Optimal. Leaf size=125 \[ -\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 125, 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 = {285, 327, 246,
218, 212, 209} \begin {gather*} \frac {5 a^3 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}-\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 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^8 \left (a+b x^4\right )^{3/4} \, dx &=\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {1}{4} a \int \frac {x^8}{\sqrt [4]{a+b x^4}} \, dx\\ &=\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}-\frac {\left (5 a^2\right ) \int \frac {x^4}{\sqrt [4]{a+b x^4}} \, dx}{32 b}\\ &=-\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt [4]{a+b x^4}} \, dx}{128 b^2}\\ &=-\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{128 b^2}\\ &=-\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{256 b^2}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{256 b^2}\\ &=-\frac {5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac {a x^5 \left (a+b x^4\right )^{3/4}}{32 b}+\frac {1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac {5 a^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 98, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt [4]{b} x \left (a+b x^4\right )^{3/4} \left (-15 a^2+12 a b x^4+32 b^2 x^8\right )+15 a^3 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+15 a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{768 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{8} \left (b \,x^{4}+a \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 189, normalized size = 1.51 \begin {gather*} -\frac {5 \, a^{3} {\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 )}}{512 \, b^{2}} - \frac {\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{3} b^{2}}{x^{3}} + \frac {42 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3} b}{x^{7}} - \frac {15 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{3}}{x^{11}}}{384 \, {\left (b^{5} - \frac {3 \, {\left (b x^{4} + a\right )} b^{4}}{x^{4}} + \frac {3 \, {\left (b x^{4} + a\right )}^{2} b^{3}}{x^{8}} - \frac {{\left (b x^{4} + a\right )}^{3} b^{2}}{x^{12}}\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. 239 vs.
\(2 (97) = 194\).
time = 0.41, size = 239, normalized size = 1.91 \begin {gather*} \frac {60 \, \left (\frac {a^{12}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {a^{12}}{b^{9}}\right )^{\frac {1}{4}} a^{9} b^{2} - \left (\frac {a^{12}}{b^{9}}\right )^{\frac {1}{4}} b^{2} x \sqrt {\frac {\sqrt {\frac {a^{12}}{b^{9}}} a^{12} b^{5} x^{2} + \sqrt {b x^{4} + a} a^{18}}{x^{2}}}}{a^{12} x}\right ) + 15 \, \left (\frac {a^{12}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\frac {125 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{9} + \left (\frac {a^{12}}{b^{9}}\right )^{\frac {3}{4}} b^{7} x\right )}}{x}\right ) - 15 \, \left (\frac {a^{12}}{b^{9}}\right )^{\frac {1}{4}} b^{2} \log \left (\frac {125 \, {\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{9} - \left (\frac {a^{12}}{b^{9}}\right )^{\frac {3}{4}} b^{7} x\right )}}{x}\right ) + 4 \, {\left (32 \, b^{2} x^{9} + 12 \, a b x^{5} - 15 \, a^{2} x\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{1536 \, b^{2}} \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 = 3.69, size = 39, normalized size = 0.31 \begin {gather*} \frac {a^{\frac {3}{4}} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{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^8\,{\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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