Optimal. Leaf size=123 \[ -\frac {x^9}{b \sqrt [4]{a+b x^4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac {45 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac {45 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {294, 327, 246,
218, 212, 209} \begin {gather*} \frac {45 a^2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac {45 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac {x^9}{b \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (a+b x^4\right )^{5/4}} \, dx &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}+\frac {9 \int \frac {x^8}{\sqrt [4]{a+b x^4}} \, dx}{b}\\ &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac {(45 a) \int \frac {x^4}{\sqrt [4]{a+b x^4}} \, dx}{8 b^2}\\ &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac {\left (45 a^2\right ) \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{32 b^3}\\ &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac {\left (45 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^3}\\ &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac {\left (45 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^3}+\frac {\left (45 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^3}\\ &=-\frac {x^9}{b \sqrt [4]{a+b x^4}}-\frac {45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac {9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac {45 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac {45 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 98, normalized size = 0.80 \begin {gather*} \frac {\frac {2 \sqrt [4]{b} x \left (-45 a^2-9 a b x^4+4 b^2 x^8\right )}{\sqrt [4]{a+b x^4}}+45 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+45 a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{12}}{\left (b \,x^{4}+a \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 172, normalized size = 1.40 \begin {gather*} -\frac {32 \, a^{2} b^{2} - \frac {81 \, {\left (b x^{4} + a\right )} a^{2} b}{x^{4}} + \frac {45 \, {\left (b x^{4} + a\right )}^{2} a^{2}}{x^{8}}}{32 \, {\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{5}}{x} - \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{4}}{x^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{3}}{x^{9}}\right )}} - \frac {45 \, 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^{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. 281 vs.
\(2 (97) = 194\).
time = 0.42, size = 281, normalized size = 2.28 \begin {gather*} \frac {180 \, {\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac {a^{8}}{b^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{6} b^{3} \left (\frac {a^{8}}{b^{13}}\right )^{\frac {1}{4}} - b^{3} x \sqrt {\frac {a^{8} b^{7} x^{2} \sqrt {\frac {a^{8}}{b^{13}}} + \sqrt {b x^{4} + a} a^{12}}{x^{2}}} \left (\frac {a^{8}}{b^{13}}\right )^{\frac {1}{4}}}{a^{8} x}\right ) + 45 \, {\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac {a^{8}}{b^{13}}\right )^{\frac {1}{4}} \log \left (\frac {91125 \, {\left (b^{10} x \left (\frac {a^{8}}{b^{13}}\right )^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) - 45 \, {\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac {a^{8}}{b^{13}}\right )^{\frac {1}{4}} \log \left (-\frac {91125 \, {\left (b^{10} x \left (\frac {a^{8}}{b^{13}}\right )^{\frac {3}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) + 4 \, {\left (4 \, b^{2} x^{9} - 9 \, a b x^{5} - 45 \, a^{2} x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, {\left (b^{4} x^{4} + a b^{3}\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 = 2.45, size = 37, normalized size = 0.30 \begin {gather*} \frac {x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{4}} \Gamma \left (\frac {17}{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^{12}}{{\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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