Integrand size = 38, antiderivative size = 163 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{8} a (2 b c+a f) x^8+\frac {1}{9} a (2 b d+a g) x^9+\frac {1}{10} a (2 b e+a h) x^{10}+\frac {1}{11} b (b c+2 a f) x^{11}+\frac {1}{12} b (b d+2 a g) x^{12}+\frac {1}{13} b (b e+2 a h) x^{13}+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \]
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Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1834} \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{11} b x^{11} (2 a f+b c)+\frac {1}{8} a x^8 (a f+2 b c)+\frac {1}{12} b x^{12} (2 a g+b d)+\frac {1}{9} a x^9 (a g+2 b d)+\frac {1}{13} b x^{13} (2 a h+b e)+\frac {1}{10} a x^{10} (a h+2 b e)+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \]
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Rule 1834
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c x^4+a^2 d x^5+a^2 e x^6+a (2 b c+a f) x^7+a (2 b d+a g) x^8+a (2 b e+a h) x^9+b (b c+2 a f) x^{10}+b (b d+2 a g) x^{11}+b (b e+2 a h) x^{12}+b^2 f x^{13}+b^2 g x^{14}+b^2 h x^{15}\right ) \, dx \\ & = \frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{8} a (2 b c+a f) x^8+\frac {1}{9} a (2 b d+a g) x^9+\frac {1}{10} a (2 b e+a h) x^{10}+\frac {1}{11} b (b c+2 a f) x^{11}+\frac {1}{12} b (b d+2 a g) x^{12}+\frac {1}{13} b (b e+2 a h) x^{13}+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{5} a^2 c x^5+\frac {1}{6} a^2 d x^6+\frac {1}{7} a^2 e x^7+\frac {1}{8} a (2 b c+a f) x^8+\frac {1}{9} a (2 b d+a g) x^9+\frac {1}{10} a (2 b e+a h) x^{10}+\frac {1}{11} b (b c+2 a f) x^{11}+\frac {1}{12} b (b d+2 a g) x^{12}+\frac {1}{13} b (b e+2 a h) x^{13}+\frac {1}{14} b^2 f x^{14}+\frac {1}{15} b^2 g x^{15}+\frac {1}{16} b^2 h x^{16} \]
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Time = 2.06 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {b^{2} h \,x^{16}}{16}+\frac {b^{2} g \,x^{15}}{15}+\frac {b^{2} f \,x^{14}}{14}+\frac {\left (2 a b h +b^{2} e \right ) x^{13}}{13}+\frac {\left (2 a b g +b^{2} d \right ) x^{12}}{12}+\frac {\left (2 a f b +b^{2} c \right ) x^{11}}{11}+\frac {\left (a^{2} h +2 a e b \right ) x^{10}}{10}+\frac {\left (a^{2} g +2 a b d \right ) x^{9}}{9}+\frac {\left (a^{2} f +2 a b c \right ) x^{8}}{8}+\frac {a^{2} e \,x^{7}}{7}+\frac {a^{2} d \,x^{6}}{6}+\frac {a^{2} c \,x^{5}}{5}\) | \(152\) |
norman | \(\frac {a^{2} c \,x^{5}}{5}+\frac {a^{2} d \,x^{6}}{6}+\frac {a^{2} e \,x^{7}}{7}+\left (\frac {1}{8} a^{2} f +\frac {1}{4} a b c \right ) x^{8}+\left (\frac {1}{9} a^{2} g +\frac {2}{9} a b d \right ) x^{9}+\left (\frac {1}{10} a^{2} h +\frac {1}{5} a e b \right ) x^{10}+\left (\frac {2}{11} a f b +\frac {1}{11} b^{2} c \right ) x^{11}+\left (\frac {1}{6} a b g +\frac {1}{12} b^{2} d \right ) x^{12}+\left (\frac {2}{13} a b h +\frac {1}{13} b^{2} e \right ) x^{13}+\frac {b^{2} f \,x^{14}}{14}+\frac {b^{2} g \,x^{15}}{15}+\frac {b^{2} h \,x^{16}}{16}\) | \(152\) |
gosper | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
risch | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
parallelrisch | \(\frac {1}{5} a^{2} c \,x^{5}+\frac {1}{6} a^{2} d \,x^{6}+\frac {1}{7} a^{2} e \,x^{7}+\frac {1}{8} x^{8} a^{2} f +\frac {1}{4} x^{8} a b c +\frac {1}{9} x^{9} a^{2} g +\frac {2}{9} a b d \,x^{9}+\frac {1}{10} x^{10} a^{2} h +\frac {1}{5} a b e \,x^{10}+\frac {2}{11} a b f \,x^{11}+\frac {1}{11} x^{11} b^{2} c +\frac {1}{6} x^{12} a b g +\frac {1}{12} x^{12} b^{2} d +\frac {2}{13} x^{13} a b h +\frac {1}{13} x^{13} b^{2} e +\frac {1}{14} b^{2} f \,x^{14}+\frac {1}{15} b^{2} g \,x^{15}+\frac {1}{16} b^{2} h \,x^{16}\) | \(158\) |
