Optimal. Leaf size=163 \[ \frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{10} b x^{10} (2 a f+b c)+\frac {1}{7} a x^7 (a f+2 b c)+\frac {1}{11} b x^{11} (2 a g+b d)+\frac {1}{8} a x^8 (a g+2 b d)+\frac {1}{12} b x^{12} (2 a h+b e)+\frac {1}{9} a x^9 (a h+2 b e)+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15} \]
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Rubi [A] time = 0.16, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1820} \[ \frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{10} b x^{10} (2 a f+b c)+\frac {1}{7} a x^7 (a f+2 b c)+\frac {1}{11} b x^{11} (2 a g+b d)+\frac {1}{8} a x^8 (a g+2 b d)+\frac {1}{12} b x^{12} (2 a h+b e)+\frac {1}{9} a x^9 (a h+2 b e)+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15} \]
Antiderivative was successfully verified.
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Rule 1820
Rubi steps
\begin {align*} \int x^3 \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 &=\int \left (a^2 c x^3+a^2 d x^4+a^2 e x^5+a (2 b c+a f) x^6+a (2 b d+a g) x^7+a (2 b e+a h) x^8+b (b c+2 a f) x^9+b (b d+2 a g) x^{10}+b (b e+2 a h) x^{11}+b^2 f x^{12}+b^2 g x^{13}+b^2 h x^{14}\right ) \, dx\\ &=\frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{7} a (2 b c+a f) x^7+\frac {1}{8} a (2 b d+a g) x^8+\frac {1}{9} a (2 b e+a h) x^9+\frac {1}{10} b (b c+2 a f) x^{10}+\frac {1}{11} b (b d+2 a g) x^{11}+\frac {1}{12} b (b e+2 a h) x^{12}+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 163, normalized size = 1.00 \[ \frac {1}{4} a^2 c x^4+\frac {1}{5} a^2 d x^5+\frac {1}{6} a^2 e x^6+\frac {1}{10} b x^{10} (2 a f+b c)+\frac {1}{7} a x^7 (a f+2 b c)+\frac {1}{11} b x^{11} (2 a g+b d)+\frac {1}{8} a x^8 (a g+2 b d)+\frac {1}{12} b x^{12} (2 a h+b e)+\frac {1}{9} a x^9 (a h+2 b e)+\frac {1}{13} b^2 f x^{13}+\frac {1}{14} b^2 g x^{14}+\frac {1}{15} b^2 h x^{15} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.35, size = 157, normalized size = 0.96 \[ \frac {1}{15} x^{15} h b^{2} + \frac {1}{14} x^{14} g b^{2} + \frac {1}{13} x^{13} f b^{2} + \frac {1}{12} x^{12} e b^{2} + \frac {1}{6} x^{12} h b a + \frac {1}{11} x^{11} d b^{2} + \frac {2}{11} x^{11} g b a + \frac {1}{10} x^{10} c b^{2} + \frac {1}{5} x^{10} f b a + \frac {2}{9} x^{9} e b a + \frac {1}{9} x^{9} h a^{2} + \frac {1}{4} x^{8} d b a + \frac {1}{8} x^{8} g a^{2} + \frac {2}{7} x^{7} c b a + \frac {1}{7} x^{7} f a^{2} + \frac {1}{6} x^{6} e a^{2} + \frac {1}{5} x^{5} d a^{2} + \frac {1}{4} x^{4} c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 160, normalized size = 0.98 \[ \frac {1}{15} \, b^{2} h x^{15} + \frac {1}{14} \, b^{2} g x^{14} + \frac {1}{13} \, b^{2} f x^{13} + \frac {1}{6} \, a b h x^{12} + \frac {1}{12} \, b^{2} x^{12} e + \frac {1}{11} \, b^{2} d x^{11} + \frac {2}{11} \, a b g x^{11} + \frac {1}{10} \, b^{2} c