Optimal. Leaf size=86 \[ \log (x) (a f+b c)+x (a g+b d)+\frac {1}{2} x^2 (a h+b e)-\frac {a c}{3 x^3}-\frac {a d}{2 x^2}-\frac {a e}{x}+\frac {1}{3} b f x^3+\frac {1}{4} b g x^4+\frac {1}{5} b h x^5 \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1820} \[ \log (x) (a f+b c)+x (a g+b d)+\frac {1}{2} x^2 (a h+b e)-\frac {a c}{3 x^3}-\frac {a d}{2 x^2}-\frac {a e}{x}+\frac {1}{3} b f x^3+\frac {1}{4} b g x^4+\frac {1}{5} b h x^5 \]
Antiderivative was successfully verified.
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Rule 1820
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x^4} \, dx &=\int \left (b d \left (1+\frac {a g}{b d}\right )+\frac {a c}{x^4}+\frac {a d}{x^3}+\frac {a e}{x^2}+\frac {b c+a f}{x}+(b e+a h) x+b f x^2+b g x^3+b h x^4\right ) \, dx\\ &=-\frac {a c}{3 x^3}-\frac {a d}{2 x^2}-\frac {a e}{x}+(b d+a g) x+\frac {1}{2} (b e+a h) x^2+\frac {1}{3} b f x^3+\frac {1}{4} b g x^4+\frac {1}{5} b h x^5+(b c+a f) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 76, normalized size = 0.88 \[ \log (x) (a f+b c)-\frac {a \left (2 c+3 x \left (d+2 e x-\left (x^3 (2 g+h x)\right )\right )\right )}{6 x^3}+\frac {1}{60} b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 81, normalized size = 0.94 \[ \frac {12 \, b h x^{8} + 15 \, b g x^{7} + 20 \, b f x^{6} + 30 \, {\left (b e + a h\right )} x^{5} + 60 \, {\left (b d + a g\right )} x^{4} + 60 \, {\left (b c + a f\right )} x^{3} \log \relax (x) - 60 \, a e x^{2} - 30 \, a d x - 20 \, a c}{60 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 79, normalized size = 0.92 \[ \frac {1}{5} \, b h x^{5} + \frac {1}{4} \, b g x^{4} + \frac {1}{3} \, b f x^{3} + \frac {1}{2} \, a h x^{2} + \frac {1}{2} \, b x^{2} e + b d x + a g x + {\left (b c + a f\right )} \log \left ({\left | x \right |}\right ) - \frac {6 \, a x^{2} e + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.88 \[ \frac {b h \,x^{5}}{5}+\frac {b g \,x^{4}}{4}+\frac {b f \,x^{3}}{3}+\frac {a h \,x^{2}}{2}+\frac {b e \,x^{2}}{2}+a f \ln \relax (x )+a g x +b c \ln \relax (x )+b d x -\frac {a e}{x}-\frac {a d}{2 x^{2}}-\frac {a c}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 75, normalized size = 0.87 \[ \frac {1}{5} \, b h x^{5} + \frac {1}{4} \, b g x^{4} + \frac {1}{3} \, b f x^{3} + \frac {1}{2} \, {\left (b e + a h\right )} x^{2} + {\left (b d + a g\right )} x + {\left (b c + a f\right )} \log \relax (x) - \frac {6 \, a e x^{2} + 3 \, a d x + 2 \, a c}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 75, normalized size = 0.87 \[ x\,\left (b\,d+a\,g\right )-\frac {a\,e\,x^2+\frac {a\,d\,x}{2}+\frac {a\,c}{3}}{x^3}+x^2\,\left (\frac {b\,e}{2}+\frac {a\,h}{2}\right )+\ln \relax (x)\,\left (b\,c+a\,f\right )+\frac {b\,h\,x^5}{5}+\frac {b\,f\,x^3}{3}+\frac {b\,g\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 83, normalized size = 0.97 \[ \frac {b f x^{3}}{3} + \frac {b g x^{4}}{4} + \frac {b h x^{5}}{5} + x^{2} \left (\frac {a h}{2} + \frac {b e}{2}\right ) + x \left (a g + b d\right ) + \left (a f + b c\right ) \log {\relax (x )} + \frac {- 2 a c - 3 a d x - 6 a e x^{2}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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