\(\int (a+b x+c x^2)^3 \, dx\) [2135]

Optimal result

Integrand size = 12, antiderivative size = 81 \[ \int \left (a+b x+c x^2\right )^3 \, dx=a^3 x+\frac {3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac {1}{4} b \left (b^2+6 a c\right ) x^4+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]

[Out]

a^3*x+3/2*a^2*b*x^2+a*(a*c+b^2)*x^3+1/4*b*(6*a*c+b^2)*x^4+3/5*c*(a*c+b^2)*x^5+1/2*b*c^2*x^6+1/7*c^3*x^7

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {625} \[ \int \left (a+b x+c x^2\right )^3 \, dx=a^3 x+\frac {3}{2} a^2 b x^2+\frac {3}{5} c x^5 \left (a c+b^2\right )+\frac {1}{4} b x^4 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]

[In]

Int[(a + b*x + c*x^2)^3,x]

[Out]

a^3*x + (3*a^2*b*x^2)/2 + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^4)/4 + (3*c*(b^2 + a*c)*x^5)/5 + (b*c^2*x^6)/
2 + (c^3*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 (a^3+3 a^2 b x+3 a b^2 \left (1+\frac {a c}{b^2}\right ) x^2+b^3 \left (1+\frac {6 a c}{b^2}\right ) x^3+3 b^2 c \left (1+\frac {a c}{b^2}\right ) x^4+3 b c^2 x^5+c^3 x^6\right ) \, dx \\ & = a^3 x+\frac {3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac {1}{4} b \left (b^2+6 a c\right ) x^4+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \left (a+b x+c x^2\right )^3 \, dx=a^3 x+\frac {3}{2} a^2 b x^2+a \left (b^2+a c\right ) x^3+\frac {1}{4} b \left (b^2+6 a c\right ) x^4+\frac {3}{5} c \left (b^2+a c\right ) x^5+\frac {1}{2} b c^2 x^6+\frac {c^3 x^7}{7} \]

[In]

Integrate[(a + b*x + c*x^2)^3,x]

[Out]

a^3*x + (3*a^2*b*x^2)/2 + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^4)/4 + (3*c*(b^2 + a*c)*x^5)/5 + (b*c^2*x^6)/
2 + (c^3*x^7)/7

Maple [A] (verified)

Time = 4.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {c^{3} x^{7}}{7}+\frac {b \,c^{2} x^{6}}{2}+\left (\frac {3}{5} c^{2} a +\frac {3}{5} b^{2} c \right ) x^{5}+\left (\frac {3}{2} a b c +\frac {1}{4} b^{3}\right ) x^{4}+\left (a^{2} c +a \,b^{2}\right ) x^{3}+\frac {3 a^{2} b \,x^{2}}{2}+a^{3} x\)	\(80\)
gosper	\(\frac {1}{7} c^{3} x^{7}+\frac {1}{2} b \,c^{2} x^{6}+\frac {3}{5} a \,c^{2} x^{5}+\frac {3}{5} b^{2} c \,x^{5}+\frac {3}{2} a b c \,x^{4}+\frac {1}{4} b^{3} x^{4}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+\frac {3}{2} a^{2} b \,x^{2}+a^{3} x\)	\(83\)
risch	\(\frac {1}{7} c^{3} x^{7}+\frac {1}{2} b \,c^{2} x^{6}+\frac {3}{5} a \,c^{2} x^{5}+\frac {3}{5} b^{2} c \,x^{5}+\frac {3}{2} a b c \,x^{4}+\frac {1}{4} b^{3} x^{4}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+\frac {3}{2} a^{2} b \,x^{2}+a^{3} x\)	\(83\)
parallelrisch	\(\frac {1}{7} c^{3} x^{7}+\frac {1}{2} b \,c^{2} x^{6}+\frac {3}{5} a \,c^{2} x^{5}+\frac {3}{5} b^{2} c \,x^{5}+\frac {3}{2} a b c \,x^{4}+\frac {1}{4} b^{3} x^{4}+a^{2} c \,x^{3}+a \,b^{2} x^{3}+\frac {3}{2} a^{2} b \,x^{2}+a^{3} x\)	\(83\)
default	\(\frac {c^{3} x^{7}}{7}+\frac {b \,c^{2} x^{6}}{2}+\frac {\left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (4 a b c +b \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right ) x^{3}}{3}+\frac {3 a^{2} b \,x^{2}}{2}+a^{3} x\)	\(108\)

