\(\int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx\) [1179]

Optimal result

Integrand size = 20, antiderivative size = 56 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {7 (2+3 x)^5}{1215}+\frac {107 (2+3 x)^6}{1458}-\frac {185}{567} (2+3 x)^7+\frac {1025 (2+3 x)^8}{1944}-\frac {250 (2+3 x)^9}{2187} \]

[Out]

-7/1215*(2+3*x)^5+107/1458*(2+3*x)^6-185/567*(2+3*x)^7+1025/1944*(2+3*x)^8-250/2187*(2+3*x)^9

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, 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} \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-\frac {250 (3 x+2)^9}{2187}+\frac {1025 (3 x+2)^8}{1944}-\frac {185}{567} (3 x+2)^7+\frac {107 (3 x+2)^6}{1458}-\frac {7 (3 x+2)^5}{1215} \]

[In]

Int[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

(-7*(2 + 3*x)^5)/1215 + (107*(2 + 3*x)^6)/1458 - (185*(2 + 3*x)^7)/567 + (1025*(2 + 3*x)^8)/1944 - (250*(2 + 3
*x)^9)/2187

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)^4+\frac {107}{81} (2+3 x)^5-\frac {185}{27} (2+3 x)^6+\frac {1025}{81} (2+3 x)^7-\frac {250}{81} (2+3 x)^8\right ) \, dx \\ & = -\frac {7 (2+3 x)^5}{1215}+\frac {107 (2+3 x)^6}{1458}-\frac {185}{567} (2+3 x)^7+\frac {1025 (2+3 x)^8}{1944}-\frac {250 (2+3 x)^9}{2187} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=432 x+1944 x^2+4296 x^3+3452 x^4-\frac {25237 x^5}{5}-\frac {32453 x^6}{2}-\frac {127845 x^7}{7}-\frac {80325 x^8}{8}-2250 x^9 \]

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

432*x + 1944*x^2 + 4296*x^3 + 3452*x^4 - (25237*x^5)/5 - (32453*x^6)/2 - (127845*x^7)/7 - (80325*x^8)/8 - 2250
*x^9

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79

method	result	size
gosper	\(-\frac {x \left (630000 x^{8}+2811375 x^{7}+5113800 x^{6}+4543420 x^{5}+1413272 x^{4}-966560 x^{3}-1202880 x^{2}-544320 x -120960\right )}{280}\)	\(44\)
default	\(-2250 x^{9}-\frac {80325}{8} x^{8}-\frac {127845}{7} x^{7}-\frac {32453}{2} x^{6}-\frac {25237}{5} x^{5}+3452 x^{4}+4296 x^{3}+1944 x^{2}+432 x\)	\(45\)
norman	\(-2250 x^{9}-\frac {80325}{8} x^{8}-\frac {127845}{7} x^{7}-\frac {32453}{2} x^{6}-\frac {25237}{5} x^{5}+3452 x^{4}+4296 x^{3}+1944 x^{2}+432 x\)	\(45\)
risch	\(-2250 x^{9}-\frac {80325}{8} x^{8}-\frac {127845}{7} x^{7}-\frac {32453}{2} x^{6}-\frac {25237}{5} x^{5}+3452 x^{4}+4296 x^{3}+1944 x^{2}+432 x\)	\(45\)
parallelrisch	\(-2250 x^{9}-\frac {80325}{8} x^{8}-\frac {127845}{7} x^{7}-\frac {32453}{2} x^{6}-\frac {25237}{5} x^{5}+3452 x^{4}+4296 x^{3}+1944 x^{2}+432 x\)	\(45\)

[In]

int((1-2*x)*(2+3*x)^4*(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/280*x*(630000*x^8+2811375*x^7+5113800*x^6+4543420*x^5+1413272*x^4-966560*x^3-1202880*x^2-544320*x-120960)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-2250 \, x^{9} - \frac {80325}{8} \, x^{8} - \frac {127845}{7} \, x^{7} - \frac {32453}{2} \, x^{6} - \frac {25237}{5} \, x^{5} + 3452 \, x^{4} + 4296 \, x^{3} + 1944 \, x^{2} + 432 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x)^3,x, algorithm="fricas")

[Out]

-2250*x^9 - 80325/8*x^8 - 127845/7*x^7 - 32453/2*x^6 - 25237/5*x^5 + 3452*x^4 + 4296*x^3 + 1944*x^2 + 432*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=- 2250 x^{9} - \frac {80325 x^{8}}{8} - \frac {127845 x^{7}}{7} - \frac {32453 x^{6}}{2} - \frac {25237 x^{5}}{5} + 3452 x^{4} + 4296 x^{3} + 1944 x^{2} + 432 x \]

[In]

integrate((1-2*x)*(2+3*x)**4*(3+5*x)**3,x)

[Out]

-2250*x**9 - 80325*x**8/8 - 127845*x**7/7 - 32453*x**6/2 - 25237*x**5/5 + 3452*x**4 + 4296*x**3 + 1944*x**2 +
432*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-2250 \, x^{9} - \frac {80325}{8} \, x^{8} - \frac {127845}{7} \, x^{7} - \frac {32453}{2} \, x^{6} - \frac {25237}{5} \, x^{5} + 3452 \, x^{4} + 4296 \, x^{3} + 1944 \, x^{2} + 432 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x)^3,x, algorithm="maxima")

[Out]

-2250*x^9 - 80325/8*x^8 - 127845/7*x^7 - 32453/2*x^6 - 25237/5*x^5 + 3452*x^4 + 4296*x^3 + 1944*x^2 + 432*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-2250 \, x^{9} - \frac {80325}{8} \, x^{8} - \frac {127845}{7} \, x^{7} - \frac {32453}{2} \, x^{6} - \frac {25237}{5} \, x^{5} + 3452 \, x^{4} + 4296 \, x^{3} + 1944 \, x^{2} + 432 \, x \]

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x)^3,x, algorithm="giac")

[Out]

-2250*x^9 - 80325/8*x^8 - 127845/7*x^7 - 32453/2*x^6 - 25237/5*x^5 + 3452*x^4 + 4296*x^3 + 1944*x^2 + 432*x

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx=-2250\,x^9-\frac {80325\,x^8}{8}-\frac {127845\,x^7}{7}-\frac {32453\,x^6}{2}-\frac {25237\,x^5}{5}+3452\,x^4+4296\,x^3+1944\,x^2+432\,x \]

[In]

int(-(2*x - 1)*(3*x + 2)^4*(5*x + 3)^3,x)

[Out]

432*x + 1944*x^2 + 4296*x^3 + 3452*x^4 - (25237*x^5)/5 - (32453*x^6)/2 - (127845*x^7)/7 - (80325*x^8)/8 - 2250
*x^9