3.3187 \(\int (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx\)
Optimal. Leaf size=91 \[ -\frac{7 (3 x+2)^{m+1}}{243 (m+1)}+\frac{107 (3 x+2)^{m+2}}{243 (m+2)}-\frac{185 (3 x+2)^{m+3}}{81 (m+3)}+\frac{1025 (3 x+2)^{m+4}}{243 (m+4)}-\frac{250 (3 x+2)^{m+5}}{243 (m+5)} \]
[Out]
(-7*(2 + 3*x)^(1 + m))/(243*(1 + m)) + (107*(2 + 3*x)^(2 + m))/(243*(2 + m)) - (185*(2 + 3*x)^(3 + m))/(81*(3
+ m)) + (1025*(2 + 3*x)^(4 + m))/(243*(4 + m)) - (250*(2 + 3*x)^(5 + m))/(243*(5 + m))
________________________________________________________________________________________
Rubi [A] time = 0.0204781, antiderivative size = 91, 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} \[ -\frac{7 (3 x+2)^{m+1}}{243 (m+1)}+\frac{107 (3 x+2)^{m+2}}{243 (m+2)}-\frac{185 (3 x+2)^{m+3}}{81 (m+3)}+\frac{1025 (3 x+2)^{m+4}}{243 (m+4)}-\frac{250 (3 x+2)^{m+5}}{243 (m+5)} \]
Antiderivative was successfully verified.
[In]
Int[(1 - 2*x)*(2 + 3*x)^m*(3 + 5*x)^3,x]
[Out]
(-7*(2 + 3*x)^(1 + m))/(243*(1 + m)) + (107*(2 + 3*x)^(2 + m))/(243*(2 + m)) - (185*(2 + 3*x)^(3 + m))/(81*(3
+ m)) + (1025*(2 + 3*x)^(4 + m))/(243*(4 + m)) - (250*(2 + 3*x)^(5 + m))/(243*(5 + m))
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 (1-2 x) (2+3 x)^m (3+5 x)^3 \, dx &=\int \left (-\frac{7}{81} (2+3 x)^m+\frac{107}{81} (2+3 x)^{1+m}-\frac{185}{27} (2+3 x)^{2+m}+\frac{1025}{81} (2+3 x)^{3+m}-\frac{250}{81} (2+3 x)^{4+m}\right ) \, dx\\ &=-\frac{7 (2+3 x)^{1+m}}{243 (1+m)}+\frac{107 (2+3 x)^{2+m}}{243 (2+m)}-\frac{185 (2+3 x)^{3+m}}{81 (3+m)}+\frac{1025 (2+3 x)^{4+m}}{243 (4+m)}-\frac{250 (2+3 x)^{5+m}}{243 (5+m)}\\ \end{align*}
Mathematica [A] time = 0.0299545, size = 75, normalized size = 0.82 \[ \frac{1}{243} (3 x+2)^{m+1} \left (-\frac{250 (3 x+2)^4}{m+5}+\frac{1025 (3 x+2)^3}{m+4}-\frac{555 (3 x+2)^2}{m+3}+\frac{107 (3 x+2)}{m+2}-\frac{7}{m+1}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(1 - 2*x)*(2 + 3*x)^m*(3 + 5*x)^3,x]
[Out]
((2 + 3*x)^(1 + m)*(-7/(1 + m) + (107*(2 + 3*x))/(2 + m) - (555*(2 + 3*x)^2)/(3 + m) + (1025*(2 + 3*x)^3)/(4 +
m) - (250*(2 + 3*x)^4)/(5 + m)))/243
________________________________________________________________________________________
Maple [B] time = 0.007, size = 187, normalized size = 2.1 \begin{align*} -{\frac{ \left ( 2+3\,x \right ) ^{1+m} \left ( 6750\,{m}^{4}{x}^{4}+8775\,{m}^{4}{x}^{3}+67500\,{m}^{3}{x}^{4}+1215\,{m}^{4}{x}^{2}+78525\,{m}^{3}{x}^{3}+236250\,{m}^{2}{x}^{4}-2187\,{m}^{4}x-2970\,{m}^{3}{x}^{2}+251775\,{m}^{2}{x}^{3}+337500\,m{x}^{4}-729\,{m}^{4}-30051\,{m}^{3}x-44865\,{m}^{2}{x}^{2}+337275\,m{x}^{3}+162000\,{x}^{4}-8748\,{m}^{3}-121833\,{m}^{2}x-95580\,m{x}^{2}+155250\,{x}^{3}-33183\,{m}^{2}-188589\,mx-54900\,{x}^{2}-49620\,m-94620\,x-24400 \right ) }{81\,{m}^{5}+1215\,{m}^{4}+6885\,{m}^{3}+18225\,{m}^{2}+22194\,m+9720}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1-2*x)*(2+3*x)^m*(3+5*x)^3,x)
[Out]
-1/81*(2+3*x)^(1+m)*(6750*m^4*x^4+8775*m^4*x^3+67500*m^3*x^4+1215*m^4*x^2+78525*m^3*x^3+236250*m^2*x^4-2187*m^
