Optimal. Leaf size=75 \[ \frac {1}{4} x \left (a x^4+b\right )^{3/4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 \sqrt [4]{a}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 \sqrt [4]{a}} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {195, 240, 212, 206, 203} \begin {gather*} \frac {1}{4} x \left (a x^4+b\right )^{3/4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 \sqrt [4]{a}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{8 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 206
Rule 212
Rule 240
Rubi steps
\begin {align*} \int \left (b+a x^4\right )^{3/4} \, dx &=\frac {1}{4} x \left (b+a x^4\right )^{3/4}+\frac {1}{4} (3 b) \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx\\ &=\frac {1}{4} x \left (b+a x^4\right )^{3/4}+\frac {1}{4} (3 b) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {1}{4} x \left (b+a x^4\right )^{3/4}+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {1}{4} x \left (b+a x^4\right )^{3/4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt [4]{a}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt [4]{a}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 0.61 \begin {gather*} \frac {x \left (a x^4+b\right )^{3/4} \, _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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{4} x \left (b+a x^4\right )^{3/4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt [4]{a}}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{8 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 192, normalized size = 2.56 \begin {gather*} \frac {1}{4} \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} x + \frac {3}{4} \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} b^{3} - \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} x \sqrt {\frac {\sqrt {\frac {b^{4}}{a}} a b^{4} x^{2} + \sqrt {a x^{4} + b} b^{6}}{x^{2}}}}{b^{4} x}\right ) + \frac {3}{16} \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{3} + \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\right ) - \frac {3}{16} \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} \log \left (\frac {27 \, {\left ({\left (a x^{4} + b\right )}^{\frac {1}{4}} b^{3} - \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a x\right )}}{x}\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*} \int {\left (a x^{4} + b\right )}^{\frac {3}{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 \[\int \left (a \,x^{4}+b \right )^{\frac {3}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 102, normalized size = 1.36 \begin {gather*} -\frac {3}{16} \, b {\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 )} - \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}} b}{4 \, {\left (a - \frac {a x^{4} + b}{x^{4}}\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 37, normalized size = 0.49 \begin {gather*} \frac {x\,{\left (a\,x^4+b\right )}^{3/4}\,{{}}_2{\mathrm {F}}_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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.18, size = 37, normalized size = 0.49 \begin {gather*} \frac {b^{\frac {3}{4}} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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