Optimal. Leaf size=86 \[ \frac {\sqrt [4]{a x^4+b} \left (a x^4+4 b\right )}{4 x}+\frac {3}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {3}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {453, 279, 331, 298, 203, 206} \begin {gather*} \frac {\left (a x^4+b\right )^{5/4}}{x}+\frac {3}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {3}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {3}{4} a x^3 \sqrt [4]{a x^4+b} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 331
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^4\right ) \sqrt [4]{b+a x^4}}{x^2} \, dx &=\frac {\left (b+a x^4\right )^{5/4}}{x}-(3 a) \int x^2 \sqrt [4]{b+a x^4} \, dx\\ &=-\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{4} (3 a b) \int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx\\ &=-\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{4} (3 a b) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}-\frac {1}{8} \left (3 \sqrt {a} b\right ) \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} \left (3 \sqrt {a} b\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {3}{4} a x^3 \sqrt [4]{b+a x^4}+\frac {\left (b+a x^4\right )^{5/4}}{x}+\frac {3}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\frac {3}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 61, normalized size = 0.71 \begin {gather*} \frac {\sqrt [4]{a x^4+b} \left (-\frac {a x^4 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {a x^4}{b}\right )}{\sqrt [4]{\frac {a x^4}{b}+1}}+a x^4+b\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 86, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b+a x^4} \left (4 b+a x^4\right )}{4 x}+\frac {3}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )-\frac {3}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}{x^{2}}\,{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 \[\int \frac {\left (a \,x^{4}-b \right ) \left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 190, normalized size = 2.21 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{a^{\frac {3}{4}}} - \frac {b \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 {3}{4}}} - \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}} b}{{\left (a - \frac {a x^{4} + b}{x^{4}}\right )} x}\right )} a - \frac {1}{4} \, {\left (2 \, a^{\frac {1}{4}} \arctan \left (\frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) - a^{\frac {1}{4}} \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 ) - \frac {4 \, {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )} b \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 {{\left (a\,x^4+b\right )}^{1/4}\,\left (b-a\,x^4\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.16, size = 83, normalized size = 0.97 \begin {gather*} \frac {a \sqrt [4]{b} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {b^{\frac {5}{4}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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