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]
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]
[Out]
Rule 200
Rule 2085
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*}
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]
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]
[Out]