Optimal. Leaf size=129 \[ \frac {3 a^{5/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b} \]
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Rubi [A] time = 0.06, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {321, 237, 335, 275, 231} \[ \frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}+\frac {3 a^{5/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 321
Rule 335
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac {(9 a) \int \frac {x^8}{\left (a+b x^4\right )^{3/4}} \, dx}{10 b}\\ &=-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac {\left (3 a^2\right ) \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx}{4 b^2}\\ &=\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac {\left (3 a^3\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{8 b^3}\\ &=\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}-\frac {\left (3 a^3 \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}{8 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac {\left (3 a^3 \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 )}{8 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac {\left (3 a^3 \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 )}{16 b^3 \left (a+b x^4\right )^{3/4}}\\ &=\frac {3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac {3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac {x^9 \sqrt [4]{a+b x^4}}{10 b}+\frac {3 a^{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 )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 90, normalized size = 0.70 \[ \frac {-15 a^3 x \left (\frac {b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^4}{a}\right )+15 a^3 x+9 a^2 b x^5-2 a b^2 x^9+4 b^3 x^{13}}{40 b^3 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{12}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {x^{12}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {x^{12}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {x^{12}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^{12}}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.14, size = 37, normalized size = 0.29 \[ \frac {x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} \Gamma \left (\frac {17}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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