Optimal. Leaf size=69 \[ \frac {(A b-2 a B) x}{b^3}+\frac {B x^2}{2 b^2}-\frac {a^2 (A b-a B)}{b^4 (a+b x)}-\frac {a (2 A b-3 a B) \log (a+b x)}{b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 69, 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, 78}
\begin {gather*} -\frac {a^2 (A b-a B)}{b^4 (a+b x)}-\frac {a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac {x (A b-2 a B)}{b^3}+\frac {B x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {x^2 (A+B x)}{(a+b x)^2} \, dx\\ &=\int \left (\frac {A b-2 a B}{b^3}+\frac {B x}{b^2}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^2}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {(A b-2 a B) x}{b^3}+\frac {B x^2}{2 b^2}-\frac {a^2 (A b-a B)}{b^4 (a+b x)}-\frac {a (2 A b-3 a B) \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 66, normalized size = 0.96 \begin {gather*} \frac {2 b (A b-2 a B) x+b^2 B x^2+\frac {2 a^2 (-A b+a B)}{a+b x}+2 a (-2 A b+3 a B) \log (a+b x)}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 67, normalized size = 0.97
method | result | size |
default | \(\frac {\frac {1}{2} b B \,x^{2}+A b x -2 B a x}{b^{3}}-\frac {a^{2} \left (A b -B a \right )}{b^{4} \left (b x +a \right )}-\frac {a \left (2 A b -3 B a \right ) \ln \left (b x +a \right )}{b^{4}}\) | \(67\) |
norman | \(\frac {\frac {B \,x^{3}}{2 b}-\frac {a \left (2 a b A -3 a^{2} B \right )}{b^{4}}+\frac {\left (2 A b -3 B a \right ) x^{2}}{2 b^{2}}}{b x +a}-\frac {a \left (2 A b -3 B a \right ) \ln \left (b x +a \right )}{b^{4}}\) | \(76\) |
risch | \(\frac {B \,x^{2}}{2 b^{2}}+\frac {A x}{b^{2}}-\frac {2 B a x}{b^{3}}-\frac {a^{2} A}{b^{3} \left (b x +a \right )}+\frac {a^{3} B}{b^{4} \left (b x +a \right )}-\frac {2 a \ln \left (b x +a \right ) A}{b^{3}}+\frac {3 a^{2} \ln \left (b x +a \right ) B}{b^{4}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 74, normalized size = 1.07 \begin {gather*} \frac {B a^{3} - A a^{2} b}{b^{5} x + a b^{4}} + \frac {B b x^{2} - 2 \, {\left (2 \, B a - A b\right )} x}{2 \, b^{3}} + \frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.35, size = 113, normalized size = 1.64 \begin {gather*} \frac {B b^{3} x^{3} + 2 \, B a^{3} - 2 \, A a^{2} b - {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} - 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x + 2 \, {\left (3 \, B a^{3} - 2 \, A a^{2} b + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 68, normalized size = 0.99 \begin {gather*} \frac {B x^{2}}{2 b^{2}} + \frac {a \left (- 2 A b + 3 B a\right ) \log {\left (a + b x \right )}}{b^{4}} + x \left (\frac {A}{b^{2}} - \frac {2 B a}{b^{3}}\right ) + \frac {- A a^{2} b + B a^{3}}{a b^{4} + b^{5} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 75, normalized size = 1.09 \begin {gather*} \frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {B b^{2} x^{2} - 4 \, B a b x + 2 \, A b^{2} x}{2 \, b^{4}} + \frac {B a^{3} - A a^{2} b}{{\left (b x + a\right )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 77, normalized size = 1.12 \begin {gather*} x\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )+\frac {B\,x^2}{2\,b^2}+\frac {B\,a^3-A\,a^2\,b}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {\ln \left (a+b\,x\right )\,\left (3\,B\,a^2-2\,A\,a\,b\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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