Optimal. Leaf size=112 \[ -\frac {2^{3/4} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{b}-\frac {2^{3/4} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{b}+\frac {2 \left (a x^4+b x^2\right )^{3/4}}{3 b x^3} \]
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Rubi [C] time = 0.37, antiderivative size = 55, normalized size of antiderivative = 0.49, number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {2056, 466, 511, 510} \begin {gather*} \frac {2 \left (a x^2+b\right ) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^2}{a x^2+b}\right )}{3 b x \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 466
Rule 510
Rule 511
Rule 2056
Rubi steps
\begin {align*} \int \frac {b+a x^2}{x^2 \left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {\left (b+a x^2\right )^{3/4}}{x^{5/2} \left (-b+a x^2\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (b+a x^4\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \left (b+a x^2\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {a x^4}{b}\right )^{3/4}}{x^4 \left (-b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\left (1+\frac {a x^2}{b}\right )^{3/4} \sqrt [4]{b x^2+a x^4}}\\ &=\frac {2 \left (b+a x^2\right ) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^2}{b+a x^2}\right )}{3 b x \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 48, normalized size = 0.43 \begin {gather*} \frac {2 \left (x^2 \left (a x^2+b\right )\right )^{3/4} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {2 a x^2}{a x^2+b}\right )}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 112, normalized size = 1.00 \begin {gather*} \frac {2 \left (b x^2+a x^4\right )^{3/4}}{3 b x^3}-\frac {2^{3/4} a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{b}-\frac {2^{3/4} a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 78.00, size = 504, normalized size = 4.50 \begin {gather*} \frac {12 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a^{4} b x^{2} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + 2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} b^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + {\left (2 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b x^{2}} a^{2} b x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (3 \, a b^{3} x^{3} + b^{4} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} a^{2} b^{2} \sqrt {\frac {a^{3}}{b^{4}}}}\right )}}{a^{5} x^{3} - a^{4} b x}\right ) - 3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} + 4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} + \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) + 3 \, \left (\frac {1}{2}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} a b^{2} x^{2} \sqrt {\frac {a^{3}}{b^{4}}} - 4 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b x^{2}} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}} a^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{3} + a b^{2} x\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{a x^{3} - b x}\right ) + 4 \, {\left (a x^{4} + b x^{2}\right )}^{\frac {3}{4}}}{6 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 208, normalized size = 1.86 \begin {gather*} -\frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{b} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, b} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, b} + \frac {2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{4}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+b}{x^{2} \left (a \,x^{2}-b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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*} \int \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )} x^{2}}\,{d x} \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 {a\,x^2+b}{x^2\,\left (b-a\,x^2\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + b}{x^{2} \sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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