Optimal. Leaf size=126 \[ -\frac {7 b^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{5/2} \sqrt [4]{a-b x^2}}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 126, 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 = {290, 325, 229, 228} \[ -\frac {7 b^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{5/2} \sqrt [4]{a-b x^2}}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}+\frac {2}{a x^3 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 228
Rule 229
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a-b x^2\right )^{5/4}} \, dx &=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}+\frac {7 \int \frac {1}{x^4 \sqrt [4]{a-b x^2}} \, dx}{a}\\ &=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}+\frac {(7 b) \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{2 a^2}\\ &=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {\left (7 b^2\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{4 a^3}\\ &=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {\left (7 b^2 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{4 a^3 \sqrt [4]{a-b x^2}}\\ &=\frac {2}{a x^3 \sqrt [4]{a-b x^2}}-\frac {7 \left (a-b x^2\right )^{3/4}}{3 a^2 x^3}-\frac {7 b \left (a-b x^2\right )^{3/4}}{2 a^3 x}-\frac {7 b^{3/2} \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 )}{2 a^{5/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 0.44 \[ -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {3}{2},\frac {5}{4};-\frac {1}{2};\frac {b x^2}{a}\right )}{3 a x^3 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b^{2} x^{8} - 2 \, a b x^{6} + a^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {5}{4}} x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (a-b\,x^2\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.25, size = 34, normalized size = 0.27 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3 a^{\frac {5}{4}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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