Optimal. Leaf size=26 \[ \frac {2 \sqrt {c x}}{a c \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {270}
\begin {gather*} \frac {2 \sqrt {c x}}{a c \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x} \left (a+b x^2\right )^{5/4}} \, dx &=\frac {2 \sqrt {c x}}{a c \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 24, normalized size = 0.92 \begin {gather*} \frac {2 x}{a \sqrt {c x} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 21, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 x}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a \sqrt {c x}}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.86, size = 31, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{a b c x^{2} + a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.52, size = 34, normalized size = 1.31 \begin {gather*} \frac {\Gamma \left (\frac {1}{4}\right )}{2 a \sqrt [4]{b} \sqrt {c} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.97, size = 29, normalized size = 1.12 \begin {gather*} \frac {2\,x\,{\left (b\,x^2+a\right )}^{3/4}}{\left (a^2+b\,a\,x^2\right )\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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