Optimal. Leaf size=78 \[ \frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 246, 218,
212, 209} \begin {gather*} -\frac {a \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}+\frac {x \left (a+b x^4\right )^{3/4}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 327
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt [4]{a+b x^4}} \, dx &=\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{4 b}\\ &=\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \text {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 b}\\ &=\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b}-\frac {a \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 b}\\ &=\frac {x \left (a+b x^4\right )^{3/4}}{4 b}-\frac {a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 73, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4}-a \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-a \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 108, normalized size = 1.38 \begin {gather*} \frac {a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {1}{4}}}\right )}}{16 \, b} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{4 \, {\left (b^{2} - \frac {{\left (b x^{4} + a\right )} b}{x^{4}}\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (58) = 116\).
time = 0.40, size = 206, normalized size = 2.64 \begin {gather*} -\frac {4 \, b \left (\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3} b \left (\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} - b x \left (\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \sqrt {\frac {a^{4} b^{3} x^{2} \sqrt {\frac {a^{4}}{b^{5}}} + \sqrt {b x^{4} + a} a^{6}}{x^{2}}}}{a^{4} x}\right ) + b \left (\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (\frac {b^{4} x \left (\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) - b \left (\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{4} x \left (\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}}{x}\right ) - 4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} x}{16 \, b} \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 = 0.62, size = 37, normalized size = 0.47 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {9}{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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (b\,x^4+a\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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