\(\int (3 a b+3 b^2 x+3 b c x^2+c^2 x^3)^2 \, dx\) [10]

Optimal result

Integrand size = 27, antiderivative size = 56 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=\frac {b^2 \left (b^2-3 a c\right )^2 x}{c^2}-\frac {b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac {(b+c x)^7}{7 c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2085, 200} \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=-\frac {b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac {b^2 x \left (b^2-3 a c\right )^2}{c^2}+\frac {(b+c x)^7}{7 c^3} \]

[In]

Int[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^2,x]

[Out]

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2085

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[1/3^p, Subst[Int[Simp[(3*a*c -
b^2)/c + c^2*(x^3/b), x]^p, x], x, c/(3*d) + x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0] && EqQ[c^2 - 3*b*d
, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \text {Subst}\left (\int \left (3 b \left (3 a-\frac {b^2}{c}\right )+3 c^2 x^3\right )^2 \, dx,x,\frac {b}{c}+x\right ) \\ & = \frac {1}{9} \text {Subst}\left (\int \left (\frac {9 \left (-b^3+3 a b c\right )^2}{c^2}-18 b c \left (b^2-3 a c\right ) x^3+9 c^4 x^6\right ) \, dx,x,\frac {b}{c}+x\right ) \\ & = \frac {b^2 \left (b^2-3 a c\right )^2 x}{c^2}-\frac {b \left (b^2-3 a c\right ) (b+c x)^4}{2 c^3}+\frac {(b+c x)^7}{7 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=9 a^2 b^2 x+9 a b^3 x^2+3 b^2 \left (b^2+2 a c\right ) x^3+\frac {3}{2} b c \left (3 b^2+a c\right ) x^4+3 b^2 c^2 x^5+b c^3 x^6+\frac {c^4 x^7}{7} \]

[In]

Integrate[(3*a*b + 3*b^2*x + 3*b*c*x^2 + c^2*x^3)^2,x]

[Out]

9*a^2*b^2*x + 9*a*b^3*x^2 + 3*b^2*(b^2 + 2*a*c)*x^3 + (3*b*c*(3*b^2 + a*c)*x^4)/2 + 3*b^2*c^2*x^5 + b*c^3*x^6
+ (c^4*x^7)/7

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46

method	result	size
norman	\(\frac {c^{4} x^{7}}{7}+c^{3} x^{6} b +3 b^{2} c^{2} x^{5}+\left (\frac {3}{2} a b \,c^{2}+\frac {9}{2} b^{3} c \right ) x^{4}+\left (6 a \,b^{2} c +3 b^{4}\right ) x^{3}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x\)	\(82\)
gosper	\(\frac {1}{7} c^{4} x^{7}+c^{3} x^{6} b +3 b^{2} c^{2} x^{5}+\frac {3}{2} x^{4} a b \,c^{2}+\frac {9}{2} x^{4} b^{3} c +6 a \,b^{2} c \,x^{3}+3 b^{4} x^{3}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x\)	\(84\)
default	\(\frac {c^{4} x^{7}}{7}+c^{3} x^{6} b +3 b^{2} c^{2} x^{5}+\frac {\left (6 a b \,c^{2}+18 b^{3} c \right ) x^{4}}{4}+\frac {\left (18 a \,b^{2} c +9 b^{4}\right ) x^{3}}{3}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x\)	\(84\)
risch	\(\frac {1}{7} c^{4} x^{7}+c^{3} x^{6} b +3 b^{2} c^{2} x^{5}+\frac {3}{2} x^{4} a b \,c^{2}+\frac {9}{2} x^{4} b^{3} c +6 a \,b^{2} c \,x^{3}+3 b^{4} x^{3}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x\)	\(84\)
parallelrisch	\(\frac {1}{7} c^{4} x^{7}+c^{3} x^{6} b +3 b^{2} c^{2} x^{5}+\frac {3}{2} x^{4} a b \,c^{2}+\frac {9}{2} x^{4} b^{3} c +6 a \,b^{2} c \,x^{3}+3 b^{4} x^{3}+9 a \,b^{3} x^{2}+9 a^{2} b^{2} x\)	\(84\)

[In]

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

[Out]

1/7*c^4*x^7+c^3*x^6*b+3*b^2*c^2*x^5+(3/2*a*b*c^2+9/2*b^3*c)*x^4+(6*a*b^2*c+3*b^4)*x^3+9*a*b^3*x^2+9*a^2*b^2*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.43 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=\frac {1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{5} + 9 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x + \frac {3}{2} \, {\left (3 \, b^{3} c + a b c^{2}\right )} x^{4} + 3 \, {\left (b^{4} + 2 \, a b^{2} c\right )} x^{3} \]

[In]

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

[Out]

1/7*c^4*x^7 + b*c^3*x^6 + 3*b^2*c^2*x^5 + 9*a*b^3*x^2 + 9*a^2*b^2*x + 3/2*(3*b^3*c + a*b*c^2)*x^4 + 3*(b^4 + 2
*a*b^2*c)*x^3

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.55 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{2} c^{2} x^{5} + b c^{3} x^{6} + \frac {c^{4} x^{7}}{7} + x^{4} \cdot \left (\frac {3 a b c^{2}}{2} + \frac {9 b^{3} c}{2}\right ) + x^{3} \cdot \left (6 a b^{2} c + 3 b^{4}\right ) \]

[In]

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

[Out]

9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**2*c**2*x**5 + b*c**3*x**6 + c**4*x**7/7 + x**4*(3*a*b*c**2/2 + 9*b**3*c/2
) + x**3*(6*a*b**2*c + 3*b**4)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.66 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=\frac {1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + \frac {9}{5} \, b^{2} c^{2} x^{5} + 3 \, b^{4} x^{3} + 9 \, a^{2} b^{2} x + \frac {3}{2} \, {\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a b + \frac {3}{10} \, {\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\right )} b^{2} \]

[In]

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

[Out]

1/7*c^4*x^7 + b*c^3*x^6 + 9/5*b^2*c^2*x^5 + 3*b^4*x^3 + 9*a^2*b^2*x + 3/2*(c^2*x^4 + 4*b*c*x^3 + 6*b^2*x^2)*a*
b + 3/10*(4*c^2*x^5 + 15*b*c*x^4)*b^2

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.48 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=\frac {1}{7} \, c^{4} x^{7} + b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{5} + \frac {9}{2} \, b^{3} c x^{4} + \frac {3}{2} \, a b c^{2} x^{4} + 3 \, b^{4} x^{3} + 6 \, a b^{2} c x^{3} + 9 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x \]

[In]

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

[Out]

1/7*c^4*x^7 + b*c^3*x^6 + 3*b^2*c^2*x^5 + 9/2*b^3*c*x^4 + 3/2*a*b*c^2*x^4 + 3*b^4*x^3 + 6*a*b^2*c*x^3 + 9*a*b^
3*x^2 + 9*a^2*b^2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2 \, dx=x^3\,\left (3\,b^4+6\,a\,c\,b^2\right )+\frac {c^4\,x^7}{7}+9\,a^2\,b^2\,x+9\,a\,b^3\,x^2+b\,c^3\,x^6+3\,b^2\,c^2\,x^5+\frac {3\,b\,c\,x^4\,\left (3\,b^2+a\,c\right )}{2} \]

[In]

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

[Out]

x^3*(3*b^4 + 6*a*b^2*c) + (c^4*x^7)/7 + 9*a^2*b^2*x + 9*a*b^3*x^2 + b*c^3*x^6 + 3*b^2*c^2*x^5 + (3*b*c*x^4*(a*
c + 3*b^2))/2