Optimal. Leaf size=101 \[ \frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}-\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}-\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 246,
218, 212, 209} \begin {gather*} -\frac {3 a^2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}-\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}+\frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^4 \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}+\frac {1}{8} (3 a) \int \frac {x^4}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}-\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{32 b}\\ &=\frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{32 b}\\ &=\frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{64 b}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{64 b}\\ &=\frac {3 a x \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{8} x^5 \left (a+b x^4\right )^{3/4}-\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}-\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 87, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (3 a+4 b x^4\right )-3 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-3 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{4} \left (b \,x^{4}+a \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 148, normalized size = 1.47 \begin {gather*} \frac {3 \, a^{2} {\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 )}}{128 \, b} + \frac {\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{2} b}{x^{3}} + \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{x^{7}}}{32 \, {\left (b^{3} - \frac {2 \, {\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2} b}{x^{8}}\right )}} \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. 218 vs.
\(2 (77) = 154\).
time = 0.39, size = 218, normalized size = 2.16 \begin {gather*} -\frac {12 \, \left (\frac {a^{8}}{b^{5}}\right )^{\frac {1}{4}} b \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {a^{8}}{b^{5}}\right )^{\frac {1}{4}} a^{6} b - \left (\frac {a^{8}}{b^{5}}\right )^{\frac {1}{4}} b x \sqrt {\frac {\sqrt {\frac {a^{8}}{b^{5}}} a^{8} b^{3} x^{2} + \sqrt {b x^{4} + a} a^{12}}{x^{2}}}}{a^{8} x}\right ) + 3 \, \left (\frac {a^{8}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {27 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{6} + \left (\frac {a^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4} x\right )}}{x}\right ) - 3 \, \left (\frac {a^{8}}{b^{5}}\right )^{\frac {1}{4}} b \log \left (\frac {27 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{6} - \left (\frac {a^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4} x\right )}}{x}\right ) - 4 \, {\left (4 \, b x^{5} + 3 \, a x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, 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 = 1.14, size = 39, normalized size = 0.39 \begin {gather*} \frac {a^{\frac {3}{4}} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \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 x^4\,{\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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