4*x-2970*m^3*x^2+251775*m^2*x^3+337500*m*x^4-729*m^4-30051*m^3*x-44865*m^2*x^2+337275*m*x^3+162000*x^4-8748*m^
3-121833*m^2*x-95580*m*x^2+155250*x^3-33183*m^2-188589*m*x-54900*x^2-49620*m-94620*x-24400)/(m^5+15*m^4+85*m^3
+225*m^2+274*m+120)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)^m*(3+5*x)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [B] time = 1.35639, size = 543, normalized size = 5.97 \begin{align*} -\frac{{\left (20250 \,{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} + 675 \,{\left (59 \, m^{4} + 549 \, m^{3} + 1819 \, m^{2} + 2499 \, m + 1170\right )} x^{4} - 1458 \, m^{4} + 45 \,{\left (471 \, m^{4} + 3292 \, m^{3} + 8199 \, m^{2} + 8618 \, m + 3240\right )} x^{3} - 17496 \, m^{3} - 9 \,{\left (459 \, m^{4} + 10677 \, m^{3} + 50581 \, m^{2} + 84103 \, m + 43740\right )} x^{2} - 66366 \, m^{2} - 3 \,{\left (2187 \, m^{4} + 28782 \, m^{3} + 114405 \, m^{2} + 175346 \, m + 87480\right )} x - 99240 \, m - 48800\right )}{\left (3 \, x + 2\right )}^{m}}{81 \,{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)^m*(3+5*x)^3,x, algorithm="fricas")
[Out]
-1/81*(20250*(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5 + 675*(59*m^4 + 549*m^3 + 1819*m^2 + 2499*m + 1170)*x^4 -
1458*m^4 + 45*(471*m^4 + 3292*m^3 + 8199*m^2 + 8618*m + 3240)*x^3 - 17496*m^3 - 9*(459*m^4 + 10677*m^3 + 5058
1*m^2 + 84103*m + 43740)*x^2 - 66366*m^2 - 3*(2187*m^4 + 28782*m^3 + 114405*m^2 + 175346*m + 87480)*x - 99240*
m - 48800)*(3*x + 2)^m/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
________________________________________________________________________________________
Sympy [A] time = 2.09324, size = 1826, normalized size = 20.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)**m*(3+5*x)**3,x)
[Out]
Piecewise((-648000*x**4*log(x + 2/3)/(629856*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416) + 909279*
x**4/(629856*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416) - 1728000*x**3*log(x + 2/3)/(629856*x**4
+ 1679616*x**3 + 1679616*x**2 + 746496*x + 124416) + 1539144*x**3/(629856*x**4 + 1679616*x**3 + 1679616*x**2 +
746496*x + 124416) - 1728000*x**2*log(x + 2/3)/(629856*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416
) + 733464*x**2/(629856*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416) - 768000*x*log(x + 2/3)/(62985
6*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416) - 128000*log(x + 2/3)/(629856*x**4 + 1679616*x**3 +
1679616*x**2 + 746496*x + 124416) - 49496/(629856*x**4 + 1679616*x**3 + 1679616*x**2 + 746496*x + 124416), Eq(
m, -5)), (-486000*x**4/(157464*x**3 + 314928*x**2 + 209952*x + 46656) + 664200*x**3*log(x + 2/3)/(157464*x**3
+ 314928*x**2 + 209952*x + 46656) - 980991*x**3/(157464*x**3 + 314928*x**2 + 209952*x + 46656) + 1328400*x**2*
log(x + 2/3)/(157464*x**3 + 314928*x**2 + 209952*x + 46656) - 546102*x**2/(157464*x**3 + 314928*x**2 + 209952*
x + 46656) + 885600*x*log(x + 2/3)/(157464*x**3 + 314928*x**2 + 209952*x + 46656) + 196800*log(x + 2/3)/(15746
