3.1207 \(\int \frac{(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{4374 x^7}{175}-\frac{21627 x^6}{250}-\frac{336798 x^5}{3125}-\frac{513783 x^4}{12500}+\frac{92592 x^3}{3125}+\frac{5740767 x^2}{156250}+\frac{5555478 x}{390625}-\frac{11}{1953125 (5 x+3)}+\frac{229 \log (5 x+3)}{1953125} \]

[Out]

(5555478*x)/390625 + (5740767*x^2)/156250 + (92592*x^3)/3125 - (513783*x^4)/12500 - (336798*x^5)/3125 - (21627
*x^6)/250 - (4374*x^7)/175 - 11/(1953125*(3 + 5*x)) + (229*Log[3 + 5*x])/1953125

________________________________________________________________________________________

Rubi [A]  time = 0.0371761, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{4374 x^7}{175}-\frac{21627 x^6}{250}-\frac{336798 x^5}{3125}-\frac{513783 x^4}{12500}+\frac{92592 x^3}{3125}+\frac{5740767 x^2}{156250}+\frac{5555478 x}{390625}-\frac{11}{1953125 (5 x+3)}+\frac{229 \log (5 x+3)}{1953125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5555478*x)/390625 + (5740767*x^2)/156250 + (92592*x^3)/3125 - (513783*x^4)/12500 - (336798*x^5)/3125 - (21627
*x^6)/250 - (4374*x^7)/175 - 11/(1953125*(3 + 5*x)) + (229*Log[3 + 5*x])/1953125

Rule 77

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*} \int \frac{(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx &=\int \left (\frac{5555478}{390625}+\frac{5740767 x}{78125}+\frac{277776 x^2}{3125}-\frac{513783 x^3}{3125}-\frac{336798 x^4}{625}-\frac{64881 x^5}{125}-\frac{4374 x^6}{25}+\frac{11}{390625 (3+5 x)^2}+\frac{229}{390625 (3+5 x)}\right ) \, dx\\ &=\frac{5555478 x}{390625}+\frac{5740767 x^2}{156250}+\frac{92592 x^3}{3125}-\frac{513783 x^4}{12500}-\frac{336798 x^5}{3125}-\frac{21627 x^6}{250}-\frac{4374 x^7}{175}-\frac{11}{1953125 (3+5 x)}+\frac{229 \log (3+5 x)}{1953125}\\ \end{align*}

Mathematica [A]  time = 0.0345407, size = 85, normalized size = 1.23 \[ \frac{-1875000 (3 x+2)^7+6781250 (3 x+2)^6+3360000 (3 x+2)^5+1273125 (3 x+2)^4+455000 (3 x+2)^3+171150 (3 x+2)^2+82320 (3 x+2)-\frac{924}{5 x+3}+19236 \log (-3 (5 x+3))}{164062500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(82320*(2 + 3*x) + 171150*(2 + 3*x)^2 + 455000*(2 + 3*x)^3 + 1273125*(2 + 3*x)^4 + 3360000*(2 + 3*x)^5 + 67812
50*(2 + 3*x)^6 - 1875000*(2 + 3*x)^7 - 924/(3 + 5*x) + 19236*Log[-3*(3 + 5*x)])/164062500

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{5555478\,x}{390625}}+{\frac{5740767\,{x}^{2}}{156250}}+{\frac{92592\,{x}^{3}}{3125}}-{\frac{513783\,{x}^{4}}{12500}}-{\frac{336798\,{x}^{5}}{3125}}-{\frac{21627\,{x}^{6}}{250}}-{\frac{4374\,{x}^{7}}{175}}-{\frac{11}{5859375+9765625\,x}}+{\frac{229\,\ln \left ( 3+5\,x \right ) }{1953125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5555478/390625*x+5740767/156250*x^2+92592/3125*x^3-513783/12500*x^4-336798/3125*x^5-21627/250*x^6-4374/175*x^7
-11/1953125/(3+5*x)+229/1953125*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.11546, size = 69, normalized size = 1. \begin{align*} -\frac{4374}{175} \, x^{7} - \frac{21627}{250} \, x^{6} - \frac{336798}{3125} \, x^{5} - \frac{513783}{12500} \, x^{4} + \frac{92592}{3125} \, x^{3} + \frac{5740767}{156250} \, x^{2} + \frac{5555478}{390625} \, x - \frac{11}{1953125 \,{\left (5 \, x + 3\right )}} + \frac{229}{1953125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4374/175*x^7 - 21627/250*x^6 - 336798/3125*x^5 - 513783/12500*x^4 + 92592/3125*x^3 + 5740767/156250*x^2 + 555
5478/390625*x - 11/1953125/(5*x + 3) + 229/1953125*log(5*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.49409, size = 266, normalized size = 3.86 \begin{align*} -\frac{6834375000 \, x^{8} + 27755156250 \, x^{7} + 43662543750 \, x^{6} + 28920898125 \, x^{5} - 1358398125 \, x^{4} - 14907422250 \, x^{3} - 9916639950 \, x^{2} - 6412 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2333300760 \, x + 308}{54687500 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54687500*(6834375000*x^8 + 27755156250*x^7 + 43662543750*x^6 + 28920898125*x^5 - 1358398125*x^4 - 149074222
50*x^3 - 9916639950*x^2 - 6412*(5*x + 3)*log(5*x + 3) - 2333300760*x + 308)/(5*x + 3)

________________________________________________________________________________________

Sympy [A]  time = 0.111641, size = 61, normalized size = 0.88 \begin{align*} - \frac{4374 x^{7}}{175} - \frac{21627 x^{6}}{250} - \frac{336798 x^{5}}{3125} - \frac{513783 x^{4}}{12500} + \frac{92592 x^{3}}{3125} + \frac{5740767 x^{2}}{156250} + \frac{5555478 x}{390625} + \frac{229 \log{\left (5 x + 3 \right )}}{1953125} - \frac{11}{9765625 x + 5859375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4374*x**7/175 - 21627*x**6/250 - 336798*x**5/3125 - 513783*x**4/12500 + 92592*x**3/3125 + 5740767*x**2/156250
+ 5555478*x/390625 + 229*log(5*x + 3)/1953125 - 11/(9765625*x + 5859375)

________________________________________________________________________________________

Giac [A]  time = 1.94223, size = 126, normalized size = 1.83 \begin{align*} \frac{3}{273437500} \,{\left (5 \, x + 3\right )}^{7}{\left (\frac{107730}{5 \, x + 3} + \frac{428652}{{\left (5 \, x + 3\right )}^{2}} + \frac{588735}{{\left (5 \, x + 3\right )}^{3}} + \frac{455700}{{\left (5 \, x + 3\right )}^{4}} + \frac{233730}{{\left (5 \, x + 3\right )}^{5}} + \frac{95060}{{\left (5 \, x + 3\right )}^{6}} - 29160\right )} - \frac{11}{1953125 \,{\left (5 \, x + 3\right )}} - \frac{229}{1953125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/273437500*(5*x + 3)^7*(107730/(5*x + 3) + 428652/(5*x + 3)^2 + 588735/(5*x + 3)^3 + 455700/(5*x + 3)^4 + 233
730/(5*x + 3)^5 + 95060/(5*x + 3)^6 - 29160) - 11/1953125/(5*x + 3) - 229/1953125*log(1/5*abs(5*x + 3)/(5*x +
3)^2)