x^{10} + \frac {1}{5} \, a b f x^{10} + \frac {1}{9} \, a^{2} h x^{9} + \frac {2}{9} \, a b x^{9} e + \frac {1}{4} \, a b d x^{8} + \frac {1}{8} \, a^{2} g x^{8} + \frac {2}{7} \, a b c x^{7} + \frac {1}{7} \, a^{2} f x^{7} + \frac {1}{6} \, a^{2} x^{6} e + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{4} \, a^{2} c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 152, normalized size = 0.93 \[ \frac {b^{2} h \,x^{15}}{15}+\frac {b^{2} g \,x^{14}}{14}+\frac {b^{2} f \,x^{13}}{13}+\frac {\left (2 a b h +b^{2} e \right ) x^{12}}{12}+\frac {\left (2 a b g +b^{2} d \right ) x^{11}}{11}+\frac {\left (2 a b f +c \,b^{2}\right ) x^{10}}{10}+\frac {a^{2} e \,x^{6}}{6}+\frac {\left (a^{2} h +2 b e a \right ) x^{9}}{9}+\frac {a^{2} d \,x^{5}}{5}+\frac {\left (a^{2} g +2 b d a \right ) x^{8}}{8}+\frac {a^{2} c \,x^{4}}{4}+\frac {\left (a^{2} f +2 a b c \right ) x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 151, normalized size = 0.93 \[ \frac {1}{15} \, b^{2} h x^{15} + \frac {1}{14} \, b^{2} g x^{14} + \frac {1}{13} \, b^{2} f x^{13} + \frac {1}{12} \, {\left (b^{2} e + 2 \, a b h\right )} x^{12} + \frac {1}{11} \, {\left (b^{2} d + 2 \, a b g\right )} x^{11} + \frac {1}{10} \, {\left (b^{2} c + 2 \, a b f\right )} x^{10} + \frac {1}{9} \, {\left (2 \, a b e + a^{2} h\right )} x^{9} + \frac {1}{6} \, a^{2} e x^{6} + \frac {1}{8} \, {\left (2 \, a b d + a^{2} g\right )} x^{8} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} f\right )} x^{7} + \frac {1}{4} \, a^{2} c x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 151, normalized size = 0.93 \[ x^7\,\left (\frac {f\,a^2}{7}+\frac {2\,b\,c\,a}{7}\right )+x^{10}\,\left (\frac {c\,b^2}{10}+\frac {a\,f\,b}{5}\right )+x^8\,\left (\frac {g\,a^2}{8}+\frac {b\,d\,a}{4}\right )+x^{11}\,\left (\frac {d\,b^2}{11}+\frac {2\,a\,g\,b}{11}\right )+x^9\,\left (\frac {h\,a^2}{9}+\frac {2\,b\,e\,a}{9}\right )+x^{12}\,\left (\frac {e\,b^2}{12}+\frac {a\,h\,b}{6}\right )+\frac {a^2\,c\,x^4}{4}+\frac {a^2\,d\,x^5}{5}+\frac {a^2\,e\,x^6}{6}+\frac {b^2\,f\,x^{13}}{13}+\frac {b^2\,g\,x^{14}}{14}+\frac {b^2\,h\,x^{15}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 167, normalized size = 1.02 \[ \frac {a^{2} c x^{4}}{4} + \frac {a^{2} d x^{5}}{5} + \frac {a^{2} e x^{6}}{6} + \frac {b^{2} f x^{13}}{13} + \frac {b^{2} g x^{14}}{14} + \frac {b^{2} h x^{15}}{15} + x^{12} \left (\frac {a b h}{6} + \frac {b^{2} e}{12}\right ) + x^{11} \left (\frac {2 a b g}{11} + \frac {b^{2} d}{11}\right ) + x^{10} \left (\frac {a b f}{5} + \frac {b^{2} c}{10}\right ) + x^{9} \left (\frac {a^{2} h}{9} + \frac {2 a b e}{9}\right ) + x^{8} \left (\frac {a^{2} g}{8} + \frac {a b d}{4}\right ) + x^{7} \left (\frac {a^{2} f}{7} + \frac {2 a b c}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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