Optimal. Leaf size=83 \[ \frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {205, 243, 342,
281, 237} \begin {gather*} \frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 237
Rule 243
Rule 281
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx &=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}+\frac {2 \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{3 a}\\ &=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}+\frac {\left (2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{3 a \left (a+b x^4\right )^{3/4}}\\ &=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\left (2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{3 a \left (a+b x^4\right )^{3/4}}\\ &=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\left (\left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{3 a \left (a+b x^4\right )^{3/4}}\\ &=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{3/2} \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 = 7.92, size = 56, normalized size = 0.67 \begin {gather*} \frac {x+2 x \left (1+\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{3 a \left (a+b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {7}{4}}}\, dx\]
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 = 32, normalized size = 0.39 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, 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 = 0.48, size = 36, normalized size = 0.43 \begin {gather*} \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \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 = 1.14, size = 37, normalized size = 0.45 \begin {gather*} \frac {x\,{\left (\frac {b\,x^4}{a}+1\right )}^{7/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {7}{4};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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