Optimal. Leaf size=97 \[ \frac {a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac {a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac {3 a (A b-2 a B)}{b^5 (a+b x)}+\frac {(A b-4 a B) \log (a+b x)}{b^5}+\frac {B x}{b^4} \]
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Rubi [A] time = 0.09, antiderivative size = 97, 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, 77} \begin {gather*} \frac {a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac {a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac {3 a (A b-2 a B)}{b^5 (a+b x)}+\frac {(A b-4 a B) \log (a+b x)}{b^5}+\frac {B x}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {x^3 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac {B}{b^4}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^4}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^3}+\frac {3 a (-A b+2 a B)}{b^4 (a+b x)^2}+\frac {A b-4 a B}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {B x}{b^4}+\frac {a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac {a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac {3 a (A b-2 a B)}{b^5 (a+b x)}+\frac {(A b-4 a B) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 97, normalized size = 1.00 \begin {gather*} \frac {-26 a^4 B+a^3 b (11 A-54 B x)+9 a^2 b^2 x (3 A-2 B x)+18 a b^3 x^2 (A+B x)+6 (a+b x)^3 (A b-4 a B) \log (a+b x)+6 b^4 B x^4}{6 b^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 193, normalized size = 1.99 \begin {gather*} \frac {6 \, B b^{4} x^{4} + 18 \, B a b^{3} x^{3} - 26 \, B a^{4} + 11 \, A a^{3} b - 18 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 27 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x - 6 \, {\left (4 \, B a^{4} - A a^{3} b + {\left (4 \, B a b^{3} - A b^{4}\right )} x^{3} + 3 \, {\left (4 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \, {\left (4 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 96, normalized size = 0.99 \begin {gather*} \frac {B x}{b^{4}} - \frac {{\left (4 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {26 \, B a^{4} - 11 \, A a^{3} b + 18 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \, {\left (20 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.30 \begin {gather*} \frac {A \,a^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {B \,a^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {3 A \,a^{2}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {2 B \,a^{3}}{\left (b x +a \right )^{2} b^{5}}+\frac {3 A a}{\left (b x +a \right ) b^{4}}+\frac {A \ln \left (b x +a \right )}{b^{4}}-\frac {6 B \,a^{2}}{\left (b x +a \right ) b^{5}}-\frac {4 B a \ln \left (b x +a \right )}{b^{5}}+\frac {B x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 120, normalized size = 1.24 \begin {gather*} -\frac {26 \, B a^{4} - 11 \, A a^{3} b + 18 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \, {\left (20 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x}{6 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {B x}{b^{4}} - \frac {{\left (4 \, B a - A b\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 118, normalized size = 1.22 \begin {gather*} \frac {B\,x}{b^4}-\frac {x\,\left (10\,B\,a^3-\frac {9\,A\,a^2\,b}{2}\right )-x^2\,\left (3\,A\,a\,b^2-6\,B\,a^2\,b\right )+\frac {26\,B\,a^4-11\,A\,a^3\,b}{6\,b}}{a^3\,b^4+3\,a^2\,b^5\,x+3\,a\,b^6\,x^2+b^7\,x^3}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-4\,B\,a\right )}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 119, normalized size = 1.23 \begin {gather*} \frac {B x}{b^{4}} + \frac {11 A a^{3} b - 26 B a^{4} + x^{2} \left (18 A a b^{3} - 36 B a^{2} b^{2}\right ) + x \left (27 A a^{2} b^{2} - 60 B a^{3} b\right )}{6 a^{3} b^{5} + 18 a^{2} b^{6} x + 18 a b^{7} x^{2} + 6 b^{8} x^{3}} - \frac {\left (- A b + 4 B a\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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