4*x**3 + 314928*x**2 + 209952*x + 46656) + 48104/(157464*x**3 + 314928*x**2 + 209952*x + 46656), Eq(m, -4)), (
-13500*x**4/(2916*x**2 + 3888*x + 1296) + 900*x**3/(2916*x**2 + 3888*x + 1296) - 6660*x**2*log(x + 2/3)/(2916*
x**2 + 3888*x + 1296) + 13221*x**2/(2916*x**2 + 3888*x + 1296) - 8880*x*log(x + 2/3)/(2916*x**2 + 3888*x + 129
6) - 2960*log(x + 2/3)/(2916*x**2 + 3888*x + 1296) - 2938/(2916*x**2 + 3888*x + 1296), Eq(m, -3)), (-13500*x**
4/(1458*x + 972) - 8325*x**3/(1458*x + 972) + 9360*x**2/(1458*x + 972) + 642*x*log(x + 2/3)/(1458*x + 972) + 4
28*log(x + 2/3)/(1458*x + 972) - 3946/(1458*x + 972), Eq(m, -2)), (-125*x**4/6 - 475*x**3/27 + 545*x**2/54 + 1
097*x/81 - 7*log(x + 2/3)/243, Eq(m, -1)), (-20250*m**4*x**5*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 1
8225*m**2 + 22194*m + 9720) - 39825*m**4*x**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 221
94*m + 9720) - 21195*m**4*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) +
4131*m**4*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 6561*m**4*x*(3*x
+ 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 1458*m**4*(3*x + 2)**m/(81*m**5 + 1
215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 202500*m**3*x**5*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 688
5*m**3 + 18225*m**2 + 22194*m + 9720) - 370575*m**3*x**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225
*m**2 + 22194*m + 9720) - 148140*m**3*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*
m + 9720) + 96093*m**3*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 863
46*m**3*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 17496*m**3*(3*x + 2)*
*m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 708750*m**2*x**5*(3*x + 2)**m/(81*m**5 +
1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 1227825*m**2*x**4*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6
885*m**3 + 18225*m**2 + 22194*m + 9720) - 368955*m**2*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 182
25*m**2 + 22194*m + 9720) + 455229*m**2*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 2219
4*m + 9720) + 343215*m**2*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 663
66*m**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 1012500*m*x**5*(3*x + 2
)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 1686825*m*x**4*(3*x + 2)**m/(81*m**5 +
1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 387810*m*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*
m**3 + 18225*m**2 + 22194*m + 9720) + 756927*m*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2
+ 22194*m + 9720) + 526038*m*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) +
99240*m*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 486000*x**5*(3*x + 2)*
*m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 789750*x**4*(3*x + 2)**m/(81*m**5 + 1215*
m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) - 145800*x**3*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 +
