Optimal. Leaf size=76 \[ -\frac {1}{a x \sqrt [4]{a+b x^2}}-\frac {3 \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {292, 203, 202}
\begin {gather*} -\frac {3 \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^2}}-\frac {1}{a x \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 203
Rule 292
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {1}{a x \sqrt [4]{a+b x^2}}-\frac {(3 b) \int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx}{2 a}\\ &=-\frac {1}{a x \sqrt [4]{a+b x^2}}-\frac {\left (3 b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{2 a^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {1}{a x \sqrt [4]{a+b x^2}}-\frac {3 \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.41, size = 52, normalized size = 0.68 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{2},\frac {5}{4};\frac {1}{2};-\frac {b x^2}{a}\right )}{a x \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 0.57, size = 27, normalized size = 0.36 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {5}{4}} x} \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 = 5.16, size = 40, normalized size = 0.53 \begin {gather*} -\frac {2\,{\left (\frac {a}{b\,x^2}+1\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{4},\frac {7}{4};\ \frac {11}{4};\ -\frac {a}{b\,x^2}\right )}{7\,x\,{\left (b\,x^2+a\right )}^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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