Optimal. Leaf size=151 \[ \frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {77 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {296, 331, 235,
234} \begin {gather*} -\frac {77 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{7/2} \sqrt [4]{a-b x^2}}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac {2}{a x^5 \sqrt [4]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 235
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a-b x^2\right )^{5/4}} \, dx &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}+\frac {11 \int \frac {1}{x^6 \sqrt [4]{a-b x^2}} \, dx}{a}\\ &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}+\frac {(77 b) \int \frac {1}{x^4 \sqrt [4]{a-b x^2}} \, dx}{10 a^2}\\ &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}+\frac {\left (77 b^2\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a^3}\\ &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {\left (77 b^3\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^4}\\ &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {\left (77 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^4 \sqrt [4]{a-b x^2}}\\ &=\frac {2}{a x^5 \sqrt [4]{a-b x^2}}-\frac {11 \left (a-b x^2\right )^{3/4}}{5 a^2 x^5}-\frac {77 b \left (a-b x^2\right )^{3/4}}{30 a^3 x^3}-\frac {77 b^2 \left (a-b x^2\right )^{3/4}}{20 a^4 x}-\frac {77 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^{7/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 55, normalized size = 0.36 \begin {gather*} -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {5}{2},\frac {5}{4};-\frac {3}{2};\frac {b x^2}{a}\right )}{5 a x^5 \sqrt [4]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{6} \left (-b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.84, size = 34, normalized size = 0.23 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {5}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac {5}{4}} x^{5}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^6\,{\left (a-b\,x^2\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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