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Time = 0.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, {\left (b^{2} e + 2 \, a b h\right )} x^{13} + \frac {1}{12} \, {\left (b^{2} d + 2 \, a b g\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} c + 2 \, a b f\right )} x^{11} + \frac {1}{10} \, {\left (2 \, a b e + a^{2} h\right )} x^{10} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{9} \, {\left (2 \, a b d + a^{2} g\right )} x^{9} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{8} \, {\left (2 \, a b c + a^{2} f\right )} x^{8} + \frac {1}{5} \, a^{2} c x^{5} \]
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Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {a^{2} c x^{5}}{5} + \frac {a^{2} d x^{6}}{6} + \frac {a^{2} e x^{7}}{7} + \frac {b^{2} f x^{14}}{14} + \frac {b^{2} g x^{15}}{15} + \frac {b^{2} h x^{16}}{16} + x^{13} \cdot \left (\frac {2 a b h}{13} + \frac {b^{2} e}{13}\right ) + x^{12} \left (\frac {a b g}{6} + \frac {b^{2} d}{12}\right ) + x^{11} \cdot \left (\frac {2 a b f}{11} + \frac {b^{2} c}{11}\right ) + x^{10} \left (\frac {a^{2} h}{10} + \frac {a b e}{5}\right ) + x^{9} \left (\frac {a^{2} g}{9} + \frac {2 a b d}{9}\right ) + x^{8} \left (\frac {a^{2} f}{8} + \frac {a b c}{4}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, {\left (b^{2} e + 2 \, a b h\right )} x^{13} + \frac {1}{12} \, {\left (b^{2} d + 2 \, a b g\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} c + 2 \, a b f\right )} x^{11} + \frac {1}{10} \, {\left (2 \, a b e + a^{2} h\right )} x^{10} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{9} \, {\left (2 \, a b d + a^{2} g\right )} x^{9} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{8} \, {\left (2 \, a b c + a^{2} f\right )} x^{8} + \frac {1}{5} \, a^{2} c x^{5} \]
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Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{16} \, b^{2} h x^{16} + \frac {1}{15} \, b^{2} g x^{15} + \frac {1}{14} \, b^{2} f x^{14} + \frac {1}{13} \, b^{2} e x^{13} + \frac {2}{13} \, a b h x^{13} + \frac {1}{12} \, b^{2} d x^{12} + \frac {1}{6} \, a b g x^{12} + \frac {1}{11} \, b^{2} c x^{11} + \frac {2}{11} \, a b f x^{11} + \frac {1}{5} \, a b e x^{10} + \frac {1}{10} \, a^{2} h x^{10} + \frac {2}{9} \, a b d x^{9} + \frac {1}{9} \, a^{2} g x^{9} + \frac {1}{4} \, a b c x^{8} + \frac {1}{8} \, a^{2} f x^{8} + \frac {1}{7} \, a^{2} e x^{7} + \frac {1}{6} \, a^{2} d x^{6} + \frac {1}{5} \, a^{2} c x^{5} \]
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Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int x^4 \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^8\,\left (\frac {f\,a^2}{8}+\frac {b\,c\,a}{4}\right )+x^{11}\,\left (\frac {c\,b^2}{11}+\frac {2\,a\,f\,b}{11}\right )+x^9\,\left (\frac {g\,a^2}{9}+\frac {2\,b\,d\,a}{9}\right )+x^{12}\,\left (\frac {d\,b^2}{12}+\frac {a\,g\,b}{6}\right )+x^{10}\,\left (\frac {h\,a^2}{10}+\frac {b\,e\,a}{5}\right )+x^{13}\,\left (\frac {e\,b^2}{13}+\frac {2\,a\,h\,b}{13}\right )+\frac {a^2\,c\,x^5}{5}+\frac {a^2\,d\,x^6}{6}+\frac {a^2\,e\,x^7}{7}+\frac {b^2\,f\,x^{14}}{14}+\frac {b^2\,g\,x^{15}}{15}+\frac {b^2\,h\,x^{16}}{16} \]
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