3.1179 \(\int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx\)

Optimal. Leaf size=56 \[ -\frac{250 (3 x+2)^9}{2187}+\frac{1025 (3 x+2)^8}{1944}-\frac{185}{567} (3 x+2)^7+\frac{107 (3 x+2)^6}{1458}-\frac{7 (3 x+2)^5}{1215} \]

[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

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Rubi [A]  time = 0.0234432, antiderivative size = 56, 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} \[ -\frac{250 (3 x+2)^9}{2187}+\frac{1025 (3 x+2)^8}{1944}-\frac{185}{567} (3 x+2)^7+\frac{107 (3 x+2)^6}{1458}-\frac{7 (3 x+2)^5}{1215} \]

Antiderivative was successfully verified.

[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 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)^4 (3+5 x)^3 \, dx &=\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]  time = 0.0019635, size = 52, normalized size = 0.93 \[ -2250 x^9-\frac{80325 x^8}{8}-\frac{127845 x^7}{7}-\frac{32453 x^6}{2}-\frac{25237 x^5}{5}+3452 x^4+4296 x^3+1944 x^2+432 x \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \begin{align*} -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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.06779, size = 59, normalized size = 1.05 \begin{align*} -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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.56031, size = 147, normalized size = 2.62 \begin{align*} -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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.065698, size = 49, normalized size = 0.88 \begin{align*} - 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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 2.05281, size = 59, normalized size = 1.05 \begin{align*} -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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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