Optimal. Leaf size=212 \[ \frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {1}{6} a^3 f x^6+\frac {1}{7} a^2 x^7 (a g+3 b d)+\frac {1}{8} a^2 x^8 (a h+3 b e)+\frac {1}{3} a^2 b f x^9+\frac {1}{13} b^2 x^{13} (3 a g+b d)+\frac {1}{14} b^2 x^{14} (3 a h+b e)+\frac {1}{4} a b^2 f x^{12}+\frac {c \left (a+b x^3\right )^4}{12 b}+\frac {3}{10} a b x^{10} (a g+b d)+\frac {3}{11} a b x^{11} (a h+b e)+\frac {1}{15} b^3 f x^{15}+\frac {1}{16} b^3 g x^{16}+\frac {1}{17} b^3 h x^{17} \]
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Rubi [A] time = 0.18, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1582, 1850} \[ \frac {1}{7} a^2 x^7 (a g+3 b d)+\frac {1}{8} a^2 x^8 (a h+3 b e)+\frac {1}{3} a^2 b f x^9+\frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {1}{6} a^3 f x^6+\frac {1}{13} b^2 x^{13} (3 a g+b d)+\frac {1}{14} b^2 x^{14} (3 a h+b e)+\frac {1}{4} a b^2 f x^{12}+\frac {c \left (a+b x^3\right )^4}{12 b}+\frac {3}{10} a b x^{10} (a g+b d)+\frac {3}{11} a b x^{11} (a h+b e)+\frac {1}{15} b^3 f x^{15}+\frac {1}{16} b^3 g x^{16}+\frac {1}{17} b^3 h x^{17} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin {align*} \int x^2 \left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx &=\frac {c \left (a+b x^3\right )^4}{12 b}+\int \left (a+b x^3\right )^3 \left (-c x^2+x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )\right ) \, dx\\ &=\frac {c \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 d x^3+a^3 e x^4+a^3 f x^5+a^2 (3 b d+a g) x^6+a^2 (3 b e+a h) x^7+3 a^2 b f x^8+3 a b (b d+a g) x^9+3 a b (b e+a h) x^{10}+3 a b^2 f x^{11}+b^2 (b d+3 a g) x^{12}+b^2 (b e+3 a h) x^{13}+b^3 f x^{14}+b^3 g x^{15}+b^3 h x^{16}\right ) \, dx\\ &=\frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {1}{6} a^3 f x^6+\frac {1}{7} a^2 (3 b d+a g) x^7+\frac {1}{8} a^2 (3 b e+a h) x^8+\frac {1}{3} a^2 b f x^9+\frac {3}{10} a b (b d+a g) x^{10}+\frac {3}{11} a b (b e+a h) x^{11}+\frac {1}{4} a b^2 f x^{12}+\frac {1}{13} b^2 (b d+3 a g) x^{13}+\frac {1}{14} b^2 (b e+3 a h) x^{14}+\frac {1}{15} b^3 f x^{15}+\frac {1}{16} b^3 g x^{16}+\frac {1}{17} b^3 h x^{17}+\frac {c \left (a+b x^3\right )^4}{12 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 223, normalized size = 1.05 \[ \frac {1}{3} a^3 c x^3+\frac {1}{4} a^3 d x^4+\frac {1}{5} a^3 e x^5+\frac {1}{6} a^2 x^6 (a f+3 b c)+\frac {1}{7} a^2 x^7 (a g+3 b d)+\frac {1}{8} a^2 x^8 (a h+3 b e)+\frac {1}{12} b^2 x^{12} (3 a f+b c)+\frac {1}{13} b^2 x^{13} (3 a g+b d)+\frac {1}{14} b^2 x^{14} (3 a h+b e)+\frac {1}{3} a b x^9 (a f+b c)+\frac {3}{10} a b x^{10} (a g+b d)+\frac {3}{11} a b x^{11} (a h+b e)+\frac {1}{15} b^3 f x^{15}+\frac {1}{16} b^3 g x^{16}+\frac {1}{17} b^3 h x^{17} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 229, normalized size = 1.08 \[ \frac {1}{17} x^{17} h b^{3} + \frac {1}{16} x^{16} g b^{3} + \frac {1}{15} x^{15} f b^{3} + \frac {1}{14} x^{14} e b^{3} + \frac {3}{14} x^{14} h b^{2} a + \frac {1}{13} x^{13} d b^{3} + \frac {3}{13} x^{13} g b^{2} a + \frac {1}{12} x^{12} c b^{3} + \frac {1}{4} x^{12} f b^{2} a + \frac {3}{11} x^{11} e b^{2} a + \frac {3}{11} x^{11} h b a^{2} + \frac {3}{10} x^{10} d b^{2} a + \frac {3}{10} x^{10} g b a^{2} + \frac {1}{3} x^{9} c b^{2} a + \frac {1}{3} x^{9} f b a^{2} + \frac {3}{8} x^{8} e b a^{2} + \frac {1}{8} x^{8} h a^{3} + \frac {3}{7} x^{7} d b a^{2} + \frac {1}{7} x^{7} g a^{3} + \frac {1}{2} x^{6} c b a^{2} + \frac {1}{6} x^{6} f a^{3} + \frac {1}{5} x^{5} e a^{3} + \frac {1}{4} x^{4} d a^{3} + \frac {1}{3} x^{3} c a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 233, normalized size = 1.10 \[ \frac {1}{17} \, b^{3} h x^{17} + \frac {1}{16} \, b^{3} g x^{16} + \frac {1}{15} \, b^{3} f x^{15} + \frac {3}{14} \, a b^{2} h x^{14} + \frac {1}{14} \, b^{3} x^{14} e + \frac {1}{13} \, b^{3} d x^{13} + \frac {3}{13} \, a b^{2} g x^{13} + \frac {1}{12} \, b^{3} c x^{12} + \frac {1}{4} \, a b^{2} f x^{12} + \frac {3}{11} \, a^{2} b h x^{11} + \frac {3}{11} \, a b^{2} x^{11} e + \frac {3}{10} \, a b^{2} d x^{10} + \frac {3}{10} \, a^{2} b g x^{10} + \frac {1}{3} \, a b^{2} c x^{9} + \frac {1}{3} \, a^{2} b f x^{9} + \frac {1}{8} \, a^{3} h x^{8} + \frac {3}{8} \, a^{2} b x^{8} e + \frac {3}{7} \, a^{2} b d x^{7} + \frac {1}{7} \, a^{3} g x^{7} + \frac {1}{2} \, a^{2} b c x^{6} + \frac {1}{6} \, a^{3} f x^{6} + \frac {1}{5} \, a^{3} x^{5} e + \frac {1}{4} \, a^{3} d x^{4} + \frac {1}{3} \, a^{3} c x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 224, normalized size = 1.06 \[ \frac {b^{3} h \,x^{17}}{17}+\frac {b^{3} g \,x^{16}}{16}+\frac {b^{3} f \,x^{15}}{15}+\frac {\left (3 a \,b^{2} h +b^{3} e \right ) x^{14}}{14}+\frac {\left (3 a \,b^{2} g +b^{3} d \right ) x^{13}}{13}+\frac {\left (3 a \,b^{2} f +b^{3} c \right ) x^{12}}{12}+\frac {\left (3 a^{2} b h +3 a e \,b^{2}\right ) x^{11}}{11}+\frac {\left (3 a^{2} b g +3 a \,b^{2} d \right ) x^{10}}{10}+\frac {a^{3} e \,x^{5}}{5}+\frac {\left (3 a^{2} b f +3 a \,b^{2} c \right ) x^{9}}{9}+\frac {a^{3} d \,x^{4}}{4}+\frac {\left (a^{3} h +3 a^{2} b e \right ) x^{8}}{8}+\frac {a^{3} c \,x^{3}}{3}+\frac {\left (a^{3} g +3 a^{2} d b \right ) x^{7}}{7}+\frac {\left (a^{3} f +3 a^{2} c b \right ) x^{6}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 217, normalized size = 1.02 \[ \frac {1}{17} \, b^{3} h x^{17} + \frac {1}{16} \, b^{3} g x^{16} + \frac {1}{15} \, b^{3} f x^{15} + \frac {1}{14} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{14} + \frac {1}{13} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{13} + \frac {1}{12} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{12} + \frac {3}{11} \, {\left (a b^{2} e + a^{2} b h\right )} x^{11} + \frac {3}{10} \, {\left (a b^{2} d + a^{2} b g\right )} x^{10} + \frac {1}{3} \, {\left (a b^{2} c + a^{2} b f\right )} x^{9} + \frac {1}{5} \, a^{3} e x^{5} + \frac {1}{8} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{8} + \frac {1}{4} \, a^{3} d x^{4} + \frac {1}{7} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{7} + \frac {1}{3} \, a^{3} c x^{3} + \frac {1}{6} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 205, normalized size = 0.97 \[ x^6\,\left (\frac {f\,a^3}{6}+\frac {b\,c\,a^2}{2}\right )+x^{12}\,\left (\frac {c\,b^3}{12}+\frac {a\,f\,b^2}{4}\right )+x^7\,\left (\frac {g\,a^3}{7}+\frac {3\,b\,d\,a^2}{7}\right )+x^{13}\,\left (\frac {d\,b^3}{13}+\frac {3\,a\,g\,b^2}{13}\right )+x^8\,\left (\frac {h\,a^3}{8}+\frac {3\,b\,e\,a^2}{8}\right )+x^{14}\,\left (\frac {e\,b^3}{14}+\frac {3\,a\,h\,b^2}{14}\right )+\frac {a^3\,c\,x^3}{3}+\frac {a^3\,d\,x^4}{4}+\frac {a^3\,e\,x^5}{5}+\frac {b^3\,f\,x^{15}}{15}+\frac {b^3\,g\,x^{16}}{16}+\frac {b^3\,h\,x^{17}}{17}+\frac {a\,b\,x^9\,\left (b\,c+a\,f\right )}{3}+\frac {3\,a\,b\,x^{10}\,\left (b\,d+a\,g\right )}{10}+\frac {3\,a\,b\,x^{11}\,\left (b\,e+a\,h\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 246, normalized size = 1.16 \[ \frac {a^{3} c x^{3}}{3} + \frac {a^{3} d x^{4}}{4} + \frac {a^{3} e x^{5}}{5} + \frac {b^{3} f x^{15}}{15} + \frac {b^{3} g x^{16}}{16} + \frac {b^{3} h x^{17}}{17} + x^{14} \left (\frac {3 a b^{2} h}{14} + \frac {b^{3} e}{14}\right ) + x^{13} \left (\frac {3 a b^{2} g}{13} + \frac {b^{3} d}{13}\right ) + x^{12} \left (\frac {a b^{2} f}{4} + \frac {b^{3} c}{12}\right ) + x^{11} \left (\frac {3 a^{2} b h}{11} + \frac {3 a b^{2} e}{11}\right ) + x^{10} \left (\frac {3 a^{2} b g}{10} + \frac {3 a b^{2} d}{10}\right ) + x^{9} \left (\frac {a^{2} b f}{3} + \frac {a b^{2} c}{3}\right ) + x^{8} \left (\frac {a^{3} h}{8} + \frac {3 a^{2} b e}{8}\right ) + x^{7} \left (\frac {a^{3} g}{7} + \frac {3 a^{2} b d}{7}\right ) + x^{6} \left (\frac {a^{3} f}{6} + \frac {a^{2} b c}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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