Optimal. Leaf size=92 \[ -\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}+\frac {\left (a x^4+b\right )^{3/4} \left (4 a x^5+3 b x\right )}{32 a} \]
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Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.10, 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 = {279, 321, 240, 212, 206, 203} \begin {gather*} -\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{64 a^{5/4}}+\frac {3 b x \left (a x^4+b\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (a x^4+b\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^4 \left (b+a x^4\right )^{3/4} \, dx &=\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}+\frac {1}{8} (3 b) \int \frac {x^4}{\sqrt [4]{b+a x^4}} \, dx\\ &=\frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx}{32 a}\\ &=\frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{32 a}\\ &=\frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{64 a}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{64 a}\\ &=\frac {3 b x \left (b+a x^4\right )^{3/4}}{32 a}+\frac {1}{8} x^5 \left (b+a x^4\right )^{3/4}-\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 62, normalized size = 0.67 \begin {gather*} \frac {x \left (a x^4+b\right )^{3/4} \left (-\frac {b \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};-\frac {a x^4}{b}\right )}{\left (\frac {a x^4}{b}+1\right )^{3/4}}+a x^4+b\right )}{8 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 92, normalized size = 1.00 \begin {gather*} \frac {\left (b+a x^4\right )^{3/4} \left (3 b x+4 a x^5\right )}{32 a}-\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{64 a^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 218, normalized size = 2.37 \begin {gather*} -\frac {12 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \arctan \left (-\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a b^{6} - \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a x \sqrt {\frac {\sqrt {\frac {b^{8}}{a^{5}}} a^{3} b^{8} x^{2} + \sqrt {a x^{4} + b} b^{12}}{x^{2}}}}{b^{8} x}\right ) + 3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} + \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) - 3 \, \left (\frac {b^{8}}{a^{5}}\right )^{\frac {1}{4}} a \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{6} - \left (\frac {b^{8}}{a^{5}}\right )^{\frac {3}{4}} a^{4} x\right )}}{x}\right ) - 4 \, {\left (4 \, a x^{5} + 3 \, b x\right )} {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{128 \, a} \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*} \int {\left (a x^{4} + b\right )}^{\frac {3}{4}} x^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int x^{4} \left (a \,x^{4}+b \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 148, normalized size = 1.61 \begin {gather*} \frac {3 \, b^{2} {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (-\frac {a^{\frac {1}{4}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}{a^{\frac {1}{4}} + \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )}}{128 \, a} + \frac {\frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} a b^{2}}{x^{3}} + \frac {3 \, {\left (a x^{4} + b\right )}^{\frac {7}{4}} b^{2}}{x^{7}}}{32 \, {\left (a^{3} - \frac {2 \, {\left (a x^{4} + b\right )} a^{2}}{x^{4}} + \frac {{\left (a x^{4} + b\right )}^{2} a}{x^{8}}\right )}} \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 (a\,x^4+b\right )}^{3/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.35, size = 39, normalized size = 0.42 \begin {gather*} \frac {b^{\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 {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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