Optimal. Leaf size=39 \[ \frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197}
\begin {gather*} \frac {2 x}{3 c^2 \sqrt {c+d x^2}}+\frac {x}{3 c \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rubi steps
\begin {align*} \int \frac {1}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {x}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x}{3 c^2 \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 29, normalized size = 0.74 \begin {gather*} \frac {3 c x+2 d x^3}{3 c^2 \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 32, normalized size = 0.82
method | result | size |
gosper | \(\frac {x \left (2 d \,x^{2}+3 c \right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} c^{2}}\) | \(26\) |
trager | \(\frac {x \left (2 d \,x^{2}+3 c \right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} c^{2}}\) | \(26\) |
default | \(\frac {x}{3 c \left (d \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {d \,x^{2}+c}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 31, normalized size = 0.79 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {d x^{2} + c} c^{2}} + \frac {x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.10, size = 47, normalized size = 1.21 \begin {gather*} \frac {{\left (2 \, d x^{3} + 3 \, c x\right )} \sqrt {d x^{2} + c}}{3 \, {\left (c^{2} d^{2} x^{4} + 2 \, c^{3} d x^{2} + c^{4}\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. 95 vs.
\(2 (32) = 64\).
time = 0.42, size = 95, normalized size = 2.44 \begin {gather*} \frac {3 c x}{3 c^{\frac {7}{2}} \sqrt {1 + \frac {d x^{2}}{c}} + 3 c^{\frac {5}{2}} d x^{2} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {2 d x^{3}}{3 c^{\frac {7}{2}} \sqrt {1 + \frac {d x^{2}}{c}} + 3 c^{\frac {5}{2}} d x^{2} \sqrt {1 + \frac {d x^{2}}{c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 27, normalized size = 0.69 \begin {gather*} \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 28, normalized size = 0.72 \begin {gather*} \frac {2\,x\,\left (d\,x^2+c\right )+c\,x}{3\,c^2\,{\left (d\,x^2+c\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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