\(\int (2 x+3 x^2)^3 \, dx\) [87]

Optimal result

Integrand size = 11, antiderivative size = 25 \[ \int \left (2 x+3 x^2\right )^3 \, dx=2 x^4+\frac {36 x^5}{5}+9 x^6+\frac {27 x^7}{7} \]

[Out]

2*x^4+36/5*x^5+9*x^6+27/7*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {625} \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27 x^7}{7}+9 x^6+\frac {36 x^5}{5}+2 x^4 \]

[In]

Int[(2*x + 3*x^2)^3,x]

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7

Rule 625

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rubi steps \begin{align*} \text {integral}& = \int \left (8 x^3+36 x^4+54 x^5+27 x^6\right ) \, dx \\ & = 2 x^4+\frac {36 x^5}{5}+9 x^6+\frac {27 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (2 x+3 x^2\right )^3 \, dx=2 x^4+\frac {36 x^5}{5}+9 x^6+\frac {27 x^7}{7} \]

[In]

Integrate[(2*x + 3*x^2)^3,x]

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {x^{4} \left (135 x^{3}+315 x^{2}+252 x +70\right )}{35}\)	\(21\)
default	\(2 x^{4}+\frac {36}{5} x^{5}+9 x^{6}+\frac {27}{7} x^{7}\)	\(22\)
norman	\(2 x^{4}+\frac {36}{5} x^{5}+9 x^{6}+\frac {27}{7} x^{7}\)	\(22\)
risch	\(2 x^{4}+\frac {36}{5} x^{5}+9 x^{6}+\frac {27}{7} x^{7}\)	\(22\)
parallelrisch	\(2 x^{4}+\frac {36}{5} x^{5}+9 x^{6}+\frac {27}{7} x^{7}\)	\(22\)

[In]

int((3*x^2+2*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/35*x^4*(135*x^3+315*x^2+252*x+70)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27}{7} \, x^{7} + 9 \, x^{6} + \frac {36}{5} \, x^{5} + 2 \, x^{4} \]

[In]

integrate((3*x^2+2*x)^3,x, algorithm="fricas")

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27 x^{7}}{7} + 9 x^{6} + \frac {36 x^{5}}{5} + 2 x^{4} \]

[In]

integrate((3*x**2+2*x)**3,x)

[Out]

27*x**7/7 + 9*x**6 + 36*x**5/5 + 2*x**4

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27}{7} \, x^{7} + 9 \, x^{6} + \frac {36}{5} \, x^{5} + 2 \, x^{4} \]

[In]

integrate((3*x^2+2*x)^3,x, algorithm="maxima")

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27}{7} \, x^{7} + 9 \, x^{6} + \frac {36}{5} \, x^{5} + 2 \, x^{4} \]

[In]

integrate((3*x^2+2*x)^3,x, algorithm="giac")

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \left (2 x+3 x^2\right )^3 \, dx=\frac {27\,x^7}{7}+9\,x^6+\frac {36\,x^5}{5}+2\,x^4 \]

[In]

int((2*x + 3*x^2)^3,x)

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7