Optimal. Leaf size=75 \[ \frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {277, 240, 212, 206, 203} \begin {gather*} \frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\frac {\left (a x^4+b\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (b+a x^4\right )^{3/4}}{x^4} \, dx &=-\frac {\left (b+a x^4\right )^{3/4}}{3 x^3}+a \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{3 x^3}+a \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {\left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.68 \begin {gather*} -\frac {\left (a x^4+b\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {3}{4};\frac {1}{4};-\frac {a x^4}{b}\right )}{3 x^3 \left (\frac {a x^4}{b}+1\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 75, normalized size = 1.00 \begin {gather*} -\frac {\left (b+a x^4\right )^{3/4}}{3 x^3}+\frac {1}{2} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )+\frac {1}{2} a^{3/4} \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 {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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \right )^{\frac {3}{4}}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 85, normalized size = 1.13 \begin {gather*} -\frac {1}{4} \, a {\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}}}{3 \, x^{3}} \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 )}^{3/4}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.24, size = 42, normalized size = 0.56 \begin {gather*} \frac {b^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {a x^{4} e^{i \pi }}{b}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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