3.26.53 \(\int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx\)

Optimal. Leaf size=73 \[ \frac {7 (3 x+2)^{m+1}}{81 (m+1)}-\frac {8 (3 x+2)^{m+2}}{9 (m+2)}+\frac {65 (3 x+2)^{m+3}}{27 (m+3)}-\frac {50 (3 x+2)^{m+4}}{81 (m+4)} \]

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Rubi [A]  time = 0.02, antiderivative size = 73, 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 {7 (3 x+2)^{m+1}}{81 (m+1)}-\frac {8 (3 x+2)^{m+2}}{9 (m+2)}+\frac {65 (3 x+2)^{m+3}}{27 (m+3)}-\frac {50 (3 x+2)^{m+4}}{81 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^(1 + m))/(81*(1 + m)) - (8*(2 + 3*x)^(2 + m))/(9*(2 + m)) + (65*(2 + 3*x)^(3 + m))/(27*(3 + m)) -
(50*(2 + 3*x)^(4 + m))/(81*(4 + 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)^2 \, dx &=\int \left (\frac {7}{27} (2+3 x)^m-\frac {8}{3} (2+3 x)^{1+m}+\frac {65}{9} (2+3 x)^{2+m}-\frac {50}{27} (2+3 x)^{3+m}\right ) \, dx\\ &=\frac {7 (2+3 x)^{1+m}}{81 (1+m)}-\frac {8 (2+3 x)^{2+m}}{9 (2+m)}+\frac {65 (2+3 x)^{3+m}}{27 (3+m)}-\frac {50 (2+3 x)^{4+m}}{81 (4+m)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{81} (3 x+2)^{m+1} \left (-\frac {50 (3 x+2)^3}{m+4}+\frac {195 (3 x+2)^2}{m+3}-\frac {72 (3 x+2)}{m+2}+\frac {7}{m+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2 + 3*x)^(1 + m)*(7/(1 + m) - (72*(2 + 3*x))/(2 + m) + (195*(2 + 3*x)^2)/(3 + m) - (50*(2 + 3*x)^3)/(4 + m))
)/81

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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (1-2 x) (2+3 x)^m (3+5 x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 120, normalized size = 1.64 \begin {gather*} -\frac {{\left (1350 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 45 \, {\left (41 \, m^{3} + 207 \, m^{2} + 334 \, m + 168\right )} x^{3} - 162 \, m^{3} + 18 \, {\left (17 \, m^{3} - 69 \, m^{2} - 302 \, m - 216\right )} x^{2} - 1314 \, m^{2} - 3 \, {\left (153 \, m^{3} + 1513 \, m^{2} + 3290 \, m + 1944\right )} x - 2644 \, m - 1520\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(1350*(m^3 + 6*m^2 + 11*m + 6)*x^4 + 45*(41*m^3 + 207*m^2 + 334*m + 168)*x^3 - 162*m^3 + 18*(17*m^3 - 69
*m^2 - 302*m - 216)*x^2 - 1314*m^2 - 3*(153*m^3 + 1513*m^2 + 3290*m + 1944)*x - 2644*m - 1520)*(3*x + 2)^m/(m^
4 + 10*m^3 + 35*m^2 + 50*m + 24)

