Optimal. Leaf size=200 \[ a^3 c \log (x)+a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {1}{4} a^2 x^4 (a g+3 b d)+\frac {1}{5} a^2 x^5 (a h+3 b e)+\frac {1}{2} a b^2 c x^6+\frac {1}{10} b^2 x^{10} (3 a g+b d)+\frac {1}{11} b^2 x^{11} (3 a h+b e)+\frac {3}{7} a b x^7 (a g+b d)+\frac {3}{8} a b x^8 (a h+b e)+\frac {f \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 c x^9+\frac {1}{13} b^3 g x^{13}+\frac {1}{14} b^3 h x^{14} \]
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Rubi [A] time = 0.15, antiderivative size = 200, 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 = {1583, 1820} \[ a^2 b c x^3+\frac {1}{4} a^2 x^4 (a g+3 b d)+\frac {1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{2} a b^2 c x^6+\frac {1}{10} b^2 x^{10} (3 a g+b d)+\frac {1}{11} b^2 x^{11} (3 a h+b e)+\frac {3}{7} a b x^7 (a g+b d)+\frac {3}{8} a b x^8 (a h+b e)+\frac {f \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 c x^9+\frac {1}{13} b^3 g x^{13}+\frac {1}{14} b^3 h x^{14} \]
Antiderivative was successfully verified.
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Rule 1583
Rule 1820
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^3 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx &=\frac {f \left (a+b x^3\right )^4}{12 b}+\int \frac {\left (a+b x^3\right )^3 \left (c+d x+e x^2+g x^4+h x^5\right )}{x} \, dx\\ &=\frac {f \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 d+\frac {a^3 c}{x}+a^3 e x+3 a^2 b c x^2+a^2 (3 b d+a g) x^3+a^2 (3 b e+a h) x^4+3 a b^2 c x^5+3 a b (b d+a g) x^6+3 a b (b e+a h) x^7+b^3 c x^8+b^2 (b d+3 a g) x^9+b^2 (b e+3 a h) x^{10}+b^3 g x^{12}+b^3 h x^{13}\right ) \, dx\\ &=a^3 d x+\frac {1}{2} a^3 e x^2+a^2 b c x^3+\frac {1}{4} a^2 (3 b d+a g) x^4+\frac {1}{5} a^2 (3 b e+a h) x^5+\frac {1}{2} a b^2 c x^6+\frac {3}{7} a b (b d+a g) x^7+\frac {3}{8} a b (b e+a h) x^8+\frac {1}{9} b^3 c x^9+\frac {1}{10} b^2 (b d+3 a g) x^{10}+\frac {1}{11} b^2 (b e+3 a h) x^{11}+\frac {1}{13} b^3 g x^{13}+\frac {1}{14} b^3 h x^{14}+\frac {f \left (a+b x^3\right )^4}{12 b}+a^3 c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.13, size = 214, normalized size = 1.07 \[ a^3 c \log (x)+a^3 d x+\frac {1}{2} a^3 e x^2+\frac {1}{3} a^2 x^3 (a f+3 b c)+\frac {1}{4} a^2 x^4 (a g+3 b d)+\frac {1}{5} a^2 x^5 (a h+3 b e)+\frac {1}{9} b^2 x^9 (3 a f+b c)+\frac {1}{10} b^2 x^{10} (3 a g+b d)+\frac {1}{11} b^2 x^{11} (3 a h+b e)+\frac {1}{2} a b x^6 (a f+b c)+\frac {3}{7} a b x^7 (a g+b d)+\frac {3}{8} a b x^8 (a h+b e)+\frac {1}{12} b^3 f x^{12}+\frac {1}{13} b^3 g x^{13}+\frac {1}{14} b^3 h x^{14} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 212, normalized size = 1.06 \[ \frac {1}{14} \, b^{3} h x^{14} + \frac {1}{13} \, b^{3} g x^{13} + \frac {1}{12} \, b^{3} f x^{12} + \frac {1}{11} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac {1}{10} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac {1}{9} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac {3}{8} \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac {3}{7} \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac {1}{2} \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac {1}{2} \, a^{3} e x^{2} + \frac {1}{5} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac {1}{4} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \relax (x) + \frac {1}{3} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 228, normalized size = 1.14 \[ \frac {1}{14} \, b^{3} h x^{14} + \frac {1}{13} \, b^{3} g x^{13} + \frac {1}{12} \, b^{3} f x^{12} + \frac {3}{11} \, a b^{2} h x^{11} + \frac {1}{11} \, b^{3} x^{11} e + \frac {1}{10} \, b^{3} d x^{10} + \frac {3}{10} \, a b^{2} g x^{10} + \frac {1}{9} \, b^{3} c