Optimal. Leaf size=81 \[ a^3 A \log (x)+a^3 B x+\frac {3}{2} a^2 A c x^2+a^2 B c x^3+\frac {3}{4} a A c^2 x^4+\frac {3}{5} a B c^2 x^5+\frac {1}{6} A c^3 x^6+\frac {1}{7} B c^3 x^7 \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {766} \begin {gather*} \frac {3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac {3}{4} a A c^2 x^4+\frac {3}{5} a B c^2 x^5+\frac {1}{6} A c^3 x^6+\frac {1}{7} B c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{x} \, dx &=\int \left (a^3 B+\frac {a^3 A}{x}+3 a^2 A c x+3 a^2 B c x^2+3 a A c^2 x^3+3 a B c^2 x^4+A c^3 x^5+B c^3 x^6\right ) \, dx\\ &=a^3 B x+\frac {3}{2} a^2 A c x^2+a^2 B c x^3+\frac {3}{4} a A c^2 x^4+\frac {3}{5} a B c^2 x^5+\frac {1}{6} A c^3 x^6+\frac {1}{7} B c^3 x^7+a^3 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 81, normalized size = 1.00 \begin {gather*} a^3 A \log (x)+a^3 B x+\frac {3}{2} a^2 A c x^2+a^2 B c x^3+\frac {3}{4} a A c^2 x^4+\frac {3}{5} a B c^2 x^5+\frac {1}{6} A c^3 x^6+\frac {1}{7} B c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 71, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{5} \, B a c^{2} x^{5} + \frac {3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac {3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 72, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{5} \, B a c^{2} x^{5} + \frac {3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac {3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 0.89 \begin {gather*} \frac {B \,c^{3} x^{7}}{7}+\frac {A \,c^{3} x^{6}}{6}+\frac {3 B a \,c^{2} x^{5}}{5}+\frac {3 A a \,c^{2} x^{4}}{4}+B \,a^{2} c \,x^{3}+\frac {3 A \,a^{2} c \,x^{2}}{2}+A \,a^{3} \ln \relax (x )+B \,a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 71, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{5} \, B a c^{2} x^{5} + \frac {3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac {3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 71, normalized size = 0.88 \begin {gather*} \frac {A\,c^3\,x^6}{6}+\frac {B\,c^3\,x^7}{7}+A\,a^3\,\ln \relax (x)+B\,a^3\,x+\frac {3\,A\,a^2\,c\,x^2}{2}+\frac {3\,A\,a\,c^2\,x^4}{4}+B\,a^2\,c\,x^3+\frac {3\,B\,a\,c^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 85, normalized size = 1.05 \begin {gather*} A a^{3} \log {\relax (x )} + \frac {3 A a^{2} c x^{2}}{2} + \frac {3 A a c^{2} x^{4}}{4} + \frac {A c^{3} x^{6}}{6} + B a^{3} x + B a^{2} c x^{3} + \frac {3 B a c^{2} x^{5}}{5} + \frac {B c^{3} x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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