Optimal. Leaf size=82 \[ -\frac {\sqrt [4]{a+b x^4}}{2 a x^2}-\frac {\sqrt {b} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {281, 331, 239,
237} \begin {gather*} -\frac {\sqrt {b} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 239
Rule 281
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^4\right )^{3/4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{a+b x^4}}{2 a x^2}-\frac {b \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{2 a x^2}-\frac {\left (b \left (1+\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{4 a \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{2 a x^2}-\frac {\sqrt {b} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \left (a+b x^4\right )^{3/4}}\\ \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 = 51, normalized size = 0.62 \begin {gather*} -\frac {\left (1+\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};-\frac {b x^4}{a}\right )}{2 x^2 \left (a+b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
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 = 0.07, size = 25, normalized size = 0.30 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b x^{7} + a x^{3}}, x\right ) \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 = 0.47, size = 31, normalized size = 0.38 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} x^{2}} \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^3\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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