Optimal. Leaf size=158 \[ \frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {1}{6} a^2 g x^6+\frac {1}{8} b x^8 (2 a f+b c)+\frac {1}{5} a x^5 (a f+2 b c)+\frac {d \left (a+b x^3\right )^3}{9 b}+\frac {1}{10} b x^{10} (2 a h+b e)+\frac {1}{7} a x^7 (a h+2 b e)+\frac {2}{9} a b g x^9+\frac {1}{11} b^2 f x^{11}+\frac {1}{12} b^2 g x^{12}+\frac {1}{13} b^2 h x^{13} \]
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Rubi [A] time = 0.13, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1582, 1850} \[ \frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {1}{6} a^2 g x^6+\frac {1}{8} b x^8 (2 a f+b c)+\frac {1}{5} a x^5 (a f+2 b c)+\frac {d \left (a+b x^3\right )^3}{9 b}+\frac {1}{10} b x^{10} (2 a h+b e)+\frac {1}{7} a x^7 (a h+2 b e)+\frac {2}{9} a b g x^9+\frac {1}{11} b^2 f x^{11}+\frac {1}{12} b^2 g x^{12}+\frac {1}{13} b^2 h x^{13} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1850
Rubi steps
\begin {align*} \int x \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 {d \left (a+b x^3\right )^3}{9 b}+\int \left (a+b x^3\right )^2 \left (-d x^2+x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )\right ) \, dx\\ &=\frac {d \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 c x+a^2 e x^3+a (2 b c+a f) x^4+a^2 g x^5+a (2 b e+a h) x^6+b (b c+2 a f) x^7+2 a b g x^8+b (b e+2 a h) x^9+b^2 f x^{10}+b^2 g x^{11}+b^2 h x^{12}\right ) \, dx\\ &=\frac {1}{2} a^2 c x^2+\frac {1}{4} a^2 e x^4+\frac {1}{5} a (2 b c+a f) x^5+\frac {1}{6} a^2 g x^6+\frac {1}{7} a (2 b e+a h) x^7+\frac {1}{8} b (b c+2 a f) x^8+\frac {2}{9} a b g x^9+\frac {1}{10} b (b e+2 a h) x^{10}+\frac {1}{11} b^2 f x^{11}+\frac {1}{12} b^2 g x^{12}+\frac {1}{13} b^2 h x^{13}+\frac {d \left (a+b x^3\right )^3}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 163, normalized size = 1.03 \[ \frac {1}{2} a^2 c x^2+\frac {1}{3} a^2 d x^3+\frac {1}{4} a^2 e x^4+\frac {1}{8} b x^8 (2 a f+b c)+\frac {1}{5} a x^5 (a f+2 b c)+\frac {1}{9} b x^9 (2 a g+b d)+\frac {1}{6} a x^6 (a g+2 b d)+\frac {1}{10} b x^{10} (2 a h+b e)+\frac {1}{7} a x^7 (a h+2 b e)+\frac {1}{11} b^2 f x^{11}+\frac {1}{12} b^2 g x^{12}+\frac {1}{13} b^2 h x^{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.36, size = 157, normalized size = 0.99 \[ \frac {1}{13} x^{13} h b^{2} + \frac {1}{12} x^{12} g b^{2} + \frac {1}{11} x^{11} f b^{2} + \frac {1}{10} x^{10} e b^{2} + \frac {1}{5} x^{10} h b a + \frac {1}{9} x^{9} d b^{2} + \frac {2}{9} x^{9} g b a + \frac {1}{8} x^{8} c b^{2} + \frac {1}{4} x^{8} f b a + \frac {2}{7} x^{7} e b a + \frac {1}{7} x^{7} h a^{2} + \frac {1}{3} x^{6} d b a + \frac {1}{6} x^{6} g a^{2} + \frac {2}{5} x^{5} c b a + \frac {1}{5} x^{5} f a^{2} + \frac {1}{4} x^{4} e a^{2} + \frac {1}{3} x^{3} d a^{2} + \frac {1}{2} x^{2} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 160, normalized size = 1.01 \[ \frac {1}{13} \, b^{2} h x^{13} + \frac {1}{12} \, b^{2} g x^{12} + \frac {1}{11} \, b^{2} f x^{11} + \frac {1}{5} \, a b h x^{10} + \frac {1}{10} \, b^{2} x^{10} e + \frac {1}{9} \, b^{2} d x^{9} + \frac {2}{9} \, a b g x^{9} + \frac {1}{8} \, b^{2} c x^{8} + \frac {1}{4} \, a b f x^{8} + \frac {1}{7} \, a^{2} h x^{7} + \frac {2}{7} \, a b x^{7} e + \frac {1}{3} \, a b d x^{6} + \frac {1}{6} \, a^{2} g x^{6} + \frac {2}{5} \, a b c x^{5} + \frac {1}{5} \, a^{2} f x^{5} + \frac {1}{4} \, a^{2} x^{4} e + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 152, normalized size = 0.96 \[ \frac {b^{2} h \,x^{13}}{13}+\frac {b^{2} g \,x^{12}}{12}+\frac {b^{2} f \,x^{11}}{11}+\frac {\left (2 a b h +b^{2} e \right ) x^{10}}{10}+\frac {\left (2 a b g +b^{2} d \right ) x^{9}}{9}+\frac {\left (2 a b f +c \,b^{2}\right ) x^{8}}{8}+\frac {a^{2} e \,x^{4}}{4}+\frac {\left (a^{2} h +2 b e a \right ) x^{7}}{7}+\frac {a^{2} d \,x^{3}}{3}+\frac {\left (a^{2} g +2 b d a \right ) x^{6}}{6}+\frac {a^{2} c \,x^{2}}{2}+\frac {\left (a^{2} f +2 a b c \right ) x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 151, normalized size = 0.96 \[ \frac {1}{13} \, b^{2} h x^{13} + \frac {1}{12} \, b^{2} g x^{12} + \frac {1}{11} \, b^{2} f x^{11} + \frac {1}{10} \, {\left (b^{2} e + 2 \, a b h\right )} x^{10} + \frac {1}{9} \, {\left (b^{2} d + 2 \, a b g\right )} x^{9} + \frac {1}{8} \, {\left (b^{2} c + 2 \, a b f\right )} x^{8} + \frac {1}{7} \, {\left (2 \, a b e + a^{2} h\right )} x^{7} + \frac {1}{4} \, a^{2} e x^{4} + \frac {1}{6} \, {\left (2 \, a b d + a^{2} g\right )} x^{6} + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} f\right )} x^{5} + \frac {1}{2} \, a^{2} c x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 151, normalized size = 0.96 \[ x^5\,\left (\frac {f\,a^2}{5}+\frac {2\,b\,c\,a}{5}\right )+x^8\,\left (\frac {c\,b^2}{8}+\frac {a\,f\,b}{4}\right )+x^6\,\left (\frac {g\,a^2}{6}+\frac {b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,b^2}{9}+\frac {2\,a\,g\,b}{9}\right )+x^7\,\left (\frac {h\,a^2}{7}+\frac {2\,b\,e\,a}{7}\right )+x^{10}\,\left (\frac {e\,b^2}{10}+\frac {a\,h\,b}{5}\right )+\frac {a^2\,c\,x^2}{2}+\frac {a^2\,d\,x^3}{3}+\frac {a^2\,e\,x^4}{4}+\frac {b^2\,f\,x^{11}}{11}+\frac {b^2\,g\,x^{12}}{12}+\frac {b^2\,h\,x^{13}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 167, normalized size = 1.06 \[ \frac {a^{2} c x^{2}}{2} + \frac {a^{2} d x^{3}}{3} + \frac {a^{2} e x^{4}}{4} + \frac {b^{2} f x^{11}}{11} + \frac {b^{2} g x^{12}}{12} + \frac {b^{2} h x^{13}}{13} + x^{10} \left (\frac {a b h}{5} + \frac {b^{2} e}{10}\right ) + x^{9} \left (\frac {2 a b g}{9} + \frac {b^{2} d}{9}\right ) + x^{8} \left (\frac {a b f}{4} + \frac {b^{2} c}{8}\right ) + x^{7} \left (\frac {a^{2} h}{7} + \frac {2 a b e}{7}\right ) + x^{6} \left (\frac {a^{2} g}{6} + \frac {a b d}{3}\right ) + x^{5} \left (\frac {a^{2} f}{5} + \frac {2 a b c}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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