18225*m**2 + 22194*m + 9720) + 393660*x**2*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*
m + 9720) + 262440*x*(3*x + 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720) + 48800*(3*x
+ 2)**m/(81*m**5 + 1215*m**4 + 6885*m**3 + 18225*m**2 + 22194*m + 9720), True))
________________________________________________________________________________________
Giac [B] time = 2.14964, size = 571, normalized size = 6.27 \begin{align*} -\frac{20250 \, m^{4}{\left (3 \, x + 2\right )}^{m} x^{5} + 39825 \, m^{4}{\left (3 \, x + 2\right )}^{m} x^{4} + 202500 \, m^{3}{\left (3 \, x + 2\right )}^{m} x^{5} + 21195 \, m^{4}{\left (3 \, x + 2\right )}^{m} x^{3} + 370575 \, m^{3}{\left (3 \, x + 2\right )}^{m} x^{4} + 708750 \, m^{2}{\left (3 \, x + 2\right )}^{m} x^{5} - 4131 \, m^{4}{\left (3 \, x + 2\right )}^{m} x^{2} + 148140 \, m^{3}{\left (3 \, x + 2\right )}^{m} x^{3} + 1227825 \, m^{2}{\left (3 \, x + 2\right )}^{m} x^{4} + 1012500 \, m{\left (3 \, x + 2\right )}^{m} x^{5} - 6561 \, m^{4}{\left (3 \, x + 2\right )}^{m} x - 96093 \, m^{3}{\left (3 \, x + 2\right )}^{m} x^{2} + 368955 \, m^{2}{\left (3 \, x + 2\right )}^{m} x^{3} + 1686825 \, m{\left (3 \, x + 2\right )}^{m} x^{4} + 486000 \,{\left (3 \, x + 2\right )}^{m} x^{5} - 1458 \, m^{4}{\left (3 \, x + 2\right )}^{m} - 86346 \, m^{3}{\left (3 \, x + 2\right )}^{m} x - 455229 \, m^{2}{\left (3 \, x + 2\right )}^{m} x^{2} + 387810 \, m{\left (3 \, x + 2\right )}^{m} x^{3} + 789750 \,{\left (3 \, x + 2\right )}^{m} x^{4} - 17496 \, m^{3}{\left (3 \, x + 2\right )}^{m} - 343215 \, m^{2}{\left (3 \, x + 2\right )}^{m} x - 756927 \, m{\left (3 \, x + 2\right )}^{m} x^{2} + 145800 \,{\left (3 \, x + 2\right )}^{m} x^{3} - 66366 \, m^{2}{\left (3 \, x + 2\right )}^{m} - 526038 \, m{\left (3 \, x + 2\right )}^{m} x - 393660 \,{\left (3 \, x + 2\right )}^{m} x^{2} - 99240 \, m{\left (3 \, x + 2\right )}^{m} - 262440 \,{\left (3 \, x + 2\right )}^{m} x - 48800 \,{\left (3 \, x + 2\right )}^{m}}{81 \,{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1-2*x)*(2+3*x)^m*(3+5*x)^3,x, algorithm="giac")
[Out]
-1/81*(20250*m^4*(3*x + 2)^m*x^5 + 39825*m^4*(3*x + 2)^m*x^4 + 202500*m^3*(3*x + 2)^m*x^5 + 21195*m^4*(3*x + 2
)^m*x^3 + 370575*m^3*(3*x + 2)^m*x^4 + 708750*m^2*(3*x + 2)^m*x^5 - 4131*m^4*(3*x + 2)^m*x^2 + 148140*m^3*(3*x
+ 2)^m*x^3 + 1227825*m^2*(3*x + 2)^m*x^4 + 1012500*m*(3*x + 2)^m*x^5 - 6561*m^4*(3*x + 2)^m*x - 96093*m^3*(3*
x + 2)^m*x^2 + 368955*m^2*(3*x + 2)^m*x^3 + 1686825*m*(3*x + 2)^m*x^4 + 486000*(3*x + 2)^m*x^5 - 1458*m^4*(3*x
+ 2)^m - 86346*m^3*(3*x + 2)^m*x - 455229*m^2*(3*x + 2)^m*x^2 + 387810*m*(3*x + 2)^m*x^3 + 789750*(3*x + 2)^m
*x^4 - 17496*m^3*(3*x + 2)^m - 343215*m^2*(3*x + 2)^m*x - 756927*m*(3*x + 2)^m*x^2 + 145800*(3*x + 2)^m*x^3 -
66366*m^2*(3*x + 2)^m - 526038*m*(3*x + 2)^m*x - 393660*(3*x + 2)^m*x^2 - 99240*m*(3*x + 2)^m - 262440*(3*x +
2)^m*x - 48800*(3*x + 2)^m)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)