Integrand size = 20, antiderivative size = 77 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8+\frac {e \left (a+b x^3\right )^3}{9 b} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1596, 1864} \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {e \left (a+b x^3\right )^3}{9 b}+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8 \]
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Rule 1596
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+b x^3\right )^3}{9 b}+\int (c+d x) \left (a+b x^3\right )^2 \, dx \\ & = \frac {e \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 c+a^2 d x+2 a b c x^3+2 a b d x^4+b^2 c x^6+b^2 d x^7\right ) \, dx \\ & = a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8+\frac {e \left (a+b x^3\right )^3}{9 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.19 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{3} a^2 e x^3+\frac {1}{2} a b c x^4+\frac {2}{5} a b d x^5+\frac {1}{3} a b e x^6+\frac {1}{7} b^2 c x^7+\frac {1}{8} b^2 d x^8+\frac {1}{9} b^2 e x^9 \]
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Time = 1.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a e b \,x^{6}+\frac {2}{5} x^{5} d b a +\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(77\) |
default | \(\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a e b \,x^{6}+\frac {2}{5} x^{5} d b a +\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(77\) |
norman | \(\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a e b \,x^{6}+\frac {2}{5} x^{5} d b a +\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(77\) |
risch | \(\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a e b \,x^{6}+\frac {2}{5} x^{5} d b a +\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(77\) |
parallelrisch | \(\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{8} b^{2} d \,x^{8}+\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{3} a e b \,x^{6}+\frac {2}{5} x^{5} d b a +\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(77\) |
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Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} e x^{9} + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {1}{3} \, a b e x^{6} + \frac {2}{5} \, a b d x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
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Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + \frac {a b c x^{4}}{2} + \frac {2 a b d x^{5}}{5} + \frac {a b e x^{6}}{3} + \frac {b^{2} c x^{7}}{7} + \frac {b^{2} d x^{8}}{8} + \frac {b^{2} e x^{9}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} e x^{9} + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {1}{3} \, a b e x^{6} + \frac {2}{5} \, a b d x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
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Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} e x^{9} + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {1}{3} \, a b e x^{6} + \frac {2}{5} \, a b d x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx=\frac {e\,a^2\,x^3}{3}+\frac {d\,a^2\,x^2}{2}+c\,a^2\,x+\frac {e\,a\,b\,x^6}{3}+\frac {2\,d\,a\,b\,x^5}{5}+\frac {c\,a\,b\,x^4}{2}+\frac {e\,b^2\,x^9}{9}+\frac {d\,b^2\,x^8}{8}+\frac {c\,b^2\,x^7}{7} \]
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