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giac [B]  time = 0.95, size = 278, normalized size = 3.81 \begin {gather*} -\frac {1350 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{4} + 1845 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{3} + 8100 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{4} + 306 \, m^{3} {\left (3 \, x + 2\right )}^{m} x^{2} + 9315 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{3} + 14850 \, m {\left (3 \, x + 2\right )}^{m} x^{4} - 459 \, m^{3} {\left (3 \, x + 2\right )}^{m} x - 1242 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{2} + 15030 \, m {\left (3 \, x + 2\right )}^{m} x^{3} + 8100 \, {\left (3 \, x + 2\right )}^{m} x^{4} - 162 \, m^{3} {\left (3 \, x + 2\right )}^{m} - 4539 \, m^{2} {\left (3 \, x + 2\right )}^{m} x - 5436 \, m {\left (3 \, x + 2\right )}^{m} x^{2} + 7560 \, {\left (3 \, x + 2\right )}^{m} x^{3} - 1314 \, m^{2} {\left (3 \, x + 2\right )}^{m} - 9870 \, m {\left (3 \, x + 2\right )}^{m} x - 3888 \, {\left (3 \, x + 2\right )}^{m} x^{2} - 2644 \, m {\left (3 \, x + 2\right )}^{m} - 5832 \, {\left (3 \, x + 2\right )}^{m} x - 1520 \, {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(1350*m^3*(3*x + 2)^m*x^4 + 1845*m^3*(3*x + 2)^m*x^3 + 8100*m^2*(3*x + 2)^m*x^4 + 306*m^3*(3*x + 2)^m*x^
2 + 9315*m^2*(3*x + 2)^m*x^3 + 14850*m*(3*x + 2)^m*x^4 - 459*m^3*(3*x + 2)^m*x - 1242*m^2*(3*x + 2)^m*x^2 + 15
030*m*(3*x + 2)^m*x^3 + 8100*(3*x + 2)^m*x^4 - 162*m^3*(3*x + 2)^m - 4539*m^2*(3*x + 2)^m*x - 5436*m*(3*x + 2)
^m*x^2 + 7560*(3*x + 2)^m*x^3 - 1314*m^2*(3*x + 2)^m - 9870*m*(3*x + 2)^m*x - 3888*(3*x + 2)^m*x^2 - 2644*m*(3
*x + 2)^m - 5832*(3*x + 2)^m*x - 1520*(3*x + 2)^m)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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maple [A]  time = 0.01, size = 120, normalized size = 1.64 \begin {gather*} -\frac {\left (450 m^{3} x^{3}+315 m^{3} x^{2}+2700 m^{2} x^{3}-108 m^{3} x +1305 m^{2} x^{2}+4950 m \,x^{3}-81 m^{3}-1284 m^{2} x +1710 m \,x^{2}+2700 x^{3}-657 m^{2}-2952 m x +720 x^{2}-1322 m -1776 x -760\right ) \left (3 x +2\right )^{m +1}}{27 \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(3*x+2)^(m+1)*(450*m^3*x^3+315*m^3*x^2+2700*m^2*x^3-108*m^3*x+1305*m^2*x^2+4950*m*x^3-81*m^3-1284*m^2*x+
1710*m*x^2+2700*x^3-657*m^2-2952*m*x+720*x^2-1322*m-1776*x-760)/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [B]  time = 0.47, size = 183, normalized size = 2.51 \begin {gather*} -\frac {50 \, {\left (27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} + 18 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} x^{3} - 36 \, {\left (m^{2} + m\right )} x^{2} + 48 \, m x - 32\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )}} - \frac {35 \, {\left (27 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 18 \, {\left (m^{2} + m\right )} x^{2} - 24 \, m x + 16\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} + \frac {4 \, {\left (9 \, {\left (m + 1\right )} x^{2} + 6 \, m x - 4\right )} {\left (3 \, x + 2\right )}^{m}}{3 \, {\left (m^{2} + 3 \, m + 2\right )}} + \frac {3 \, {\left (3 \, x + 2\right )}^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50/27*(27*(m^3 + 6*m^2 + 11*m + 6)*x^4 + 18*(m^3 + 3*m^2 + 2*m)*x^3 - 36*(m^2 + m)*x^2 + 48*m*x - 32)*(3*x +
2)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24) - 35/27*(27*(m^2 + 3*m + 2)*x^3 + 18*(m^2 + m)*x^2 - 24*m*x + 16)*(3*
x + 2)^m/(m^3 + 6*m^2 + 11*m + 6) + 4/3*(9*(m + 1)*x^2 + 6*m*x - 4)*(3*x + 2)^m/(m^2 + 3*m + 2) + 3*(3*x + 2)^
(m + 1)/(m + 1)

