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

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

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 52, normalized size = 0.93 \begin {gather*} -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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1-2 x) (2+3 x)^4 (3+5 x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.25, size = 44, normalized size = 0.79 \begin {gather*} -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 {gather*}

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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giac [A]  time = 1.19, size = 44, normalized size = 0.79 \begin {gather*} -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 {gather*}

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

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maple [A]  time = 0.00, size = 45, normalized size = 0.80 \begin {gather*} -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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3*x+2)^4*(5*x+3)^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 = 0.60, size = 44, normalized size = 0.79 \begin {gather*} -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 {gather*}

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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mupad [B]  time = 0.03, size = 44, normalized size = 0.79 \begin {gather*} -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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.07, size = 49, normalized size = 0.88 \begin {gather*} - 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 {gather*}

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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