Optimal. Leaf size=130 \[ \frac {4 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b} \]
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Rubi [A] time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {275, 279, 321, 233, 232} \[ -\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}+\frac {4 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{5/2} \left (a-b x^4\right )^{3/4}}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b} \]
Antiderivative was successfully verified.
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Rule 232
Rule 233
Rule 275
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^9 \sqrt [4]{a-b x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^4 \sqrt [4]{a-b x^2} \, dx,x,x^2\right )\\ &=\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {1}{22} a \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b}\\ &=-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2}\\ &=-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {\left (2 a^3 \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {2 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {a x^6 \sqrt [4]{a-b x^4}}{77 b}+\frac {1}{11} x^{10} \sqrt [4]{a-b x^4}+\frac {4 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{5/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 98, normalized size = 0.75 \[ \frac {x^2 \sqrt [4]{a-b x^4} \left (6 a^2 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {b x^4}{a}\right )-\sqrt [4]{1-\frac {b x^4}{a}} \left (6 a^2+a b x^4-7 b^2 x^8\right )\right )}{77 b^2 \sqrt [4]{1-\frac {b x^4}{a}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9}, x\right ) \]
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 \[ \int {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{9}\, dx \]
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 \[ \int {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{9}\,{d x} \]
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 \[ \int x^9\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.94, size = 31, normalized size = 0.24 \[ \frac {\sqrt [4]{a} x^{10} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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