Optimal. Leaf size=107 \[ -\frac {4 b^{5/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac {\sqrt [4]{a+b x^4}}{7 a x^7} \]
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Rubi [A] time = 0.05, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {325, 237, 335, 275, 231} \[ -\frac {4 b^{5/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac {\sqrt [4]{a+b x^4}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}-\frac {(6 b) \int \frac {1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx}{7 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}+\frac {\left (4 b^2\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{7 a^2}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}+\frac {\left (4 b^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}{7 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac {\left (4 b^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 )}{7 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac {\left (2 b^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 )}{7 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{7 a x^7}+\frac {2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac {4 b^{5/2} \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 )}{7 a^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.48 \[ -\frac {\left (\frac {b x^4}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {7}{4},\frac {3}{4};-\frac {3}{4};-\frac {b x^4}{a}\right )}{7 x^7 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b x^{12} + a x^{8}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{8}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{8}}\,{d x} \]
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 \[ \int \frac {1}{x^8\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.99, size = 44, normalized size = 0.41 \[ \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{4} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} x^{7} \Gamma \left (- \frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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