Optimal. Leaf size=19 \[ -\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, 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 = {643}
\begin {gather*} -\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int \frac {b d+2 c d x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d}{3 \left (a+b x+c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} -\frac {2 d}{3 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 16, normalized size = 0.84
method | result | size |
gosper | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
default | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
trager | \(-\frac {2 d}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2 \, d}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \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. 52 vs.
\(2 (15) = 30\).
time = 3.79, size = 52, normalized size = 2.74 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x + a} d}{3 \, {\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. 60 vs.
\(2 (19) = 38\).
time = 0.71, size = 60, normalized size = 3.16 \begin {gather*} - \frac {2 d}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 30, normalized size = 1.58 \begin {gather*} -\frac {2 \, d^{5}}{3 \, {\left (a d^{2} + {\left (c d x^{2} + b d x\right )} d\right )}^{\frac {3}{2}} {\left | d \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2\,d}{3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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