Optimal. Leaf size=102 \[ \frac {a^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \sqrt {b} \left (a+b x^4\right )^{3/4}}+\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, 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, 237, 335, 275, 231} \[ \frac {a^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \sqrt {b} \left (a+b x^4\right )^{3/4}}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}+\frac {a x \sqrt [4]{a+b x^4}}{12 b} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 279
Rule 321
Rule 335
Rubi steps
\begin {align*} \int x^4 \sqrt [4]{a+b x^4} \, dx &=\frac {1}{6} x^5 \sqrt [4]{a+b x^4}+\frac {1}{6} a \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}-\frac {a^2 \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{12 b}\\ &=\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}-\frac {\left (a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{12 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}+\frac {\left (a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{12 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}+\frac {\left (a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{24 b \left (a+b x^4\right )^{3/4}}\\ &=\frac {a x \sqrt [4]{a+b x^4}}{12 b}+\frac {1}{6} x^5 \sqrt [4]{a+b x^4}+\frac {a^{3/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \sqrt {b} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 62, normalized size = 0.61 \[ \frac {x \sqrt [4]{a+b x^4} \left (-\frac {a \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt [4]{\frac {b x^4}{a}+1}}+a+b x^4\right )}{6 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{4}, x\right ) \]
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 \[ \int {\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{4}+a \right )^{\frac {1}{4}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{4}\,{d x} \]
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 \[ \int x^4\,{\left (b\,x^4+a\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.04, size = 39, normalized size = 0.38 \[ \frac {\sqrt [4]{a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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