Optimal. Leaf size=198 \[ -\frac {a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac {1}{2} a^2 x^2 (a f+3 b c)+a^2 b d x^3+\frac {1}{4} a^2 x^4 (a h+3 b e)+\frac {1}{8} b^2 x^8 (3 a f+b c)+\frac {1}{2} a b^2 d x^6+\frac {1}{10} b^2 x^{10} (3 a h+b e)+\frac {3}{5} a b x^5 (a f+b c)+\frac {3}{7} a b x^7 (a h+b e)+\frac {g \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 d x^9+\frac {1}{11} b^3 f x^{11}+\frac {1}{13} b^3 h x^{13} \]
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Rubi [A] time = 0.18, antiderivative size = 198, 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} \[ \frac {1}{2} a^2 x^2 (a f+3 b c)+a^2 b d x^3+\frac {1}{4} a^2 x^4 (a h+3 b e)-\frac {a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac {1}{8} b^2 x^8 (3 a f+b c)+\frac {1}{2} a b^2 d x^6+\frac {1}{10} b^2 x^{10} (3 a h+b e)+\frac {3}{5} a b x^5 (a f+b c)+\frac {3}{7} a b x^7 (a h+b e)+\frac {g \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 d x^9+\frac {1}{11} b^3 f x^{11}+\frac {1}{13} b^3 h x^{13} \]
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^2} \, dx &=\frac {g \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+f x^3+h x^5\right )}{x^2} \, dx\\ &=\frac {g \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 e+\frac {a^3 c}{x^2}+\frac {a^3 d}{x}+a^2 (3 b c+a f) x+3 a^2 b d x^2+a^2 (3 b e+a h) x^3+3 a b (b c+a f) x^4+3 a b^2 d x^5+3 a b (b e+a h) x^6+b^2 (b c+3 a f) x^7+b^3 d x^8+b^2 (b e+3 a h) x^9+b^3 f x^{10}+b^3 h x^{12}\right ) \, dx\\ &=-\frac {a^3 c}{x}+a^3 e x+\frac {1}{2} a^2 (3 b c+a f) x^2+a^2 b d x^3+\frac {1}{4} a^2 (3 b e+a h) x^4+\frac {3}{5} a b (b c+a f) x^5+\frac {1}{2} a b^2 d x^6+\frac {3}{7} a b (b e+a h) x^7+\frac {1}{8} b^2 (b c+3 a f) x^8+\frac {1}{9} b^3 d x^9+\frac {1}{10} b^2 (b e+3 a h) x^{10}+\frac {1}{11} b^3 f x^{11}+\frac {1}{13} b^3 h x^{13}+\frac {g \left (a+b x^3\right )^4}{12 b}+a^3 d \log (x)\\ \end {align*}
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Mathematica [A] time = 0.21, size = 172, normalized size = 0.87 \[ a^3 \left (-\frac {c}{x}+e x+\frac {1}{12} x^2 \left (6 f+4 g x+3 h x^2\right )\right )+a^3 d \log (x)+\frac {1}{140} a^2 b x^2 \left (210 c+x \left (140 d+x \left (105 e+84 f x+70 g x^2+60 h x^3\right )\right )\right )+\frac {1}{840} a b^2 x^5 \left (504 c+x \left (420 d+x \left (360 e+315 f x+280 g x^2+252 h x^3\right )\right )\right )+\frac {b^3 x^8 \left (6435 c+5720 d x+6 x^2 \left (858 e+780 f x+715 g x^2+660 h x^3\right )\right )}{51480} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 219, normalized size = 1.11 \[ \frac {27720 \, b^{3} h x^{14} + 30030 \, b^{3} g x^{13} + 32760 \, b^{3} f x^{12} + 36036 \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 40040 \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 45045 \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 154440 \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + 180180 \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + 216216 \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} + 360360 \, a^{3} e x^{2} + 90090 \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + 360360 \, a^{3} d x \log \relax (x) + 120120 \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} - 360360 \, a^{3} c + 180180 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{360360 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 228, normalized size = 1.15 \[ \frac {1}{13} \, b^{3} h x^{13} + \frac {1}{12} \, b^{3} g x^{12} + \frac {1}{11} \, b^{3} f x^{11} + \frac {3}{10} \, a b^{2} h x^{10} + \frac {1}{10} \, b^{3} x^{10} e + \frac {1}{9} \, b^{3} d x^{9} + \frac {1}{3} \, a b^{2} g x^{9} + \frac {1}{8} \, b^{3} c x^{8} + \frac {3}{8} \, a b^{2} f