Optimal. Leaf size=209 \[ -\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac {1}{2} a^2 x^2 (a h+3 b e)+\frac {1}{6} b^2 x^6 (3 a f+b c)+\frac {1}{7} b^2 x^7 (3 a g+b d)+\frac {1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac {3}{4} a b x^4 (a g+b d)+\frac {3}{5} a b x^5 (a h+b e)+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11} \]
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Rubi [A] time = 0.18, antiderivative size = 209, 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} \[ a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac {1}{2} a^2 x^2 (a h+3 b e)-\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+\frac {1}{6} b^2 x^6 (3 a f+b c)+\frac {1}{7} b^2 x^7 (3 a g+b d)+\frac {1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac {3}{4} a b x^4 (a g+b d)+\frac {3}{5} a b x^5 (a h+b e)+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11} \]
Antiderivative was successfully verified.
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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^4} \, dx &=\int \left (a^2 (3 b d+a g)+\frac {a^3 c}{x^4}+\frac {a^3 d}{x^3}+\frac {a^3 e}{x^2}+\frac {a^2 (3 b c+a f)}{x}+a^2 (3 b e+a h) x+3 a b (b c+a f) x^2+3 a b (b d+a g) x^3+3 a b (b e+a h) x^4+b^2 (b c+3 a f) x^5+b^2 (b d+3 a g) x^6+b^2 (b e+3 a h) x^7+b^3 f x^8+b^3 g x^9+b^3 h x^{10}\right ) \, dx\\ &=-\frac {a^3 c}{3 x^3}-\frac {a^3 d}{2 x^2}-\frac {a^3 e}{x}+a^2 (3 b d+a g) x+\frac {1}{2} a^2 (3 b e+a h) x^2+a b (b c+a f) x^3+\frac {3}{4} a b (b d+a g) x^4+\frac {3}{5} a b (b e+a h) x^5+\frac {1}{6} b^2 (b c+3 a f) x^6+\frac {1}{7} b^2 (b d+3 a g) x^7+\frac {1}{8} b^2 (b e+3 a h) x^8+\frac {1}{9} b^3 f x^9+\frac {1}{10} b^3 g x^{10}+\frac {1}{11} b^3 h x^{11}+a^2 (3 b c+a f) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 172, normalized size = 0.82 \[ -\frac {a^3 \left (2 c+3 x \left (d+2 e x-\left (x^3 (2 g+h x)\right )\right )\right )}{6 x^3}+a^2 \log (x) (a f+3 b c)+\frac {1}{20} a^2 b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right )+\frac {1}{280} a b^2 x^3 \left (280 c+x \left (210 d+x \left (168 e+140 f x+120 g x^2+105 h x^3\right )\right )\right )+\frac {b^3 x^6 \left (4620 c+x \left (3960 d+7 x \left (495 e+4 x \left (110 f+99 g x+90 h x^2\right )\right )\right )\right )}{27720} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 219, normalized size = 1.05 \[ \frac {2520 \, b^{3} h x^{14} + 2772 \, b^{3} g x^{13} + 3080 \, b^{3} f x^{12} + 3465 \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3960 \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 4620 \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 16632 \, {\left (a b^{2} e + a^{2} b h\right )} x^{8} + 20790 \, {\left (a b^{2} d + a^{2} b g\right )} x^{7} + 27720 \, {\left (a b^{2} c + a^{2} b f\right )} x^{6} - 27720 \, a^{3} e x^{2} + 13860 \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 13860 \, a^{3} d x + 27720 \, {\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + 27720 \, {\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \log \relax (x) - 9240 \, a^{3} c}{27720 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 225, normalized size = 1.08 \[ \frac {1}{11} \, b^{3} h x^{11} + \frac {1}{10} \, b^{3} g x^{10} + \frac {1}{9} \, b^{3} f x^{9} + \frac {3}{8} \, a b^{2} h x^{8} + \frac {1}{8} \, b^{3} x^{8} e + \frac {1}{7} \, b^{3} d x^{7} + \frac {3}{7} \, a b^{2} g x^{7} + \frac {1}{6} \, b^{3} c x^{6} + \frac {1}{2} \, a b^{2} f x^{6} + \frac {3}{5} \, a^{2} b h x^{5} + \frac {3}{5} \, a b^{2} x^{5} e + \frac {3}{4} \, a b^{2} d x^{4} + \frac {3}{4} \, a^{2} b g x^{4} + a b^{2} c x^{3} + a^{2} b f x^{3} + \frac {1}{2} \, a^{3} h x^{2} + \frac {3}{2} \, a^{2} b x^{2} e + 3 \, a^{2} b d x + a^{3} g x + {\left (3 \, a^{2} b c + a^{3} f\right )} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{3} x^{2} e + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 220, normalized size = 1.05 \[ \frac {b^{3} h \,x^{11}}{11}+\frac {b^{3} g \,x^{10}}{10}+\frac {b^{3} f \,x^{9}}{9}+\frac {3 a \,b^{2} h \,x^{8}}{8}+\frac {b^{3} e \,x^{8}}{8}+\frac {3 a \,b^{2} g \,x^{7}}{7}+\frac {b^{3} d \,x^{7}}{7}+\frac {a \,b^{2} f \,x^{6}}{2}+\frac {b^{3} c \,x^{6}}{6}+\frac {3 a^{2} b h \,x^{5}}{5}+\frac {3 a \,b^{2} e \,x^{5}}{5}+\frac {3 a^{2} b g \,x^{4}}{4}+\frac {3 a \,b^{2} d \,x^{4}}{4}+a^{2} b f \,x^{3}+a \,b^{2} c \,x^{3}+\frac {a^{3} h \,x^{2}}{2}+\frac {3 a^{2} b e \,x^{2}}{2}+a^{3} f \ln \relax (x )+a^{3} g x +3 a^{2} b c \ln \relax (x )+3 a^{2} b d x -\frac {a^{3} e}{x}-\frac {a^{3} d}{2 x^{2}}-\frac {a^{3} c}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 212, normalized size = 1.01 \[ \frac {1}{11} \, b^{3} h x^{11} + \frac {1}{10} \, b^{3} g x^{10} + \frac {1}{9} \, b^{3} f x^{9} + \frac {1}{8} \, {\left (b^{3} e + 3 \, a b^{2} h\right )} x^{8} + \frac {1}{7} \, {\left (b^{3} d + 3 \, a b^{2} g\right )} x^{7} + \frac {1}{6} \, {\left (b^{3} c + 3 \, a b^{2} f\right )} x^{6} + \frac {3}{5} \, {\left (a b^{2} e + a^{2} b h\right )} x^{5} + \frac {3}{4} \, {\left (a b^{2} d + a^{2} b g\right )} x^{4} + {\left (a b^{2} c + a^{2} b f\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b e + a^{3} h\right )} x^{2} + {\left (3 \, a^{2} b d + a^{3} g\right )} x + {\left (3 \, a^{2} b c + a^{3} f\right )} \log \relax (x) - \frac {6 \, a^{3} e x^{2} + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 199, normalized size = 0.95 \[ x^6\,\left (\frac {c\,b^3}{6}+\frac {a\,f\,b^2}{2}\right )+x^7\,\left (\frac {d\,b^3}{7}+\frac {3\,a\,g\,b^2}{7}\right )+x^2\,\left (\frac {h\,a^3}{2}+\frac {3\,b\,e\,a^2}{2}\right )+x^8\,\left (\frac {e\,b^3}{8}+\frac {3\,a\,h\,b^2}{8}\right )+\ln \relax (x)\,\left (f\,a^3+3\,b\,c\,a^2\right )-\frac {e\,a^3\,x^2+\frac {d\,a^3\,x}{2}+\frac {c\,a^3}{3}}{x^3}+x\,\left (g\,a^3+3\,b\,d\,a^2\right )+\frac {b^3\,f\,x^9}{9}+\frac {b^3\,g\,x^{10}}{10}+\frac {b^3\,h\,x^{11}}{11}+a\,b\,x^3\,\left (b\,c+a\,f\right )+\frac {3\,a\,b\,x^4\,\left (b\,d+a\,g\right )}{4}+\frac {3\,a\,b\,x^5\,\left (b\,e+a\,h\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 236, normalized size = 1.13 \[ a^{2} \left (a f + 3 b c\right ) \log {\relax (x )} + \frac {b^{3} f x^{9}}{9} + \frac {b^{3} g x^{10}}{10} + \frac {b^{3} h x^{11}}{11} + x^{8} \left (\frac {3 a b^{2} h}{8} + \frac {b^{3} e}{8}\right ) + x^{7} \left (\frac {3 a b^{2} g}{7} + \frac {b^{3} d}{7}\right ) + x^{6} \left (\frac {a b^{2} f}{2} + \frac {b^{3} c}{6}\right ) + x^{5} \left (\frac {3 a^{2} b h}{5} + \frac {3 a b^{2} e}{5}\right ) + x^{4} \left (\frac {3 a^{2} b g}{4} + \frac {3 a b^{2} d}{4}\right ) + x^{3} \left (a^{2} b f + a b^{2} c\right ) + x^{2} \left (\frac {a^{3} h}{2} + \frac {3 a^{2} b e}{2}\right ) + x \left (a^{3} g + 3 a^{2} b d\right ) + \frac {- 2 a^{3} c - 3 a^{3} d x - 6 a^{3} e x^{2}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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