Optimal. Leaf size=105 \[ -\frac {a^4 A}{2 x^2}-\frac {a^4 B}{x}+4 a^3 A c \log (x)+4 a^3 B c x+3 a^2 A c^2 x^2+2 a^2 B c^2 x^3+a A c^3 x^4+\frac {4}{5} a B c^3 x^5+\frac {1}{6} A c^4 x^6+\frac {1}{7} B c^4 x^7 \]
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Rubi [A] time = 0.05, antiderivative size = 105, 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*} 3 a^2 A c^2 x^2+4 a^3 A c \log (x)-\frac {a^4 A}{2 x^2}+2 a^2 B c^2 x^3+4 a^3 B c x-\frac {a^4 B}{x}+a A c^3 x^4+\frac {4}{5} a B c^3 x^5+\frac {1}{6} A c^4 x^6+\frac {1}{7} B c^4 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 )^4}{x^3} \, dx &=\int \left (4 a^3 B c+\frac {a^4 A}{x^3}+\frac {a^4 B}{x^2}+\frac {4 a^3 A c}{x}+6 a^2 A c^2 x+6 a^2 B c^2 x^2+4 a A c^3 x^3+4 a B c^3 x^4+A c^4 x^5+B c^4 x^6\right ) \, dx\\ &=-\frac {a^4 A}{2 x^2}-\frac {a^4 B}{x}+4 a^3 B c x+3 a^2 A c^2 x^2+2 a^2 B c^2 x^3+a A c^3 x^4+\frac {4}{5} a B c^3 x^5+\frac {1}{6} A c^4 x^6+\frac {1}{7} B c^4 x^7+4 a^3 A c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a^4 A}{2 x^2}-\frac {a^4 B}{x}+4 a^3 A c \log (x)+4 a^3 B c x+3 a^2 A c^2 x^2+2 a^2 B c^2 x^3+a A c^3 x^4+\frac {4}{5} a B c^3 x^5+\frac {1}{6} A c^4 x^6+\frac {1}{7} B c^4 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 )^4}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 103, normalized size = 0.98 \begin {gather*} \frac {30 \, B c^{4} x^{9} + 35 \, A c^{4} x^{8} + 168 \, B a c^{3} x^{7} + 210 \, A a c^{3} x^{6} + 420 \, B a^{2} c^{2} x^{5} + 630 \, A a^{2} c^{2} x^{4} + 840 \, B a^{3} c x^{3} + 840 \, A a^{3} c x^{2} \log \relax (x) - 210 \, B a^{4} x - 105 \, A a^{4}}{210 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 98, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, B c^{4} x^{7} + \frac {1}{6} \, A c^{4} x^{6} + \frac {4}{5} \, B a c^{3} x^{5} + A a c^{3} x^{4} + 2 \, B a^{2} c^{2} x^{3} + 3 \, A a^{2} c^{2} x^{2} + 4 \, B a^{3} c x + 4 \, A a^{3} c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{4} x + A a^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 98, normalized size = 0.93 \begin {gather*} \frac {B \,c^{4} x^{7}}{7}+\frac {A \,c^{4} x^{6}}{6}+\frac {4 B a \,c^{3} x^{5}}{5}+A a \,c^{3} x^{4}+2 B \,a^{2} c^{2} x^{3}+3 A \,a^{2} c^{2} x^{2}+4 A \,a^{3} c \ln \relax (x )+4 B \,a^{3} c x -\frac {B \,a^{4}}{x}-\frac {A \,a^{4}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 97, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, B c^{4} x^{7} + \frac {1}{6} \, A c^{4} x^{6} + \frac {4}{5} \, B a c^{3} x^{5} + A a c^{3} x^{4} + 2 \, B a^{2} c^{2} x^{3} + 3 \, A a^{2} c^{2} x^{2} + 4 \, B a^{3} c x + 4 \, A a^{3} c \log \relax (x) - \frac {2 \, B a^{4} x + A a^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 97, normalized size = 0.92 \begin {gather*} \frac {A\,c^4\,x^6}{6}-\frac {\frac {A\,a^4}{2}+B\,a^4\,x}{x^2}+\frac {B\,c^4\,x^7}{7}+4\,B\,a^3\,c\,x+A\,a\,c^3\,x^4+\frac {4\,B\,a\,c^3\,x^5}{5}+4\,A\,a^3\,c\,\ln \relax (x)+3\,A\,a^2\,c^2\,x^2+2\,B\,a^2\,c^2\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 112, normalized size = 1.07 \begin {gather*} 4 A a^{3} c \log {\relax (x )} + 3 A a^{2} c^{2} x^{2} + A a c^{3} x^{4} + \frac {A c^{4} x^{6}}{6} + 4 B a^{3} c x + 2 B a^{2} c^{2} x^{3} + \frac {4 B a c^{3} x^{5}}{5} + \frac {B c^{4} x^{7}}{7} + \frac {- A a^{4} - 2 B a^{4} x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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