Optimal. Leaf size=162 \[ -\frac {a^4 c}{x}+a^4 d \log (x)+a^4 e x+2 a^3 b c x^2+\frac {4}{3} a^3 b d x^3+a^3 b e x^4+\frac {6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac {6}{7} a^2 b^2 e x^7+\frac {1}{2} a b^3 c x^8+\frac {4}{9} a b^3 d x^9+\frac {2}{5} a b^3 e x^{10}+\frac {1}{11} b^4 c x^{11}+\frac {1}{12} b^4 d x^{12}+\frac {1}{13} b^4 e x^{13} \]
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Rubi [A] time = 0.13, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1628} \[ \frac {6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac {6}{7} a^2 b^2 e x^7+2 a^3 b c x^2+\frac {4}{3} a^3 b d x^3+a^3 b e x^4-\frac {a^4 c}{x}+a^4 d \log (x)+a^4 e x+\frac {1}{2} a b^3 c x^8+\frac {4}{9} a b^3 d x^9+\frac {2}{5} a b^3 e x^{10}+\frac {1}{11} b^4 c x^{11}+\frac {1}{12} b^4 d x^{12}+\frac {1}{13} b^4 e x^{13} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^2} \, dx &=\int \left (a^4 e+\frac {a^4 c}{x^2}+\frac {a^4 d}{x}+4 a^3 b c x+4 a^3 b d x^2+4 a^3 b e x^3+6 a^2 b^2 c x^4+6 a^2 b^2 d x^5+6 a^2 b^2 e x^6+4 a b^3 c x^7+4 a b^3 d x^8+4 a b^3 e x^9+b^4 c x^{10}+b^4 d x^{11}+b^4 e x^{12}\right ) \, dx\\ &=-\frac {a^4 c}{x}+a^4 e x+2 a^3 b c x^2+\frac {4}{3} a^3 b d x^3+a^3 b e x^4+\frac {6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac {6}{7} a^2 b^2 e x^7+\frac {1}{2} a b^3 c x^8+\frac {4}{9} a b^3 d x^9+\frac {2}{5} a b^3 e x^{10}+\frac {1}{11} b^4 c x^{11}+\frac {1}{12} b^4 d x^{12}+\frac {1}{13} b^4 e x^{13}+a^4 d \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 162, normalized size = 1.00 \[ -\frac {a^4 c}{x}+a^4 d \log (x)+a^4 e x+2 a^3 b c x^2+\frac {4}{3} a^3 b d x^3+a^3 b e x^4+\frac {6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac {6}{7} a^2 b^2 e x^7+\frac {1}{2} a b^3 c x^8+\frac {4}{9} a b^3 d x^9+\frac {2}{5} a b^3 e x^{10}+\frac {1}{11} b^4 c x^{11}+\frac {1}{12} b^4 d x^{12}+\frac {1}{13} b^4 e x^{13} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 153, normalized size = 0.94 \[ \frac {13860 \, b^{4} e x^{14} + 15015 \, b^{4} d x^{13} + 16380 \, b^{4} c x^{12} + 72072 \, a b^{3} e x^{11} + 80080 \, a b^{3} d x^{10} + 90090 \, a b^{3} c x^{9} + 154440 \, a^{2} b^{2} e x^{8} + 180180 \, a^{2} b^{2} d x^{7} + 216216 \, a^{2} b^{2} c x^{6} + 180180 \, a^{3} b e x^{5} + 240240 \, a^{3} b d x^{4} + 360360 \, a^{3} b c x^{3} + 180180 \, a^{4} e x^{2} + 180180 \, a^{4} d x \log \relax (x) - 180180 \, a^{4} c}{180180 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 150, normalized size = 0.93 \[ \frac {1}{13} \, b^{4} x^{13} e + \frac {1}{12} \, b^{4} d x^{12} + \frac {1}{11} \, b^{4} c x^{11} + \frac {2}{5} \, a b^{3} x^{10} e + \frac {4}{9} \, a b^{3} d x^{9} + \frac {1}{2} \, a b^{3} c x^{8} + \frac {6}{7} \, a^{2} b^{2} x^{7} e + a^{2} b^{2} d x^{6} + \frac {6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b x^{4} e + \frac {4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + a^{4} x e + a^{4} d \log \left ({\left | x \right |}\right ) - \frac {a^{4} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 145, normalized size = 0.90 \[ \frac {b^{4} e \,x^{13}}{13}+\frac {b^{4} d \,x^{12}}{12}+\frac {b^{4} c \,x^{11}}{11}+\frac {2 a \,b^{3} e \,x^{10}}{5}+\frac {4 a \,b^{3} d \,x^{9}}{9}+\frac {a \,b^{3} c \,x^{8}}{2}+\frac {6 a^{2} b^{2} e \,x^{7}}{7}+a^{2} b^{2} d \,x^{6}+\frac {6 a^{2} b^{2} c \,x^{5}}{5}+a^{3} b e \,x^{4}+\frac {4 a^{3} b d \,x^{3}}{3}+2 a^{3} b c \,x^{2}+a^{4} d \ln \relax (x )+a^{4} e x -\frac {a^{4} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 144, normalized size = 0.89 \[ \frac {1}{13} \, b^{4} e x^{13} + \frac {1}{12} \, b^{4} d x^{12} + \frac {1}{11} \, b^{4} c x^{11} + \frac {2}{5} \, a b^{3} e x^{10} + \frac {4}{9} \, a b^{3} d x^{9} + \frac {1}{2} \, a b^{3} c x^{8} + \frac {6}{7} \, a^{2} b^{2} e x^{7} + a^{2} b^{2} d x^{6} + \frac {6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b e x^{4} + \frac {4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + a^{4} e x + a^{4} d \log \relax (x) - \frac {a^{4} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 144, normalized size = 0.89 \[ \frac {b^4\,c\,x^{11}}{11}-\frac {a^4\,c}{x}+\frac {b^4\,d\,x^{12}}{12}+\frac {b^4\,e\,x^{13}}{13}+a^4\,d\,\ln \relax (x)+a^4\,e\,x+\frac {6\,a^2\,b^2\,c\,x^5}{5}+a^2\,b^2\,d\,x^6+\frac {6\,a^2\,b^2\,e\,x^7}{7}+2\,a^3\,b\,c\,x^2+\frac {a\,b^3\,c\,x^8}{2}+\frac {4\,a^3\,b\,d\,x^3}{3}+\frac {4\,a\,b^3\,d\,x^9}{9}+a^3\,b\,e\,x^4+\frac {2\,a\,b^3\,e\,x^{10}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 168, normalized size = 1.04 \[ - \frac {a^{4} c}{x} + a^{4} d \log {\relax (x )} + a^{4} e x + 2 a^{3} b c x^{2} + \frac {4 a^{3} b d x^{3}}{3} + a^{3} b e x^{4} + \frac {6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac {6 a^{2} b^{2} e x^{7}}{7} + \frac {a b^{3} c x^{8}}{2} + \frac {4 a b^{3} d x^{9}}{9} + \frac {2 a b^{3} e x^{10}}{5} + \frac {b^{4} c x^{11}}{11} + \frac {b^{4} d x^{12}}{12} + \frac {b^{4} e x^{13}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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