x^{9} + \frac {1}{3} \, a b^{2} f x^{9} + \frac {3}{8} \, a^{2} b h x^{8} + \frac {3}{8} \, a b^{2} x^{8} e + \frac {3}{7} \, a b^{2} d x^{7} + \frac {3}{7} \, a^{2} b g x^{7} + \frac {1}{2} \, a b^{2} c x^{6} + \frac {1}{2} \, a^{2} b f x^{6} + \frac {1}{5} \, a^{3} h x^{5} + \frac {3}{5} \, a^{2} b x^{5} e + \frac {3}{4} \, a^{2} b d x^{4} + \frac {1}{4} \, a^{3} g x^{4} + a^{2} b c x^{3} + \frac {1}{3} \, a^{3} f x^{3} + \frac {1}{2} \, a^{3} x^{2} e + a^{3} d x + a^{3} c \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 224, normalized size = 1.12 \[ \frac {b^{3} h \,x^{14}}{14}+\frac {b^{3} g \,x^{13}}{13}+\frac {b^{3} f \,x^{12}}{12}+\frac {3 a \,b^{2} h \,x^{11}}{11}+\frac {b^{3} e \,x^{11}}{11}+\frac {3 a \,b^{2} g \,x^{10}}{10}+\frac {b^{3} d \,x^{10}}{10}+\frac {a \,b^{2} f \,x^{9}}{3}+\frac {b^{3} c \,x^{9}}{9}+\frac {3 a^{2} b h \,x^{8}}{8}+\frac {3 a \,b^{2} e \,x^{8}}{8}+\frac {3 a^{2} b g \,x^{7}}{7}+\frac {3 a \,b^{2} d \,x^{7}}{7}+\frac {a^{2} b f \,x^{6}}{2}+\frac {a \,b^{2} c \,x^{6}}{2}+\frac {a^{3} h \,x^{5}}{5}+\frac {3 a^{2} b e \,x^{5}}{5}+\frac {a^{3} g \,x^{4}}{4}+\frac {3 a^{2} b d \,x^{4}}{4}+\frac {a^{3} f \,x^{3}}{3}+a^{2} b c \,x^{3}+\frac {a^{3} e \,x^{2}}{2}+a^{3} c \ln \relax (x )+a^{3} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 212, normalized size = 1.06 \[ \frac {1}{14} \, b^{3} h x^{14} + \frac {1}{13} \, b^{3} g x^{13} + \frac {1}{12} \, b^{3} f x^{12} + \frac {1}{11} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac {1}{10} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac {1}{9} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac {3}{8} \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac {3}{7} \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac {1}{2} \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac {1}{2} \, a^{3} e x^{2} + \frac {1}{5} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac {1}{4} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \relax (x) + \frac {1}{3} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.11, size = 199, normalized size = 1.00 \[ x^3\,\left (\frac {f\,a^3}{3}+b\,c\,a^2\right )+x^9\,\left (\frac {c\,b^3}{9}+\frac {a\,f\,b^2}{3}\right )+x^4\,\left (\frac {g\,a^3}{4}+\frac {3\,b\,d\,a^2}{4}\right )+x^{10}\,\left (\frac {d\,b^3}{10}+\frac {3\,a\,g\,b^2}{10}\right )+x^5\,\left (\frac {h\,a^3}{5}+\frac {3\,b\,e\,a^2}{5}\right )+x^{11}\,\left (\frac {e\,b^3}{11}+\frac {3\,a\,h\,b^2}{11}\right )+\frac {a^3\,e\,x^2}{2}+\frac {b^3\,f\,x^{12}}{12}+\frac {b^3\,g\,x^{13}}{13}+\frac {b^3\,h\,x^{14}}{14}+a^3\,c\,\ln \relax (x)+a^3\,d\,x+\frac {a\,b\,x^6\,\left (b\,c+a\,f\right )}{2}+\frac {3\,a\,b\,x^7\,\left (b\,d+a\,g\right )}{7}+\frac {3\,a\,b\,x^8\,\left (b\,e+a\,h\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 240, normalized size = 1.20 \[ a^{3} c \log {\relax (x )} + a^{3} d x + \frac {a^{3} e x^{2}}{2} + \frac {b^{3} f x^{12}}{12} + \frac {b^{3} g x^{13}}{13} + \frac {b^{3} h x^{14}}{14} + x^{11} \left (\frac {3 a b^{2} h}{11} + \frac {b^{3} e}{11}\right ) + x^{10} \left (\frac {3 a b^{2} g}{10} + \frac {b^{3} d}{10}\right ) + x^{9} \left (\frac {a b^{2} f}{3} + \frac {b^{3} c}{9}\right ) + x^{8} \left (\frac {3 a^{2} b h}{8} + \frac {3 a b^{2} e}{8}\right ) + x^{7} \left (\frac {3 a^{2} b g}{7} + \frac {3 a b^{2} d}{7}\right ) + x^{6} \left (\frac {a^{2} b f}{2} + \frac {a b^{2} c}{2}\right ) + x^{5} \left (\frac {a^{3} h}{5} + \frac {3 a^{2} b e}{5}\right ) + x^{4} \left (\frac {a^{3} g}{4} + \frac {3 a^{2} b d}{4}\right ) + x^{3} \left (\frac {a^{3} f}{3} + a^{2} b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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