Optimal. Leaf size=56 \[ -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3+2 a A c \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 56, 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 = {780}
\begin {gather*} -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a A c \log (x)+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 780
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^2}{x^3} \, dx &=\int \left (2 a B c+\frac {a^2 A}{x^3}+\frac {a^2 B}{x^2}+\frac {2 a A c}{x}+A c^2 x+B c^2 x^2\right ) \, dx\\ &=-\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3+2 a A c \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} -\frac {a^2 A}{2 x^2}-\frac {a^2 B}{x}+2 a B c x+\frac {1}{2} A c^2 x^2+\frac {1}{3} B c^2 x^3+2 a A c \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 51, normalized size = 0.91
method | result | size |
default | \(-\frac {a^{2} A}{2 x^{2}}-\frac {a^{2} B}{x}+2 a B c x +\frac {A \,c^{2} x^{2}}{2}+\frac {B \,c^{2} x^{3}}{3}+2 a A c \ln \left (x \right )\) | \(51\) |
risch | \(\frac {B \,c^{2} x^{3}}{3}+\frac {A \,c^{2} x^{2}}{2}+2 a B c x +\frac {-a^{2} B x -\frac {1}{2} a^{2} A}{x^{2}}+2 a A c \ln \left (x \right )\) | \(51\) |
norman | \(\frac {-\frac {1}{2} a^{2} A +\frac {1}{2} A \,c^{2} x^{4}+\frac {1}{3} B \,c^{2} x^{5}-a^{2} B x +2 a B c \,x^{3}}{x^{2}}+2 a A c \ln \left (x \right )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \left (x\right ) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.07, size = 55, normalized size = 0.98 \begin {gather*} \frac {2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 12 \, A a c x^{2} \log \left (x\right ) - 6 \, B a^{2} x - 3 \, A a^{2}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 58, normalized size = 1.04 \begin {gather*} 2 A a c \log {\left (x \right )} + \frac {A c^{2} x^{2}}{2} + 2 B a c x + \frac {B c^{2} x^{3}}{3} + \frac {- A a^{2} - 2 B a^{2} x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 51, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, B c^{2} x^{3} + \frac {1}{2} \, A c^{2} x^{2} + 2 \, B a c x + 2 \, A a c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{2} x + A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 50, normalized size = 0.89 \begin {gather*} \frac {A\,c^2\,x^2}{2}-\frac {\frac {A\,a^2}{2}+B\,a^2\,x}{x^2}+\frac {B\,c^2\,x^3}{3}+2\,A\,a\,c\,\ln \left (x\right )+2\,B\,a\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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