Optimal. Leaf size=116 \[ -\frac {a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac {a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac {2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}-\frac {3 a x (A b-2 a B)}{b^5}+\frac {x^2 (A b-3 a B)}{2 b^4}+\frac {B x^3}{3 b^3} \]
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Rubi [A] time = 0.11, antiderivative size = 116, 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} \[ -\frac {a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac {a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac {2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}+\frac {x^2 (A b-3 a B)}{2 b^4}-\frac {3 a x (A b-2 a B)}{b^5}+\frac {B x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{(a+b x)^3} \, dx &=\int \left (\frac {3 a (-A b+2 a B)}{b^5}+\frac {(A b-3 a B) x}{b^4}+\frac {B x^2}{b^3}-\frac {a^4 (-A b+a B)}{b^5 (a+b x)^3}+\frac {a^3 (-4 A b+5 a B)}{b^5 (a+b x)^2}-\frac {2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac {3 a (A b-2 a B) x}{b^5}+\frac {(A b-3 a B) x^2}{2 b^4}+\frac {B x^3}{3 b^3}-\frac {a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac {a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac {2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 108, normalized size = 0.93 \[ \frac {\frac {3 a^4 (a B-A b)}{(a+b x)^2}+\frac {6 a^3 (4 A b-5 a B)}{a+b x}-12 a^2 (5 a B-3 A b) \log (a+b x)+3 b^2 x^2 (A b-3 a B)+18 a b x (2 a B-A b)+2 b^3 B x^3}{6 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 197, normalized size = 1.70 \[ \frac {2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \, {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \, {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \, {\left (B a^{4} b + A a^{3} b^{2}\right )} x - 12 \, {\left (5 \, B a^{5} - 3 \, A a^{4} b + {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 125, normalized size = 1.08 \[ -\frac {2 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {9 \, B a^{5} - 7 \, A a^{4} b + 2 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, B b^{6} x^{3} - 9 \, B a b^{5} x^{2} + 3 \, A b^{6} x^{2} + 36 \, B a^{2} b^{4} x - 18 \, A a b^{5} x}{6 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 142, normalized size = 1.22 \[ \frac {B \,x^{3}}{3 b^{3}}-\frac {A \,a^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {A \,x^{2}}{2 b^{3}}+\frac {B \,a^{5}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {3 B a \,x^{2}}{2 b^{4}}+\frac {4 A \,a^{3}}{\left (b x +a \right ) b^{5}}+\frac {6 A \,a^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {3 A a x}{b^{4}}-\frac {5 B \,a^{4}}{\left (b x +a \right ) b^{6}}-\frac {10 B \,a^{3} \ln \left (b x +a \right )}{b^{6}}+\frac {6 B \,a^{2} x}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 133, normalized size = 1.15 \[ -\frac {9 \, B a^{5} - 7 \, A a^{4} b + 2 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, B b^{2} x^{3} - 3 \, {\left (3 \, B a b - A b^{2}\right )} x^{2} + 18 \, {\left (2 \, B a^{2} - A a b\right )} x}{6 \, b^{5}} - \frac {2 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 147, normalized size = 1.27 \[ x^2\,\left (\frac {A}{2\,b^3}-\frac {3\,B\,a}{2\,b^4}\right )-x\,\left (\frac {3\,a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b}+\frac {3\,B\,a^2}{b^5}\right )-\frac {x\,\left (5\,B\,a^4-4\,A\,a^3\,b\right )+\frac {9\,B\,a^5-7\,A\,a^4\,b}{2\,b}}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}-\frac {\ln \left (a+b\,x\right )\,\left (10\,B\,a^3-6\,A\,a^2\,b\right )}{b^6}+\frac {B\,x^3}{3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 136, normalized size = 1.17 \[ \frac {B x^{3}}{3 b^{3}} - \frac {2 a^{2} \left (- 3 A b + 5 B a\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{2} \left (\frac {A}{2 b^{3}} - \frac {3 B a}{2 b^{4}}\right ) + x \left (- \frac {3 A a}{b^{4}} + \frac {6 B a^{2}}{b^{5}}\right ) + \frac {7 A a^{4} b - 9 B a^{5} + x \left (8 A a^{3} b^{2} - 10 B a^{4} b\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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