Optimal. Leaf size=86 \[ -\frac {a^4 A}{x}+a^3 \log (x) (a B+4 A b)+2 a^2 b x (2 a B+3 A b)+\frac {1}{3} b^3 x^3 (4 a B+A b)+a b^2 x^2 (3 a B+2 A b)+\frac {1}{4} b^4 B x^4 \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 76} \[ 2 a^2 b x (2 a B+3 A b)+a^3 \log (x) (a B+4 A b)-\frac {a^4 A}{x}+a b^2 x^2 (3 a B+2 A b)+\frac {1}{3} b^3 x^3 (4 a B+A b)+\frac {1}{4} b^4 B x^4 \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^2} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^2} \, dx\\ &=\int \left (2 a^2 b (3 A b+2 a B)+\frac {a^4 A}{x^2}+\frac {a^3 (4 A b+a B)}{x}+2 a b^2 (2 A b+3 a B) x+b^3 (A b+4 a B) x^2+b^4 B x^3\right ) \, dx\\ &=-\frac {a^4 A}{x}+2 a^2 b (3 A b+2 a B) x+a b^2 (2 A b+3 a B) x^2+\frac {1}{3} b^3 (A b+4 a B) x^3+\frac {1}{4} b^4 B x^4+a^3 (4 A b+a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.99 \[ -\frac {a^4 A}{x}+a^3 \log (x) (a B+4 A b)+4 a^3 b B x+3 a^2 b^2 x (2 A+B x)+\frac {2}{3} a b^3 x^2 (3 A+2 B x)+\frac {1}{12} b^4 x^3 (4 A+3 B x) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 101, normalized size = 1.17 \[ \frac {3 \, B b^{4} x^{5} - 12 \, A a^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 24 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x \log \relax (x)}{12 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 95, normalized size = 1.10 \[ \frac {1}{4} \, B b^{4} x^{4} + \frac {4}{3} \, B a b^{3} x^{3} + \frac {1}{3} \, A b^{4} x^{3} + 3 \, B a^{2} b^{2} x^{2} + 2 \, A a b^{3} x^{2} + 4 \, B a^{3} b x + 6 \, A a^{2} b^{2} x - \frac {A a^{4}}{x} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 95, normalized size = 1.10 \[ \frac {B \,b^{4} x^{4}}{4}+\frac {A \,b^{4} x^{3}}{3}+\frac {4 B a \,b^{3} x^{3}}{3}+2 A a \,b^{3} x^{2}+3 B \,a^{2} b^{2} x^{2}+4 A \,a^{3} b \ln \relax (x )+6 A \,a^{2} b^{2} x +B \,a^{4} \ln \relax (x )+4 B \,a^{3} b x -\frac {A \,a^{4}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 94, normalized size = 1.09 \[ \frac {1}{4} \, B b^{4} x^{4} - \frac {A a^{4}}{x} + \frac {1}{3} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x + {\left (B a^{4} + 4 \, A a^{3} b\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 86, normalized size = 1.00 \[ x^3\,\left (\frac {A\,b^4}{3}+\frac {4\,B\,a\,b^3}{3}\right )+\ln \relax (x)\,\left (B\,a^4+4\,A\,b\,a^3\right )-\frac {A\,a^4}{x}+\frac {B\,b^4\,x^4}{4}+2\,a^2\,b\,x\,\left (3\,A\,b+2\,B\,a\right )+a\,b^2\,x^2\,\left (2\,A\,b+3\,B\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 94, normalized size = 1.09 \[ - \frac {A a^{4}}{x} + \frac {B b^{4} x^{4}}{4} + a^{3} \left (4 A b + B a\right ) \log {\relax (x )} + x^{3} \left (\frac {A b^{4}}{3} + \frac {4 B a b^{3}}{3}\right ) + x^{2} \left (2 A a b^{3} + 3 B a^{2} b^{2}\right ) + x \left (6 A a^{2} b^{2} + 4 B a^{3} b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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