Optimal. Leaf size=38 \[ -\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b x)^4} \, dx &=\int \left (\frac {b c-a d}{b (a+b x)^4}+\frac {d}{b (a+b x)^3}\right ) \, dx\\ &=-\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {2 b c+a d+3 b d x}{6 b^2 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 35, normalized size = 0.92
method | result | size |
gosper | \(-\frac {3 b d x +a d +2 b c}{6 b^{2} \left (b x +a \right )^{3}}\) | \(26\) |
risch | \(\frac {-\frac {d x}{2 b}-\frac {a d +2 b c}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(30\) |
norman | \(\frac {-\frac {d x}{2 b}+\frac {-a b d -2 b^{2} c}{6 b^{3}}}{\left (b x +a \right )^{3}}\) | \(34\) |
default | \(-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}-\frac {-a d +b c}{3 b^{2} \left (b x +a \right )^{3}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 50, normalized size = 1.32 \begin {gather*} -\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 50, normalized size = 1.32 \begin {gather*} -\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 53, normalized size = 1.39 \begin {gather*} \frac {- a d - 2 b c - 3 b d x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 25, normalized size = 0.66 \begin {gather*} -\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b x + a\right )}^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 52, normalized size = 1.37 \begin {gather*} -\frac {\frac {a\,d+2\,b\,c}{6\,b^2}+\frac {d\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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