Optimal. Leaf size=126 \[ -\frac {2 a \sqrt {a+\frac {b}{x^4}}}{7 x}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{7 x}-\frac {2 a^{7/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}} \]
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Rubi [A]
time = 0.04, antiderivative size = 126, 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 = {342, 201, 226}
\begin {gather*} -\frac {2 a^{7/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{7 x}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{7 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 226
Rule 342
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^2} \, dx &=-\text {Subst}\left (\int \left (a+b x^4\right )^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{7 x}-\frac {1}{7} (6 a) \text {Subst}\left (\int \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{7 x}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{7 x}-\frac {1}{7} \left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{7 x}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{7 x}-\frac {2 a^{7/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{b} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 52, normalized size = 0.41 \begin {gather*} -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {7}{4},-\frac {3}{2};-\frac {3}{4};-\frac {a x^4}{b}\right )}{7 x^5 \sqrt {1+\frac {a x^4}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.06, size = 157, normalized size = 1.25
method | result | size |
risch | \(-\frac {\left (3 a \,x^{4}+b \right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{7 x^{5}}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{7 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \left (a \,x^{4}+b \right )}\) | \(117\) |
default | \(-\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (-4 a^{2} \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) x^{7}+3 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{8}+4 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a b \,x^{4}+\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{2}\right )}{7 x \left (a \,x^{4}+b \right )^{2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(157\) |
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 = 0.08, size = 58, normalized size = 0.46 \begin {gather*} -\frac {4 \, a \sqrt {b} x^{5} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{7 \, x^{5}} \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 = 0.60, size = 39, normalized size = 0.31 \begin {gather*} - \frac {a^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 x \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 = 1.57, size = 39, normalized size = 0.31 \begin {gather*} -\frac {{\left (a\,x^4+b\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ -\frac {b}{a\,x^4}\right )}{x\,{\left (\frac {b}{a}+x^4\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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