Optimal. Leaf size=28 \[ -\frac {(A+B x)^2}{2 (A b-a B) (a+b x)^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {37}
\begin {gather*} -\frac {(A+B x)^2}{2 (a+b x)^2 (A b-a B)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^3} \, dx &=-\frac {(A+B x)^2}{2 (A b-a B) (a+b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} -\frac {A b+B (a+2 b x)}{2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 35, normalized size = 1.25
method | result | size |
gosper | \(-\frac {2 b B x +A b +B a}{2 b^{2} \left (b x +a \right )^{2}}\) | \(25\) |
norman | \(\frac {-\frac {B x}{b}-\frac {A b +B a}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
risch | \(\frac {-\frac {B x}{b}-\frac {A b +B a}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
default | \(-\frac {B}{b^{2} \left (b x +a \right )}-\frac {A b -B a}{2 b^{2} \left (b x +a \right )^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, B b x + B a + A b}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, B b x + B a + A b}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 39, normalized size = 1.39 \begin {gather*} \frac {- A b - B a - 2 B b x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 24, normalized size = 0.86 \begin {gather*} -\frac {2 \, B b x + B a + A b}{2 \, {\left (b x + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\frac {A\,b+B\,a}{2\,b^2}+\frac {B\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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