Optimal. Leaf size=28 \[ -\frac {(c+d x)^2}{2 (b c-a d) (a+b x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 37}
\begin {gather*} -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 37
Rubi steps
\begin {align*} \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx &=\frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^3} \, dx}{b^2}\\ &=-\frac {(c+d x)^2}{2 (b c-a d) (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 d+b (c+2 d x)}{2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 35, normalized size = 1.25
method | result | size |
gosper | \(-\frac {2 x b d +a d +b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(25\) |
risch | \(\frac {-\frac {d x}{b}-\frac {a d +b c}{2 b^{2}}}{\left (b x +a \right )^{2}}\) | \(29\) |
default | \(-\frac {d}{b^{2} \left (b x +a \right )}-\frac {-a d +b c}{2 b^{2} \left (b x +a \right )^{2}}\) | \(35\) |
norman | \(\frac {-d \,x^{2}+\frac {a \left (-a b d -b^{2} c \right )}{2 b^{3}}+\frac {\left (-3 a b d -b^{2} c \right ) x}{2 b^{2}}}{\left (b x +a \right )^{3}}\) | \(52\) |
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 d x + b c + a d}{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 = 2.72, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{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 = 0.16, size = 39, normalized size = 1.39 \begin {gather*} \frac {- a d - b c - 2 b d 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 = 0.75, size = 24, normalized size = 0.86 \begin {gather*} -\frac {2 \, b d x + b c + a d}{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 = 0.55, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,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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