Optimal. Leaf size=90 \[ \frac {a^3 (A b-a B)}{b^5 (a+b x)}+\frac {a^2 (3 A b-4 a B) \log (a+b x)}{b^5}-\frac {a x (2 A b-3 a B)}{b^4}+\frac {x^2 (A b-2 a B)}{2 b^3}+\frac {B x^3}{3 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 90, 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} \[ \frac {a^3 (A b-a B)}{b^5 (a+b x)}+\frac {a^2 (3 A b-4 a B) \log (a+b x)}{b^5}+\frac {x^2 (A b-2 a B)}{2 b^3}-\frac {a x (2 A b-3 a B)}{b^4}+\frac {B x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{(a+b x)^2} \, dx &=\int \left (\frac {a (-2 A b+3 a B)}{b^4}+\frac {(A b-2 a B) x}{b^3}+\frac {B x^2}{b^2}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^2}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac {a (2 A b-3 a B) x}{b^4}+\frac {(A b-2 a B) x^2}{2 b^3}+\frac {B x^3}{3 b^2}+\frac {a^3 (A b-a B)}{b^5 (a+b x)}+\frac {a^2 (3 A b-4 a B) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 87, normalized size = 0.97 \[ \frac {\frac {6 a^3 (A b-a B)}{a+b x}+6 a^2 (3 A b-4 a B) \log (a+b x)+3 b^2 x^2 (A b-2 a B)+6 a b x (3 a B-2 A b)+2 b^3 B x^3}{6 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 140, normalized size = 1.56 \[ \frac {2 \, B b^{4} x^{4} - 6 \, B a^{4} + 6 \, A a^{3} b - {\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} x^{3} + 3 \, {\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{2} + 6 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x - 6 \, {\left (4 \, B a^{4} - 3 \, A a^{3} b + {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{6} x + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 144, normalized size = 1.60 \[ \frac {{\left (b x + a\right )}^{3} {\left (2 \, B - \frac {3 \, {\left (4 \, B a b - A b^{2}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}}{6 \, b^{5}} + \frac {{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5}} - \frac {\frac {B a^{4} b^{3}}{b x + a} - \frac {A a^{3} b^{4}}{b x + a}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 109, normalized size = 1.21 \[ \frac {B \,x^{3}}{3 b^{2}}+\frac {A \,x^{2}}{2 b^{2}}-\frac {B a \,x^{2}}{b^{3}}+\frac {A \,a^{3}}{\left (b x +a \right ) b^{4}}+\frac {3 A \,a^{2} \ln \left (b x +a \right )}{b^{4}}-\frac {2 A a x}{b^{3}}-\frac {B \,a^{4}}{\left (b x +a \right ) b^{5}}-\frac {4 B \,a^{3} \ln \left (b x +a \right )}{b^{5}}+\frac {3 B \,a^{2} x}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 101, normalized size = 1.12 \[ -\frac {B a^{4} - A a^{3} b}{b^{6} x + a b^{5}} + \frac {2 \, B b^{2} x^{3} - 3 \, {\left (2 \, B a b - A b^{2}\right )} x^{2} + 6 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} x}{6 \, b^{4}} - \frac {{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 115, normalized size = 1.28 \[ x^2\,\left (\frac {A}{2\,b^2}-\frac {B\,a}{b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )-\frac {\ln \left (a+b\,x\right )\,\left (4\,B\,a^3-3\,A\,a^2\,b\right )}{b^5}+\frac {B\,x^3}{3\,b^2}-\frac {B\,a^4-A\,a^3\,b}{b\,\left (x\,b^5+a\,b^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 92, normalized size = 1.02 \[ \frac {B x^{3}}{3 b^{2}} - \frac {a^{2} \left (- 3 A b + 4 B a\right ) \log {\left (a + b x \right )}}{b^{5}} + x^{2} \left (\frac {A}{2 b^{2}} - \frac {B a}{b^{3}}\right ) + x \left (- \frac {2 A a}{b^{3}} + \frac {3 B a^{2}}{b^{4}}\right ) + \frac {A a^{3} b - B a^{4}}{a b^{5} + b^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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