Optimal. Leaf size=78 \[ \frac {2 \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 \sqrt {a} \sqrt {b} \left (a+b x^2\right )^{3/4}}+\frac {2 x}{3 a \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {199, 233, 231} \[ \frac {2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac {2 \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {a} \sqrt {b} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 231
Rule 233
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{7/4}} \, dx &=\frac {2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac {\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{3 a}\\ &=\frac {2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac {\left (1+\frac {b x^2}{a}\right )^{3/4} \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{3 a \left (a+b x^2\right )^{3/4}}\\ &=\frac {2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac {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 )}{3 \sqrt {a} \sqrt {b} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 55, normalized size = 0.71 \[ \frac {x \left (\left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {b x^2}{a}\right )+2\right )}{3 a \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.88, size = 37, normalized size = 0.47 \[ \frac {x\,{\left (\frac {b\,x^2}{a}+1\right )}^{7/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (b\,x^2+a\right )}^{7/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.04, size = 24, normalized size = 0.31 \[ \frac {x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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