Optimal. Leaf size=198 \[ -\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+a^3 e \log (x)+a^2 x (a f+3 b c)+\frac {1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3+\frac {1}{7} b^2 x^7 (3 a f+b c)+\frac {1}{8} b^2 x^8 (3 a g+b d)+\frac {1}{2} a b^2 e x^6+\frac {3}{4} a b x^4 (a f+b c)+\frac {3}{5} a b x^5 (a g+b d)+\frac {h \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 e x^9+\frac {1}{10} b^3 f x^{10}+\frac {1}{11} b^3 g x^{11} \]
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Rubi [A] time = 0.20, 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} \[ a^2 x (a f+3 b c)+\frac {1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3-\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+a^3 e \log (x)+\frac {1}{7} b^2 x^7 (3 a f+b c)+\frac {1}{8} b^2 x^8 (3 a g+b d)+\frac {1}{2} a b^2 e x^6+\frac {3}{4} a b x^4 (a f+b c)+\frac {3}{5} a b x^5 (a g+b d)+\frac {h \left (a+b x^3\right )^4}{12 b}+\frac {1}{9} b^3 e x^9+\frac {1}{10} b^3 f x^{10}+\frac {1}{11} b^3 g x^{11} \]
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^3} \, dx &=\frac {h \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+g x^4\right )}{x^3} \, dx\\ &=\frac {h \left (a+b x^3\right )^4}{12 b}+\int \left (a^2 (3 b c+a f)+\frac {a^3 c}{x^3}+\frac {a^3 d}{x^2}+\frac {a^3 e}{x}+a^2 (3 b d+a g) x+3 a^2 b e x^2+3 a b (b c+a f) x^3+3 a b (b d+a g) x^4+3 a b^2 e x^5+b^2 (b c+3 a f) x^6+b^2 (b d+3 a g) x^7+b^3 e x^8+b^3 f x^9+b^3 g x^{10}\right ) \, dx\\ &=-\frac {a^3 c}{2 x^2}-\frac {a^3 d}{x}+a^2 (3 b c+a f) x+\frac {1}{2} a^2 (3 b d+a g) x^2+a^2 b e x^3+\frac {3}{4} a b (b c+a f) x^4+\frac {3}{5} a b (b d+a g) x^5+\frac {1}{2} a b^2 e x^6+\frac {1}{7} b^2 (b c+3 a f) x^7+\frac {1}{8} b^2 (b d+3 a g) x^8+\frac {1}{9} b^3 e x^9+\frac {1}{10} b^3 f x^{10}+\frac {1}{11} b^3 g x^{11}+\frac {h \left (a+b x^3\right )^4}{12 b}+a^3 e \log (x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 174, normalized size = 0.88 \[ \frac {a^3 \left (-3 c-6 d x+x^3 \left (6 f+3 g x+2 h x^2\right )\right )}{6 x^2}+a^3 e \log (x)+\frac {1}{20} a^2 b x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+\frac {1}{840} a b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+\frac {b^3 x^7 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )}{27720} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 219, normalized size = 1.11 \[ \frac {2310 \, b^{3} h x^{14} + 2520 \, b^{3} g x^{13} + 2772 \, b^{3} f x^{12} + 3080 \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3465 \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 3960 \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 13860 \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + 16632 \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + 20790 \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} + 27720 \, a^{3} e x^{2} \log \relax (x) + 9240 \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 27720 \, a^{3} d x + 13860 \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} - 13860 \, a^{3} c + 27720 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{27720 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 226, normalized size = 1.14 \[ \frac {1}{12} \, b^{3} h x^{12} + \frac {1}{11} \, b^{3} g x^{11} + \frac {1}{10} \, b^{3} f x^{10} + \frac {1}{3} \, a b^{2} h x^{9} + \frac {1}{9} \, b^{3} x^{9} e + \frac {1}{8} \, b^{3} d x^{8} + \frac {3}{8} \, a b^{2} g x^{8} + \frac {1}{7} \, b^{3} c x^{7} + \frac {3}{7} \, a b^{2} f x^{7} + \frac {1}{2} \, a^{2} b h x^{6} + \frac {1}{2} \, a b^{2} x^{6} e + \frac {3}{5} \, a b^{2} d x^{5} + \frac {3}{5} \, a^{2} b g x^{5} + \frac {3}{4} \, a b^{2} c x^{4} + \frac {3}{4} \, a^{2} b f x^{4} + \frac {1}{3} \, a^{3} h x^{3} + a^{2} b x^{3} e + \frac {3}{2} \, a^{2} b d x^{2} + \frac {1}{2} \, a^{3} g x^{2} + 3 \, a^{2} b c x + a^{3} f x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac {2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 222, normalized size = 1.12 \[ \frac {b^{3} h \,x^{12}}{12}+\frac {b^{3} g \,x^{11}}{11}+\frac {b^{3} f \,x^{10}}{10}+\frac {a \,b^{2} h \,x^{9}}{3}+\frac {b^{3} e \,x^{9}}{9}+\frac {3 a \,b^{2} g \,x^{8}}{8}+\frac {b^{3} d \,x^{8}}{8}+\frac {3 a \,b^{2} f \,x^{7}}{7}+\frac {b^{3} c \,x^{7}}{7}+\frac {a^{2} b h \,x^{6}}{2}+\frac {a \,b^{2} e \,x^{6}}{2}+\frac {3 a^{2} b g \,x^{5}}{5}+\frac {3 a \,b^{2} d \,x^{5}}{5}+\frac {3 a^{2} b f \,x^{4}}{4}+\frac {3 a \,b^{2} c \,x^{4}}{4}+\frac {a^{3} h \,x^{3}}{3}+a^{2} b e \,x^{3}+\frac {a^{3} g \,x^{2}}{2}+\frac {3 a^{2} b d \,x^{2}}{2}+a^{3} e \ln \relax (x )+a^{3} f x +3 a^{2} b c x -\frac {a^{3} d}{x}-\frac {a^{3} c}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 212, normalized size = 1.07 \[ \frac {1}{12} \, b^{3} h x^{12} + \frac {1}{11} \, b^{3} g x^{11} + \frac {1}{10} \, b^{3} f x^{10} + \frac {1}{9} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{9} + \frac {1}{8} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{8} + \frac {1}{7} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{7} + \frac {1}{2} \, {\left (a b^{2} e + a^{2} b h\right )} x^{6} + \frac {3}{5} \, {\left (a b^{2} d + a^{2} b g\right )} x^{5} + \frac {3}{4} \, {\left (a b^{2} c + a^{2} b f\right )} x^{4} + a^{3} e \log \relax (x) + \frac {1}{3} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{2} + {\left (3 \, a^{2} b c + a^{3} f\right )} x - \frac {2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 199, normalized size = 1.01 \[ x^7\,\left (\frac {c\,b^3}{7}+\frac {3\,a\,f\,b^2}{7}\right )+x^2\,\left (\frac {g\,a^3}{2}+\frac {3\,b\,d\,a^2}{2}\right )+x^8\,\left (\frac {d\,b^3}{8}+\frac {3\,a\,g\,b^2}{8}\right )+x^3\,\left (\frac {h\,a^3}{3}+b\,e\,a^2\right )+x^9\,\left (\frac {e\,b^3}{9}+\frac {a\,h\,b^2}{3}\right )-\frac {\frac {a^3\,c}{2}+a^3\,d\,x}{x^2}+x\,\left (f\,a^3+3\,b\,c\,a^2\right )+\frac {b^3\,f\,x^{10}}{10}+\frac {b^3\,g\,x^{11}}{11}+\frac {b^3\,h\,x^{12}}{12}+a^3\,e\,\ln \relax (x)+\frac {3\,a\,b\,x^4\,\left (b\,c+a\,f\right )}{4}+\frac {3\,a\,b\,x^5\,\left (b\,d+a\,g\right )}{5}+\frac {a\,b\,x^6\,\left (b\,e+a\,h\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 238, normalized size = 1.20 \[ a^{3} e \log {\relax (x )} + \frac {b^{3} f x^{10}}{10} + \frac {b^{3} g x^{11}}{11} + \frac {b^{3} h x^{12}}{12} + x^{9} \left (\frac {a b^{2} h}{3} + \frac {b^{3} e}{9}\right ) + x^{8} \left (\frac {3 a b^{2} g}{8} + \frac {b^{3} d}{8}\right ) + x^{7} \left (\frac {3 a b^{2} f}{7} + \frac {b^{3} c}{7}\right ) + x^{6} \left (\frac {a^{2} b h}{2} + \frac {a b^{2} e}{2}\right ) + x^{5} \left (\frac {3 a^{2} b g}{5} + \frac {3 a b^{2} d}{5}\right ) + x^{4} \left (\frac {3 a^{2} b f}{4} + \frac {3 a b^{2} c}{4}\right ) + x^{3} \left (\frac {a^{3} h}{3} + a^{2} b e\right ) + x^{2} \left (\frac {a^{3} g}{2} + \frac {3 a^{2} b d}{2}\right ) + x \left (a^{3} f + 3 a^{2} b c\right ) + \frac {- a^{3} c - 2 a^{3} d x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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