Optimal. Leaf size=149 \[ \frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {45 a^4 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac {45 a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {45 a^4 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac {45 a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}+\frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b} \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^{12} \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac {1}{16} (3 a) \int \frac {x^{12}}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {\left (9 a^2\right ) \int \frac {x^8}{\sqrt [4]{a+b x^4}} \, dx}{64 b}\\ &=-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac {\left (45 a^3\right ) \int \frac {x^4}{\sqrt [4]{a+b x^4}} \, dx}{512 b^2}\\ &=\frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {\left (45 a^4\right ) \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{2048 b^3}\\ &=\frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {\left (45 a^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{2048 b^3}\\ &=\frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {\left (45 a^4\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^3}-\frac {\left (45 a^4\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^3}\\ &=\frac {45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac {9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac {a x^9 \left (a+b x^4\right )^{3/4}}{64 b}+\frac {1}{16} x^{13} \left (a+b x^4\right )^{3/4}-\frac {45 a^4 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac {45 a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 109, normalized size = 0.73 \begin {gather*} \frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (45 a^3-36 a^2 b x^4+32 a b^2 x^8+128 b^3 x^{12}\right )-45 a^4 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-45 a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{12} \left (b \,x^{4}+a \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 225, normalized size = 1.51 \begin {gather*} \frac {45 \, a^{4} {\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 )}}{8192 \, b^{3}} + \frac {\frac {15 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4} b^{3}}{x^{3}} + \frac {239 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{4} b^{2}}{x^{7}} - \frac {171 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{4} b}{x^{11}} + \frac {45 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a^{4}}{x^{15}}}{2048 \, {\left (b^{7} - \frac {4 \, {\left (b x^{4} + a\right )} b^{6}}{x^{4}} + \frac {6 \, {\left (b x^{4} + a\right )}^{2} b^{5}}{x^{8}} - \frac {4 \, {\left (b x^{4} + a\right )}^{3} b^{4}}{x^{12}} + \frac {{\left (b x^{4} + a\right )}^{4} b^{3}}{x^{16}}\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. 250 vs.
\(2 (117) = 234\).
time = 0.40, size = 250, normalized size = 1.68 \begin {gather*} -\frac {180 \, \left (\frac {a^{16}}{b^{13}}\right )^{\frac {1}{4}} b^{3} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {a^{16}}{b^{13}}\right )^{\frac {1}{4}} a^{12} b^{3} - \left (\frac {a^{16}}{b^{13}}\right )^{\frac {1}{4}} b^{3} x \sqrt {\frac {\sqrt {\frac {a^{16}}{b^{13}}} a^{16} b^{7} x^{2} + \sqrt {b x^{4} + a} a^{24}}{x^{2}}}}{a^{16} x}\right ) + 45 \, \left (\frac {a^{16}}{b^{13}}\right )^{\frac {1}{4}} b^{3} \log \left (\frac {91125 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{12} + \left (\frac {a^{16}}{b^{13}}\right )^{\frac {3}{4}} b^{10} x\right )}}{x}\right ) - 45 \, \left (\frac {a^{16}}{b^{13}}\right )^{\frac {1}{4}} b^{3} \log \left (\frac {91125 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{12} - \left (\frac {a^{16}}{b^{13}}\right )^{\frac {3}{4}} b^{10} x\right )}}{x}\right ) - 4 \, {\left (128 \, b^{3} x^{13} + 32 \, a b^{2} x^{9} - 36 \, a^{2} b x^{5} + 45 \, a^{3} x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{8192 \, b^{3}} \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 = 19.62, size = 39, normalized size = 0.26 \begin {gather*} \frac {a^{\frac {3}{4}} x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{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 x^{12}\,{\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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