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mupad [B]  time = 2.54, size = 211, normalized size = 2.89 \begin {gather*} {\left (3\,x+2\right )}^m\,\left (\frac {162\,m^3+1314\,m^2+2644\,m+1520}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}+\frac {x\,\left (459\,m^3+4539\,m^2+9870\,m+5832\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}+\frac {x^2\,\left (-306\,m^3+1242\,m^2+5436\,m+3888\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}-\frac {x^4\,\left (1350\,m^3+8100\,m^2+14850\,m+8100\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}-\frac {x^3\,\left (1845\,m^3+9315\,m^2+15030\,m+7560\right )}{27\,m^4+270\,m^3+945\,m^2+1350\,m+648}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x + 2)^m*((2644*m + 1314*m^2 + 162*m^3 + 1520)/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) + (x*(9870*m + 4
539*m^2 + 459*m^3 + 5832))/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) + (x^2*(5436*m + 1242*m^2 - 306*m^3 + 3
888))/(1350*m + 945*m^2 + 270*m^3 + 27*m^4 + 648) - (x^4*(14850*m + 8100*m^2 + 1350*m^3 + 8100))/(1350*m + 945
*m^2 + 270*m^3 + 27*m^4 + 648) - (x^3*(15030*m + 9315*m^2 + 1845*m^3 + 7560))/(1350*m + 945*m^2 + 270*m^3 + 27
*m^4 + 648))

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sympy [A]  time = 1.76, size = 1017, normalized size = 13.93

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4050*x**3*log(x + 2/3)/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 8100*x**2*log(x + 2/3)/(6561*x**
3 + 13122*x**2 + 8748*x + 1944) - 5265*x**2/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 5400*x*log(x + 2/3)/(65
61*x**3 + 13122*x**2 + 8748*x + 1944) - 6696*x/(6561*x**3 + 13122*x**2 + 8748*x + 1944) - 1200*log(x + 2/3)/(6
561*x**3 + 13122*x**2 + 8748*x + 1944) - 2131/(6561*x**3 + 13122*x**2 + 8748*x + 1944), Eq(m, -4)), (-900*x**3
/(486*x**2 + 648*x + 216) + 1170*x**2*log(x + 2/3)/(486*x**2 + 648*x + 216) + 1560*x*log(x + 2/3)/(486*x**2 +
648*x + 216) + 1344*x/(486*x**2 + 648*x + 216) + 520*log(x + 2/3)/(486*x**2 + 648*x + 216) + 627/(486*x**2 + 6
48*x + 216), Eq(m, -3)), (-75*x**3/(27*x + 18) + 45*x**2/(27*x + 18) - 24*x*log(x + 2/3)/(27*x + 18) - 16*log(
x + 2/3)/(27*x + 18) - 43/(27*x + 18), Eq(m, -2)), (-50*x**3/9 - 5*x**2/18 + 118*x/27 + 7*log(x + 2/3)/81, Eq(
m, -1)), (-1350*m**3*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 1845*m**3*x**3*(3*x +
2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 306*m**3*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m*
*2 + 1350*m + 648) + 459*m**3*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 162*m**3*(3*x +
2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 8100*m**2*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m
**2 + 1350*m + 648) - 9315*m**2*x**3*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 1242*m**2*x
**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 4539*m**2*x*(3*x + 2)**m/(27*m**4 + 270*m**3
+ 945*m**2 + 1350*m + 648) + 1314*m**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 14850*m*
x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 15030*m*x**3*(3*x + 2)**m/(27*m**4 + 270*m*
*3 + 945*m**2 + 1350*m + 648) + 5436*m*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 9870
*m*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 2644*m*(3*x + 2)**m/(27*m**4 + 270*m**3 + 9
45*m**2 + 1350*m + 648) - 8100*x**4*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) - 7560*x**3*(3
*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 3888*x**2*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m
**2 + 1350*m + 648) + 5832*x*(3*x + 2)**m/(27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648) + 1520*(3*x + 2)**m/(
27*m**4 + 270*m**3 + 945*m**2 + 1350*m + 648), True))

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