Optimal. Leaf size=87 \[ -\frac {a^3 (A b-a B) \log (a+b x)}{b^5}+\frac {a^2 x (A b-a B)}{b^4}-\frac {a x^2 (A b-a B)}{2 b^3}+\frac {x^3 (A b-a B)}{3 b^2}+\frac {B x^4}{4 b} \]
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Rubi [A] time = 0.06, antiderivative size = 87, 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} \[ \frac {a^2 x (A b-a B)}{b^4}-\frac {a^3 (A b-a B) \log (a+b x)}{b^5}+\frac {x^3 (A b-a B)}{3 b^2}-\frac {a x^2 (A b-a B)}{2 b^3}+\frac {B x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{a+b x} \, dx &=\int \left (-\frac {a^2 (-A b+a B)}{b^4}+\frac {a (-A b+a B) x}{b^3}+\frac {(A b-a B) x^2}{b^2}+\frac {B x^3}{b}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^2}{2 b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^4}{4 b}-\frac {a^3 (A b-a B) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 80, normalized size = 0.92 \[ \frac {12 a^3 (a B-A b) \log (a+b x)+b x \left (-12 a^3 B+6 a^2 b (2 A+B x)-2 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )}{12 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 94, normalized size = 1.08 \[ \frac {3 \, B b^{4} x^{4} - 4 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 12 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x + 12 \, {\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 94, normalized size = 1.08 \[ \frac {3 \, B b^{3} x^{4} - 4 \, B a b^{2} x^{3} + 4 \, A b^{3} x^{3} + 6 \, B a^{2} b x^{2} - 6 \, A a b^{2} x^{2} - 12 \, B a^{3} x + 12 \, A a^{2} b x}{12 \, b^{4}} + \frac {{\left (B a^{4} - A a^{3} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 100, normalized size = 1.15 \[ \frac {B \,x^{4}}{4 b}+\frac {A \,x^{3}}{3 b}-\frac {B a \,x^{3}}{3 b^{2}}-\frac {A a \,x^{2}}{2 b^{2}}+\frac {B \,a^{2} x^{2}}{2 b^{3}}-\frac {A \,a^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {A \,a^{2} x}{b^{3}}+\frac {B \,a^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {B \,a^{3} x}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 92, normalized size = 1.06 \[ \frac {3 \, B b^{3} x^{4} - 4 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} + 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} - 12 \, {\left (B a^{3} - A a^{2} b\right )} x}{12 \, b^{4}} + \frac {{\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 94, normalized size = 1.08 \[ x^3\,\left (\frac {A}{3\,b}-\frac {B\,a}{3\,b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4-A\,a^3\,b\right )}{b^5}+\frac {B\,x^4}{4\,b}-\frac {a\,x^2\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 85, normalized size = 0.98 \[ \frac {B x^{4}}{4 b} + \frac {a^{3} \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{5}} + x^{3} \left (\frac {A}{3 b} - \frac {B a}{3 b^{2}}\right ) + x^{2} \left (- \frac {A a}{2 b^{2}} + \frac {B a^{2}}{2 b^{3}}\right ) + x \left (\frac {A a^{2}}{b^{3}} - \frac {B a^{3}}{b^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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