Optimal. Leaf size=66 \[ \frac {a^2 (A b-a B) \log (a+b x)}{b^4}-\frac {a x (A b-a B)}{b^3}+\frac {x^2 (A b-a B)}{2 b^2}+\frac {B x^3}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \frac {a^2 (A b-a B) \log (a+b x)}{b^4}+\frac {x^2 (A b-a B)}{2 b^2}-\frac {a x (A b-a B)}{b^3}+\frac {B x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{a+b x} \, dx &=\int \left (\frac {a (-A b+a B)}{b^3}+\frac {(A b-a B) x}{b^2}+\frac {B x^2}{b}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^3}{3 b}+\frac {a^2 (A b-a B) \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.92 \[ \frac {b x \left (6 a^2 B-3 a b (2 A+B x)+b^2 x (3 A+2 B x)\right )+6 a^2 (A b-a B) \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 71, normalized size = 1.08 \[ \frac {2 \, B b^{3} x^{3} - 3 \, {\left (B a b^{2} - A b^{3}\right )} x^{2} + 6 \, {\left (B a^{2} b - A a b^{2}\right )} x - 6 \, {\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 71, normalized size = 1.08 \[ \frac {2 \, B b^{2} x^{3} - 3 \, B a b x^{2} + 3 \, A b^{2} x^{2} + 6 \, B a^{2} x - 6 \, A a b x}{6 \, b^{3}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 1.15 \[ \frac {B \,x^{3}}{3 b}+\frac {A \,x^{2}}{2 b}-\frac {B a \,x^{2}}{2 b^{2}}+\frac {A \,a^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {A a x}{b^{2}}-\frac {B \,a^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {B \,a^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 70, normalized size = 1.06 \[ \frac {2 \, B b^{2} x^{3} - 3 \, {\left (B a b - A b^{2}\right )} x^{2} + 6 \, {\left (B a^{2} - A a b\right )} x}{6 \, b^{3}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 72, normalized size = 1.09 \[ x^2\,\left (\frac {A}{2\,b}-\frac {B\,a}{2\,b^2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^3-A\,a^2\,b\right )}{b^4}+\frac {B\,x^3}{3\,b}-\frac {a\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 61, normalized size = 0.92 \[ \frac {B x^{3}}{3 b} - \frac {a^{2} \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{4}} + x^{2} \left (\frac {A}{2 b} - \frac {B a}{2 b^{2}}\right ) + x \left (- \frac {A a}{b^{2}} + \frac {B a^{2}}{b^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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