x^{8} + \frac {3}{7} \, a^{2} b h x^{7} + \frac {3}{7} \, a b^{2} x^{7} e + \frac {1}{2} \, a b^{2} d x^{6} + \frac {1}{2} \, a^{2} b g x^{6} + \frac {3}{5} \, a b^{2} c x^{5} + \frac {3}{5} \, a^{2} b f x^{5} + \frac {1}{4} \, a^{3} h x^{4} + \frac {3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac {1}{3} \, a^{3} g x^{3} + \frac {3}{2} \, a^{2} b c x^{2} + \frac {1}{2} \, a^{3} f x^{2} + a^{3} x e + a^{3} d \log \left ({\left | x \right |}\right ) - \frac {a^{3} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 224, normalized size = 1.13 \[ \frac {b^{3} h \,x^{13}}{13}+\frac {b^{3} g \,x^{12}}{12}+\frac {b^{3} f \,x^{11}}{11}+\frac {3 a \,b^{2} h \,x^{10}}{10}+\frac {b^{3} e \,x^{10}}{10}+\frac {a \,b^{2} g \,x^{9}}{3}+\frac {b^{3} d \,x^{9}}{9}+\frac {3 a \,b^{2} f \,x^{8}}{8}+\frac {b^{3} c \,x^{8}}{8}+\frac {3 a^{2} b h \,x^{7}}{7}+\frac {3 a \,b^{2} e \,x^{7}}{7}+\frac {a^{2} b g \,x^{6}}{2}+\frac {a \,b^{2} d \,x^{6}}{2}+\frac {3 a^{2} b f \,x^{5}}{5}+\frac {3 a \,b^{2} c \,x^{5}}{5}+\frac {a^{3} h \,x^{4}}{4}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {a^{3} g \,x^{3}}{3}+a^{2} b d \,x^{3}+\frac {a^{3} f \,x^{2}}{2}+\frac {3 a^{2} b c \,x^{2}}{2}+a^{3} d \ln \relax (x )+a^{3} e x -\frac {a^{3} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 212, normalized size = 1.07 \[ \frac {1}{13} \, b^{3} h x^{13} + \frac {1}{12} \, b^{3} g x^{12} + \frac {1}{11} \, b^{3} f x^{11} + \frac {1}{10} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{10} + \frac {1}{9} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{9} + \frac {1}{8} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{8} + \frac {3}{7} \, {\left (a b^{2} e + a^{2} b h\right )} x^{7} + \frac {1}{2} \, {\left (a b^{2} d + a^{2} b g\right )} x^{6} + \frac {3}{5} \, {\left (a b^{2} c + a^{2} b f\right )} x^{5} + a^{3} e x + \frac {1}{4} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{4} + a^{3} d \log \relax (x) + \frac {1}{3} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{3} - \frac {a^{3} c}{x} + \frac {1}{2} \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 199, normalized size = 1.01 \[ x^2\,\left (\frac {f\,a^3}{2}+\frac {3\,b\,c\,a^2}{2}\right )+x^8\,\left (\frac {c\,b^3}{8}+\frac {3\,a\,f\,b^2}{8}\right )+x^3\,\left (\frac {g\,a^3}{3}+b\,d\,a^2\right )+x^9\,\left (\frac {d\,b^3}{9}+\frac {a\,g\,b^2}{3}\right )+x^4\,\left (\frac {h\,a^3}{4}+\frac {3\,b\,e\,a^2}{4}\right )+x^{10}\,\left (\frac {e\,b^3}{10}+\frac {3\,a\,h\,b^2}{10}\right )-\frac {a^3\,c}{x}+\frac {b^3\,f\,x^{11}}{11}+\frac {b^3\,g\,x^{12}}{12}+\frac {b^3\,h\,x^{13}}{13}+a^3\,d\,\ln \relax (x)+a^3\,e\,x+\frac {3\,a\,b\,x^5\,\left (b\,c+a\,f\right )}{5}+\frac {a\,b\,x^6\,\left (b\,d+a\,g\right )}{2}+\frac {3\,a\,b\,x^7\,\left (b\,e+a\,h\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 236, normalized size = 1.19 \[ - \frac {a^{3} c}{x} + a^{3} d \log {\relax (x )} + a^{3} e x + \frac {b^{3} f x^{11}}{11} + \frac {b^{3} g x^{12}}{12} + \frac {b^{3} h x^{13}}{13} + x^{10} \left (\frac {3 a b^{2} h}{10} + \frac {b^{3} e}{10}\right ) + x^{9} \left (\frac {a b^{2} g}{3} + \frac {b^{3} d}{9}\right ) + x^{8} \left (\frac {3 a b^{2} f}{8} + \frac {b^{3} c}{8}\right ) + x^{7} \left (\frac {3 a^{2} b h}{7} + \frac {3 a b^{2} e}{7}\right ) + x^{6} \left (\frac {a^{2} b g}{2} + \frac {a b^{2} d}{2}\right ) + x^{5} \left (\frac {3 a^{2} b f}{5} + \frac {3 a b^{2} c}{5}\right ) + x^{4} \left (\frac {a^{3} h}{4} + \frac {3 a^{2} b e}{4}\right ) + x^{3} \left (\frac {a^{3} g}{3} + a^{2} b d\right ) + x^{2} \left (\frac {a^{3} f}{2} + \frac {3 a^{2} b c}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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