Optimal. Leaf size=128 \[ \frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {277, 325, 229, 228} \[ \frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 277
Rule 325
Rubi steps
\begin {align*} \int \frac {\left (a-b x^2\right )^{3/4}}{x^6} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}-\frac {1}{10} (3 b) \int \frac {1}{x^4 \sqrt [4]{a-b x^2}} \, dx\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}-\frac {\left (3 b^2\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {\left (3 b^3\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^2}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {\left (3 b^3 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{40 a^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 x^5}+\frac {b \left (a-b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a-b x^2\right )^{3/4}}{20 a^2 x}+\frac {3 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.41 \[ -\frac {\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac {5}{2},-\frac {3}{4};-\frac {3}{2};\frac {b x^2}{a}\right )}{5 x^5 \left (1-\frac {b x^2}{a}\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}, x\right ) \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{x^{6}}\, dx \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (a-b\,x^2\right )}^{3/4}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.28, size = 36, normalized size = 0.28 \[ - \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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