Optimal. Leaf size=38 \[ -\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \]
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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}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \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)^6} \, dx &=\frac {\int \frac {b^2 c+b^2 d x}{(a+b x)^5} \, dx}{b^2}\\ &=\frac {\int \left (\frac {b (b c-a d)}{(a+b x)^5}+\frac {b d}{(a+b x)^4}\right ) \, dx}{b^2}\\ &=-\frac {b c-a d}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {3 b c+a d+4 b d x}{12 b^2 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 35, normalized size = 0.92
method | result | size |
gosper | \(-\frac {4 x b d +a d +3 b c}{12 b^{2} \left (b x +a \right )^{4}}\) | \(26\) |
risch | \(\frac {-\frac {d x}{3 b}-\frac {a d +3 b c}{12 b^{2}}}{\left (b x +a \right )^{4}}\) | \(30\) |
default | \(-\frac {d}{3 b^{2} \left (b x +a \right )^{3}}-\frac {-a d +b c}{4 b^{2} \left (b x +a \right )^{4}}\) | \(35\) |
norman | \(\frac {\frac {a \left (-a d \,b^{3}-3 c \,b^{4}\right )}{12 b^{5}}-\frac {d \,x^{2}}{3}+\frac {\left (-5 a d \,b^{3}-3 c \,b^{4}\right ) x}{12 b^{4}}}{\left (b x +a \right )^{5}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.25, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (32) = 64\).
time = 0.27, size = 65, normalized size = 1.71 \begin {gather*} \frac {- a d - 3 b c - 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 25, normalized size = 0.66 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b x + a\right )}^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 63, normalized size = 1.66 \begin {gather*} -\frac {\frac {a\,d+3\,b\,c}{12\,b^2}+\frac {d\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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