Optimal. Leaf size=89 \[ -\frac {a^4 A}{3 x^3}-\frac {a^3 (a B+4 A b)}{2 x^2}-\frac {2 a^2 b (2 a B+3 A b)}{x}+b^3 x (4 a B+A b)+2 a b^2 \log (x) (3 a B+2 A b)+\frac {1}{2} b^4 B x^2 \]
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Rubi [A] time = 0.05, antiderivative size = 89, 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} \[ -\frac {a^3 (a B+4 A b)}{2 x^2}-\frac {2 a^2 b (2 a B+3 A b)}{x}-\frac {a^4 A}{3 x^3}+b^3 x (4 a B+A b)+2 a b^2 \log (x) (3 a B+2 A b)+\frac {1}{2} b^4 B x^2 \]
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^4} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{x^4} \, dx\\ &=\int \left (b^3 (A b+4 a B)+\frac {a^4 A}{x^4}+\frac {a^3 (4 A b+a B)}{x^3}+\frac {2 a^2 b (3 A b+2 a B)}{x^2}+\frac {2 a b^2 (2 A b+3 a B)}{x}+b^4 B x\right ) \, dx\\ &=-\frac {a^4 A}{3 x^3}-\frac {a^3 (4 A b+a B)}{2 x^2}-\frac {2 a^2 b (3 A b+2 a B)}{x}+b^3 (A b+4 a B) x+\frac {1}{2} b^4 B x^2+2 a b^2 (2 A b+3 a B) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 86, normalized size = 0.97 \[ -\frac {a^4 (2 A+3 B x)}{6 x^3}-\frac {2 a^3 b (A+2 B x)}{x^2}-\frac {6 a^2 A b^2}{x}+2 a b^2 \log (x) (3 a B+2 A b)+4 a b^3 B x+\frac {1}{2} b^4 x (2 A+B x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 101, normalized size = 1.13 \[ \frac {3 \, B b^{4} x^{5} - 2 \, A a^{4} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} \log \relax (x) - 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 96, normalized size = 1.08 \[ \frac {1}{2} \, B b^{4} x^{2} + 4 \, B a b^{3} x + A b^{4} x + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{4} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 95, normalized size = 1.07 \[ \frac {B \,b^{4} x^{2}}{2}+4 A a \,b^{3} \ln \relax (x )+A \,b^{4} x +6 B \,a^{2} b^{2} \ln \relax (x )+4 B a \,b^{3} x -\frac {6 A \,a^{2} b^{2}}{x}-\frac {4 B \,a^{3} b}{x}-\frac {2 A \,a^{3} b}{x^{2}}-\frac {B \,a^{4}}{2 x^{2}}-\frac {A \,a^{4}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 96, normalized size = 1.08 \[ \frac {1}{2} \, B b^{4} x^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} x + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \log \relax (x) - \frac {2 \, A a^{4} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 94, normalized size = 1.06 \[ x\,\left (A\,b^4+4\,B\,a\,b^3\right )+\ln \relax (x)\,\left (6\,B\,a^2\,b^2+4\,A\,a\,b^3\right )-\frac {x\,\left (\frac {B\,a^4}{2}+2\,A\,b\,a^3\right )+\frac {A\,a^4}{3}+x^2\,\left (4\,B\,a^3\,b+6\,A\,a^2\,b^2\right )}{x^3}+\frac {B\,b^4\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 99, normalized size = 1.11 \[ \frac {B b^{4} x^{2}}{2} + 2 a b^{2} \left (2 A b + 3 B a\right ) \log {\relax (x )} + x \left (A b^{4} + 4 B a b^{3}\right ) + \frac {- 2 A a^{4} + x^{2} \left (- 36 A a^{2} b^{2} - 24 B a^{3} b\right ) + x \left (- 12 A a^{3} b - 3 B a^{4}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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