 3013741)/(3*x + 2)^6 + 35
2/823543*log(abs(3*x + 2)) - 352/823543*log(abs(2*x - 1))

Mupad [B] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {3+5 x}{(1-2 x) (2+3 x)^7} \, dx=\frac {704\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {176\,x^5}{352947}+\frac {2068\,x^4}{1058841}+\frac {30668\,x^3}{9529569}+\frac {6149\,x^2}{2117682}+\frac {75647\,x}{47647845}+\frac {3013741}{7718950890}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \]

[In]

int(-(5*x + 3)/((2*x - 1)*(3*x + 2)^7),x)

[Out]

(704*atanh((12*x)/7 + 1/7))/823543 - ((75647*x)/47647845 + (6149*x^2)/2117682 + (30668*x^3)/9529569 + (2068*x^
4)/1058841 + (176*x^5)/352947 + 3013741/7718950890)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x
^5 + x^6 + 64/729)