3.1372 \(\int (1-2 x)^3 (2+3 x)^7 (3+5 x)^3 \, dx\)

Optimal. Leaf size=78 \[ -\frac{500 (3 x+2)^{14}}{15309}+\frac{3700 (3 x+2)^{13}}{9477}-\frac{7195 (3 x+2)^{12}}{4374}+\frac{66193 (3 x+2)^{11}}{24057}-\frac{10073 (3 x+2)^{10}}{7290}+\frac{1813 (3 x+2)^9}{6561}-\frac{343 (3 x+2)^8}{17496} \]

[Out]

(-343*(2 + 3*x)^8)/17496 + (1813*(2 + 3*x)^9)/6561 - (10073*(2 + 3*x)^10)/7290 + (66193*(2 + 3*x)^11)/24057 -
(7195*(2 + 3*x)^12)/4374 + (3700*(2 + 3*x)^13)/9477 - (500*(2 + 3*x)^14)/15309

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Rubi [A]  time = 0.0355359, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{500 (3 x+2)^{14}}{15309}+\frac{3700 (3 x+2)^{13}}{9477}-\frac{7195 (3 x+2)^{12}}{4374}+\frac{66193 (3 x+2)^{11}}{24057}-\frac{10073 (3 x+2)^{10}}{7290}+\frac{1813 (3 x+2)^9}{6561}-\frac{343 (3 x+2)^8}{17496} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3*(2 + 3*x)^7*(3 + 5*x)^3,x]

[Out]

(-343*(2 + 3*x)^8)/17496 + (1813*(2 + 3*x)^9)/6561 - (10073*(2 + 3*x)^10)/7290 + (66193*(2 + 3*x)^11)/24057 -
(7195*(2 + 3*x)^12)/4374 + (3700*(2 + 3*x)^13)/9477 - (500*(2 + 3*x)^14)/15309

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int (1-2 x)^3 (2+3 x)^7 (3+5 x)^3 \, dx &=\int \left (-\frac{343}{729} (2+3 x)^7+\frac{1813}{243} (2+3 x)^8-\frac{10073}{243} (2+3 x)^9+\frac{66193}{729} (2+3 x)^{10}-\frac{14390}{243} (2+3 x)^{11}+\frac{3700}{243} (2+3 x)^{12}-\frac{1000}{729} (2+3 x)^{13}\right ) \, dx\\ &=-\frac{343 (2+3 x)^8}{17496}+\frac{1813 (2+3 x)^9}{6561}-\frac{10073 (2+3 x)^{10}}{7290}+\frac{66193 (2+3 x)^{11}}{24057}-\frac{7195 (2+3 x)^{12}}{4374}+\frac{3700 (2+3 x)^{13}}{9477}-\frac{500 (2+3 x)^{14}}{15309}\\ \end{align*}

Mathematica [A]  time = 0.0030788, size = 85, normalized size = 1.09 \[ -\frac{1093500 x^{14}}{7}-\frac{10862100 x^{13}}{13}-\frac{3595185 x^{12}}{2}-\frac{19532907 x^{11}}{11}-\frac{2909493 x^{10}}{10}+1119837 x^9+\frac{8511675 x^8}{8}+\frac{1241998 x^7}{7}-299014 x^6-\frac{1022472 x^5}{5}-20732 x^4+31200 x^3+16416 x^2+3456 x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3*(2 + 3*x)^7*(3 + 5*x)^3,x]

[Out]

3456*x + 16416*x^2 + 31200*x^3 - 20732*x^4 - (1022472*x^5)/5 - 299014*x^6 + (1241998*x^7)/7 + (8511675*x^8)/8
+ 1119837*x^9 - (2909493*x^10)/10 - (19532907*x^11)/11 - (3595185*x^12)/2 - (10862100*x^13)/13 - (1093500*x^14
)/7

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Maple [A]  time = 0.002, size = 70, normalized size = 0.9 \begin{align*} -{\frac{1093500\,{x}^{14}}{7}}-{\frac{10862100\,{x}^{13}}{13}}-{\frac{3595185\,{x}^{12}}{2}}-{\frac{19532907\,{x}^{11}}{11}}-{\frac{2909493\,{x}^{10}}{10}}+1119837\,{x}^{9}+{\frac{8511675\,{x}^{8}}{8}}+{\frac{1241998\,{x}^{7}}{7}}-299014\,{x}^{6}-{\frac{1022472\,{x}^{5}}{5}}-20732\,{x}^{4}+31200\,{x}^{3}+16416\,{x}^{2}+3456\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^7*(3+5*x)^3,x)

[Out]

