Optimal. Leaf size=145 \[ -\frac {3 b^3 x}{20 a^2 \sqrt [4]{a+b x^2}}-\frac {\left (a+b x^2\right )^{3/4}}{5 x^5}-\frac {b \left (a+b x^2\right )^{3/4}}{10 a x^3}+\frac {3 b^2 \left (a+b x^2\right )^{3/4}}{20 a^2 x}+\frac {3 b^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {283, 331, 235,
233, 202} \begin {gather*} \frac {3 b^{5/2} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a+b x^2}}-\frac {3 b^3 x}{20 a^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} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 283
Rule 331
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 {3 b^3 x}{20 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 {\left (3 b^3 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{40 a^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {3 b^3 x}{20 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} \tan ^{-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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 51, normalized size = 0.35 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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 = 0.68, size = 34, normalized size = 0.23 \begin {gather*} - \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^{i \pi }}{a}} \right )}}{5 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 {{\left (b\,x^2+a\right )}^{3/4}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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