Optimal. Leaf size=90 \[ -\frac {a^2 A}{4 x^4}-\frac {a (2 A b+a B)}{3 x^3}-\frac {2 a b B+A \left (b^2+2 a c\right )}{2 x^2}-\frac {b^2 B+2 A b c+2 a B c}{x}+B c^2 x+c (2 b B+A c) \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779}
\begin {gather*} -\frac {a^2 A}{4 x^4}-\frac {A \left (2 a c+b^2\right )+2 a b B}{2 x^2}-\frac {2 a B c+2 A b c+b^2 B}{x}-\frac {a (a B+2 A b)}{3 x^3}+c \log (x) (A c+2 b B)+B c^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^5} \, dx &=\int \left (B c^2+\frac {a^2 A}{x^5}+\frac {a (2 A b+a B)}{x^4}+\frac {2 a b B+A \left (b^2+2 a c\right )}{x^3}+\frac {b^2 B+2 A b c+2 a B c}{x^2}+\frac {c (2 b B+A c)}{x}\right ) \, dx\\ &=-\frac {a^2 A}{4 x^4}-\frac {a (2 A b+a B)}{3 x^3}-\frac {2 a b B+A \left (b^2+2 a c\right )}{2 x^2}-\frac {b^2 B+2 A b c+2 a B c}{x}+B c^2 x+c (2 b B+A c) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 92, normalized size = 1.02 \begin {gather*} -\frac {a^2 (3 A+4 B x)+4 a x (3 B x (b+2 c x)+A (2 b+3 c x))+6 x^2 \left (A b (b+4 c x)+2 B x \left (b^2-c^2 x^2\right )\right )-12 c (2 b B+A c) x^4 \log (x)}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 85, normalized size = 0.94
method | result | size |
default | \(B \,c^{2} x -\frac {a^{2} A}{4 x^{4}}-\frac {2 A a c +b^{2} A +2 a b B}{2 x^{2}}-\frac {a \left (2 A b +B a \right )}{3 x^{3}}+c \left (A c +2 B b \right ) \ln \left (x \right )-\frac {2 A b c +2 a B c +b^{2} B}{x}\) | \(85\) |
risch | \(B \,c^{2} x +\frac {\left (-2 A b c -2 a B c -b^{2} B \right ) x^{3}+\left (-A a c -\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x -\frac {a^{2} A}{4}}{x^{4}}+A \ln \left (x \right ) c^{2}+2 B \ln \left (x \right ) b c\) | \(89\) |
norman | \(\frac {\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x +\left (-A a c -\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-2 A b c -2 a B c -b^{2} B \right ) x^{3}+B \,c^{2} x^{5}-\frac {a^{2} A}{4}}{x^{4}}+\left (A \,c^{2}+2 b B c \right ) \ln \left (x \right )\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 89, normalized size = 0.99 \begin {gather*} B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \left (x\right ) - \frac {12 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 95, normalized size = 1.06 \begin {gather*} \frac {12 \, B c^{2} x^{5} + 12 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} \log \left (x\right ) - 12 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.61, size = 99, normalized size = 1.10 \begin {gather*} B c^{2} x + c \left (A c + 2 B b\right ) \log {\left (x \right )} + \frac {- 3 A a^{2} + x^{3} \left (- 24 A b c - 24 B a c - 12 B b^{2}\right ) + x^{2} \left (- 12 A a c - 6 A b^{2} - 12 B a b\right ) + x \left (- 8 A a b - 4 B a^{2}\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 90, normalized size = 1.00 \begin {gather*} B c^{2} x + {\left (2 \, B b c + A c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, {\left (B b^{2} + 2 \, B a c + 2 \, A b c\right )} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 86, normalized size = 0.96 \begin {gather*} \ln \left (x\right )\,\left (A\,c^2+2\,B\,b\,c\right )-\frac {\frac {A\,a^2}{4}+x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b+A\,a\,c\right )+x^3\,\left (B\,b^2+2\,A\,c\,b+2\,B\,a\,c\right )+x\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )}{x^4}+B\,c^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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