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 = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 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 24
Rule 45
Rubi steps
\begin {align*} \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx &=\frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^4} \, dx}{b^2}\\ &=\frac {\int \left (\frac {b (b c-a d)}{(a+b x)^4}+\frac {b d}{(a+b x)^3}\right ) \, dx}{b^2}\\ &=-\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 = 0.45, size = 35, normalized size = 0.92
method | result | size |
gosper | \(-\frac {3 x b d +a d +2 b c}{6 b^{2} \left (b x +a \right )^{3}}\) | \(26\) |
risch | \(\frac {-\frac {d x}{2 b}-\frac {a d +2 b c}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(30\) |
default | \(-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}-\frac {-a d +b c}{3 b^{2} \left (b x +a \right )^{3}}\) | \(35\) |
norman | \(\frac {\frac {a \left (-a \,b^{2} d -2 b^{3} c \right )}{6 b^{4}}-\frac {d \,x^{2}}{2}+\frac {\left (-2 a \,b^{2} d -b^{3} c \right ) x}{3 b^{3}}}{\left (b x +a \right )^{4}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, 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 = 3.27, 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 = 0.22, 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 = 0.90, size = 41, normalized size = 1.08 \begin {gather*} -\frac {c}{3 \, {\left (b x + a\right )}^{3} b} - \frac {d}{2 \, {\left (b x + a\right )}^{2} b^{2}} + \frac {a d}{3 \, {\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 = 0.55, 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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