[In]

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

[Out]

1/7*c^3*x^7+1/2*b*c^2*x^6+(3/5*c^2*a+3/5*b^2*c)*x^5+(3/2*a*b*c+1/4*b^3)*x^4+(a^2*c+a*b^2)*x^3+3/2*a^2*b*x^2+a^
3*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, {\left (b^{2} c + a c^{2}\right )} x^{5} + \frac {3}{2} \, a^{2} b x^{2} + \frac {1}{4} \, {\left (b^{3} + 6 \, a b c\right )} x^{4} + a^{3} x + {\left (a b^{2} + a^{2} c\right )} x^{3} \]

[In]

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

[Out]

1/7*c^3*x^7 + 1/2*b*c^2*x^6 + 3/5*(b^2*c + a*c^2)*x^5 + 3/2*a^2*b*x^2 + 1/4*(b^3 + 6*a*b*c)*x^4 + a^3*x + (a*b
^2 + a^2*c)*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \left (a+b x+c x^2\right )^3 \, dx=a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + x^{5} \cdot \left (\frac {3 a c^{2}}{5} + \frac {3 b^{2} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a b c}{2} + \frac {b^{3}}{4}\right ) + x^{3} \left (a^{2} c + a b^{2}\right ) \]

[In]

integrate((c*x**2+b*x+a)**3,x)

[Out]

a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2/5 + 3*b**2*c/5) + x**4*(3*a*b*c/2 + b*
*3/4) + x**3*(a**2*c + a*b**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {1}{4} \, b^{3} x^{4} + a^{3} x + \frac {1}{2} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a^{2} + \frac {1}{10} \, {\left (6 \, c^{2} x^{5} + 15 \, b c x^{4} + 10 \, b^{2} x^{3}\right )} a \]

[In]

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

[Out]

1/7*c^3*x^7 + 1/2*b*c^2*x^6 + 3/5*b^2*c*x^5 + 1/4*b^3*x^4 + a^3*x + 1/2*(2*c*x^3 + 3*b*x^2)*a^2 + 1/10*(6*c^2*
x^5 + 15*b*c*x^4 + 10*b^2*x^3)*a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int \left (a+b x+c x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} x^{7} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{5} \, b^{2} c x^{5} + \frac {3}{5} \, a c^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b c x^{4} + a b^{2} x^{3} + a^{2} c x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \]

[In]

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

[Out]

1/7*c^3*x^7 + 1/2*b*c^2*x^6 + 3/5*b^2*c*x^5 + 3/5*a*c^2*x^5 + 1/4*b^3*x^4 + 3/2*a*b*c*x^4 + a*b^2*x^3 + a^2*c*
x^3 + 3/2*a^2*b*x^2 + a^3*x

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \left (a+b x+c x^2\right )^3 \, dx=x^4\,\left (\frac {b^3}{4}+\frac {3\,a\,c\,b}{2}\right )+a^3\,x+\frac {c^3\,x^7}{7}+\frac {3\,a^2\,b\,x^2}{2}+\frac {b\,c^2\,x^6}{2}+a\,x^3\,\left (b^2+a\,c\right )+\frac {3\,c\,x^5\,\left (b^2+a\,c\right )}{5} \]

[In]

int((a + b*x + c*x^2)^3,x)

[Out]

x^4*(b^3/4 + (3*a*b*c)/2) + a^3*x + (c^3*x^7)/7 + (3*a^2*b*x^2)/2 + (b*c^2*x^6)/2 + a*x^3*(a*c + b^2) + (3*c*x
^5*(a*c + b^2))/5