Optimal. Leaf size=102 \[ -\frac {1}{3 a x^3 \sqrt [4]{a+b x^2}}+\frac {7 b}{6 a^2 x \sqrt [4]{a+b x^2}}+\frac {7 b^{3/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{5/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {7 b^{3/2} \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 )}{2 a^{5/2} \sqrt [4]{a+b x^2}}+\frac {7 b}{6 a^2 x \sqrt [4]{a+b x^2}}-\frac {1}{3 a x^3 \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^4 \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {1}{3 a x^3 \sqrt [4]{a+b x^2}}-\frac {(7 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{5/4}} \, dx}{6 a}\\ &=-\frac {1}{3 a x^3 \sqrt [4]{a+b x^2}}+\frac {7 b}{6 a^2 x \sqrt [4]{a+b x^2}}+\frac {\left (7 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx}{4 a^2}\\ &=-\frac {1}{3 a x^3 \sqrt [4]{a+b x^2}}+\frac {7 b}{6 a^2 x \sqrt [4]{a+b x^2}}+\frac {\left (7 b^2 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{4 a^3 \sqrt [4]{a+b x^2}}\\ &=-\frac {1}{3 a x^3 \sqrt [4]{a+b x^2}}+\frac {7 b}{6 a^2 x \sqrt [4]{a+b x^2}}+\frac {7 b^{3/2} \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 )}{2 a^{5/2} \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 = 10.01, size = 54, normalized size = 0.53 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {3}{2},\frac {5}{4};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 a x^3 \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 \[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
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 = 0.66, size = 32, normalized size = 0.31 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{4}} x^{3}} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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