Integrand size = 15, antiderivative size = 56 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {1}{7} c^3 d x^7+\frac {e \left (a+c x^2\right )^4}{8 c} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200} \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {e \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} c^3 d x^7 \]
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Rule 200
Rule 655
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a+c x^2\right )^4}{8 c}+d \int \left (a+c x^2\right )^3 \, dx \\ & = \frac {e \left (a+c x^2\right )^4}{8 c}+d \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx \\ & = a^3 d x+a^2 c d x^3+\frac {3}{5} a c^2 d x^5+\frac {1}{7} c^3 d x^7+\frac {e \left (a+c x^2\right )^4}{8 c} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 d x+\frac {1}{2} a^3 e x^2+a^2 c d x^3+\frac {3}{4} a^2 c e x^4+\frac {3}{5} a c^2 d x^5+\frac {1}{2} a c^2 e x^6+\frac {1}{7} c^3 d x^7+\frac {1}{8} c^3 e x^8 \]
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Time = 2.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32
method | result | size |
gosper | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
default | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
norman | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
risch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
parallelrisch | \(\frac {1}{8} c^{3} e \,x^{8}+\frac {1}{7} c^{3} d \,x^{7}+\frac {1}{2} a \,c^{2} e \,x^{6}+\frac {3}{5} a \,c^{2} d \,x^{5}+\frac {3}{4} a^{2} c e \,x^{4}+a^{2} c d \,x^{3}+\frac {1}{2} a^{3} e \,x^{2}+a^{3} d x\) | \(74\) |
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Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=a^{3} d x + \frac {a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac {3 a^{2} c e x^{4}}{4} + \frac {3 a c^{2} d x^{5}}{5} + \frac {a c^{2} e x^{6}}{2} + \frac {c^{3} d x^{7}}{7} + \frac {c^{3} e x^{8}}{8} \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]
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Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{8} \, c^{3} e x^{8} + \frac {1}{7} \, c^{3} d x^{7} + \frac {1}{2} \, a c^{2} e x^{6} + \frac {3}{5} \, a c^{2} d x^{5} + \frac {3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac {1}{2} \, a^{3} e x^{2} + a^{3} d x \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {e\,a^3\,x^2}{2}+d\,a^3\,x+\frac {3\,e\,a^2\,c\,x^4}{4}+d\,a^2\,c\,x^3+\frac {e\,a\,c^2\,x^6}{2}+\frac {3\,d\,a\,c^2\,x^5}{5}+\frac {e\,c^3\,x^8}{8}+\frac {d\,c^3\,x^7}{7} \]
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