Optimal. Leaf size=37 \[ -\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {696}
\begin {gather*} -\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 696
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.03 \begin {gather*} -\frac {d^3 \left (b^2+8 b c x+4 c \left (a+2 c x^2\right )\right )}{2 (a+x (b+c x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 40, normalized size = 1.08
method | result | size |
gosper | \(-\frac {d^{3} \left (8 c^{2} x^{2}+8 b c x +4 a c +b^{2}\right )}{2 \left (c \,x^{2}+b x +a \right )^{2}}\) | \(39\) |
default | \(\frac {d^{3} \left (-4 c^{2} x^{2}-4 b c x -2 a c -\frac {1}{2} b^{2}\right )}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(40\) |
risch | \(\frac {-4 c^{2} d^{3} x^{2}-4 b c \,d^{3} x -\frac {d^{3} \left (4 a c +b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(47\) |
norman | \(\frac {-4 c^{2} d^{3} x^{2}+\frac {-4 a \,c^{3} d^{3}-b^{2} c^{2} d^{3}}{2 c^{2}}-4 b c \,d^{3} x}{\left (c \,x^{2}+b x +a \right )^{2}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (35) = 70\).
time = 0.30, size = 71, normalized size = 1.92 \begin {gather*} -\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (35) = 70\).
time = 1.35, size = 71, normalized size = 1.92 \begin {gather*} -\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{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. 80 vs.
\(2 (34) = 68\).
time = 0.84, size = 80, normalized size = 2.16 \begin {gather*} \frac {- 4 a c d^{3} - b^{2} d^{3} - 8 b c d^{3} x - 8 c^{2} d^{3} x^{2}}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 50, normalized size = 1.35 \begin {gather*} -\frac {b^{2} d^{5} + 4 \, a c d^{5} + 8 \, {\left (c d x^{2} + b d x\right )} c d^{4}}{2 \, {\left (c d x^{2} + b d x + a d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 74, normalized size = 2.00 \begin {gather*} -\frac {\frac {b^2\,d^3}{2}+4\,b\,c\,d^3\,x+4\,c^2\,d^3\,x^2+2\,a\,c\,d^3}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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