Optimal. Leaf size=126 \[ -\frac {\sqrt [4]{a+b x^2}}{5 a x^5}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac {3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}-\frac {3 b^{5/2} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {331, 239, 237}
\begin {gather*} -\frac {3 b^{5/2} \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac {\sqrt [4]{a+b x^2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 239
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a+b x^2}}{5 a x^5}-\frac {(9 b) \int \frac {1}{x^4 \left (a+b x^2\right )^{3/4}} \, dx}{10 a}\\ &=-\frac {\sqrt [4]{a+b x^2}}{5 a x^5}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}+\frac {\left (3 b^2\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx}{4 a^2}\\ &=-\frac {\sqrt [4]{a+b x^2}}{5 a x^5}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac {3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}-\frac {\left (3 b^3\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{8 a^3}\\ &=-\frac {\sqrt [4]{a+b x^2}}{5 a x^5}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac {3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}-\frac {\left (3 b^3 \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{8 a^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^2}}{5 a x^5}+\frac {3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac {3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}-\frac {3 b^{5/2} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}\\ \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 = 51, normalized size = 0.40 \begin {gather*} -\frac {\left (1+\frac {b x^2}{a}\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 (a+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{6} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]
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 \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 = 0.70, size = 32, normalized size = 0.25 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {3}{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 (b\,x^2+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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