-1093500/7*x^14-10862100/13*x^13-3595185/2*x^12-19532907/11*x^11-2909493/10*x^10+1119837*x^9+8511675/8*x^8+124
1998/7*x^7-299014*x^6-1022472/5*x^5-20732*x^4+31200*x^3+16416*x^2+3456*x

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Maxima [A]  time = 1.02746, size = 93, normalized size = 1.19 \begin{align*} -\frac{1093500}{7} \, x^{14} - \frac{10862100}{13} \, x^{13} - \frac{3595185}{2} \, x^{12} - \frac{19532907}{11} \, x^{11} - \frac{2909493}{10} \, x^{10} + 1119837 \, x^{9} + \frac{8511675}{8} \, x^{8} + \frac{1241998}{7} \, x^{7} - 299014 \, x^{6} - \frac{1022472}{5} \, x^{5} - 20732 \, x^{4} + 31200 \, x^{3} + 16416 \, x^{2} + 3456 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^7*(3+5*x)^3,x, algorithm="maxima")

[Out]

-1093500/7*x^14 - 10862100/13*x^13 - 3595185/2*x^12 - 19532907/11*x^11 - 2909493/10*x^10 + 1119837*x^9 + 85116
75/8*x^8 + 1241998/7*x^7 - 299014*x^6 - 1022472/5*x^5 - 20732*x^4 + 31200*x^3 + 16416*x^2 + 3456*x

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Fricas [A]  time = 1.16343, size = 284, normalized size = 3.64 \begin{align*} -\frac{1093500}{7} x^{14} - \frac{10862100}{13} x^{13} - \frac{3595185}{2} x^{12} - \frac{19532907}{11} x^{11} - \frac{2909493}{10} x^{10} + 1119837 x^{9} + \frac{8511675}{8} x^{8} + \frac{1241998}{7} x^{7} - 299014 x^{6} - \frac{1022472}{5} x^{5} - 20732 x^{4} + 31200 x^{3} + 16416 x^{2} + 3456 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^7*(3+5*x)^3,x, algorithm="fricas")

[Out]

-1093500/7*x^14 - 10862100/13*x^13 - 3595185/2*x^12 - 19532907/11*x^11 - 2909493/10*x^10 + 1119837*x^9 + 85116
75/8*x^8 + 1241998/7*x^7 - 299014*x^6 - 1022472/5*x^5 - 20732*x^4 + 31200*x^3 + 16416*x^2 + 3456*x

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Sympy [A]  time = 0.076897, size = 82, normalized size = 1.05 \begin{align*} - \frac{1093500 x^{14}}{7} - \frac{10862100 x^{13}}{13} - \frac{3595185 x^{12}}{2} - \frac{19532907 x^{11}}{11} - \frac{2909493 x^{10}}{10} + 1119837 x^{9} + \frac{8511675 x^{8}}{8} + \frac{1241998 x^{7}}{7} - 299014 x^{6} - \frac{1022472 x^{5}}{5} - 20732 x^{4} + 31200 x^{3} + 16416 x^{2} + 3456 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(2+3*x)**7*(3+5*x)**3,x)

[Out]

-1093500*x**14/7 - 10862100*x**13/13 - 3595185*x**12/2 - 19532907*x**11/11 - 2909493*x**10/10 + 1119837*x**9 +
8511675*x**8/8 + 1241998*x**7/7 - 299014*x**6 - 1022472*x**5/5 - 20732*x**4 + 31200*x**3 + 16416*x**2 + 3456*
x

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Giac [A]  time = 3.16659, size = 93, normalized size = 1.19 \begin{align*} -\frac{1093500}{7} \, x^{14} - \frac{10862100}{13} \, x^{13} - \frac{3595185}{2} \, x^{12} - \frac{19532907}{11} \, x^{11} - \frac{2909493}{10} \, x^{10} + 1119837 \, x^{9} + \frac{8511675}{8} \, x^{8} + \frac{1241998}{7} \, x^{7} - 299014 \, x^{6} - \frac{1022472}{5} \, x^{5} - 20732 \, x^{4} + 31200 \, x^{3} + 16416 \, x^{2} + 3456 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(2+3*x)^7*(3+5*x)^3,x, algorithm="giac")

[Out]

-1093500/7*x^14 - 10862100/13*x^13 - 3595185/2*x^12 - 19532907/11*x^11 - 2909493/10*x^10 + 1119837*x^9 + 85116
75/8*x^8 + 1241998/7*x^7 - 299014*x^6 - 1022472/5*x^5 - 20732*x^4 + 31200*x^3 + 16416*x^2 + 3456*x