Optimal. Leaf size=100 \[ \frac {5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}-\frac {5 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {285, 338, 304,
209, 212} \begin {gather*} -\frac {5 a^2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac {5}{32} a x^3 \sqrt [4]{a+b x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 338
Rubi steps
\begin {align*} \int x^2 \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac {1}{8} (5 a) \int x^2 \sqrt [4]{a+b x^4} \, dx\\ &=\frac {5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac {1}{32} \left (5 a^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac {5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac {1}{32} \left (5 a^2\right ) \text {Subst}\left (\int \frac {x^2}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac {5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}+\frac {\left (5 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 \sqrt {b}}-\frac {\left (5 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 \sqrt {b}}\\ &=\frac {5}{32} a x^3 \sqrt [4]{a+b x^4}+\frac {1}{8} x^3 \left (a+b x^4\right )^{5/4}-\frac {5 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 91, normalized size = 0.91 \begin {gather*} \frac {1}{32} x^3 \sqrt [4]{a+b x^4} \left (9 a+4 b x^4\right )-\frac {5 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}}+\frac {5 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (b \,x^{4}+a \right )^{\frac {5}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 144, normalized size = 1.44 \begin {gather*} \frac {5 \, a^{2} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{64 \, b^{\frac {3}{4}}} - \frac {5 \, a^{2} \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 )}{128 \, b^{\frac {3}{4}}} - \frac {\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b}{x} - \frac {9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{x^{5}}}{32 \, {\left (b^{2} - \frac {2 \, {\left (b x^{4} + a\right )} b}{x^{4}} + \frac {{\left (b x^{4} + a\right )}^{2}}{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. 209 vs.
\(2 (76) = 152\).
time = 0.39, size = 209, normalized size = 2.09 \begin {gather*} \frac {1}{32} \, {\left (4 \, b x^{7} + 9 \, a x^{3}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}} - \frac {5}{32} \, \left (\frac {a^{8}}{b^{3}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (\frac {a^{8}}{b^{3}}\right )^{\frac {3}{4}} {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b^{2} - \left (\frac {a^{8}}{b^{3}}\right )^{\frac {3}{4}} b^{2} x \sqrt {\frac {\sqrt {b x^{4} + a} a^{4} + \sqrt {\frac {a^{8}}{b^{3}}} b^{2} x^{2}}{x^{2}}}}{a^{8} x}\right ) + \frac {5}{128} \, \left (\frac {a^{8}}{b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {5 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} + \left (\frac {a^{8}}{b^{3}}\right )^{\frac {1}{4}} b x\right )}}{x}\right ) - \frac {5}{128} \, \left (\frac {a^{8}}{b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {5 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} - \left (\frac {a^{8}}{b^{3}}\right )^{\frac {1}{4}} b x\right )}}{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 = 1.00, size = 39, normalized size = 0.39 \begin {gather*} \frac {a^{\frac {5}{4}} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{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^2\,{\left (b\,x^4+a\right )}^{5/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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