Optimal. Leaf size=39 \[ \frac {2 x}{3 a^2 \sqrt {a+c x^2}}+\frac {x}{3 a \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, 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 = {192, 191} \[ \frac {2 x}{3 a^2 \sqrt {a+c x^2}}+\frac {x}{3 a \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \frac {1}{\left (a+c x^2\right )^{5/2}} \, dx &=\frac {x}{3 a \left (a+c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac {x}{3 a \left (a+c x^2\right )^{3/2}}+\frac {2 x}{3 a^2 \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.74 \[ \frac {x \left (3 a+2 c x^2\right )}{3 a^2 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 47, normalized size = 1.21 \[ \frac {{\left (2 \, c x^{3} + 3 \, a x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 27, normalized size = 0.69 \[ \frac {x {\left (\frac {2 \, c x^{2}}{a^{2}} + \frac {3}{a}\right )}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.67 \[ \frac {\left (2 c \,x^{2}+3 a \right ) x}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 31, normalized size = 0.79 \[ \frac {2 \, x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 28, normalized size = 0.72 \[ \frac {2\,x\,\left (c\,x^2+a\right )+a\,x}{3\,a^2\,{\left (c\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.86, size = 95, normalized size = 2.44 \[ \frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {2 c x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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