Optimal. Leaf size=28 \[ \frac {(a+b x)^2}{2 (b c-a d) (c+d x)^2} \]
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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 = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 37}
\begin {gather*} \frac {(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 640
Rubi steps
\begin {align*} \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac {a+b x}{(c+d x)^3} \, dx\\ &=\frac {(a+b x)^2}{2 (b c-a d) (c+d 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 d^2 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 35, normalized size = 1.25
method | result | size |
gosper | \(-\frac {2 x b d +a d +b c}{2 \left (d x +c \right )^{2} d^{2}}\) | \(25\) |
risch | \(\frac {-\frac {b x}{d}-\frac {a d +b c}{2 d^{2}}}{\left (d x +c \right )^{2}}\) | \(29\) |
default | \(-\frac {b}{d^{2} \left (d x +c \right )}-\frac {a d -b c}{2 d^{2} \left (d x +c \right )^{2}}\) | \(35\) |
norman | \(\frac {-\frac {b^{3} x^{3}}{d}-\frac {\left (a \,b^{2} d +b^{3} c \right ) a^{2}}{2 b^{2} d^{2}}+\frac {\left (-5 a d \,b^{4}-b^{5} c \right ) x^{2}}{2 b^{2} d^{2}}-\frac {a \left (2 a d \,b^{3}+c \,b^{4}\right ) x}{b^{2} d^{2}}}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}\) | \(99\) |
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 (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.49, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 39, normalized size = 1.39 \begin {gather*} \frac {- a d - b c - 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 24, normalized size = 0.86 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (d x + c\right )}^{2} d^{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\,d+b\,c}{2\,d^2}+\frac {b\,x}{d}}{c^2+2\,c\,d\,x+d^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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