Optimal. Leaf size=166 \[ -\frac {7 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{2 a^{3/4}}+\frac {3 \sqrt [4]{2} b \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{a^{3/4}}+\frac {7 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{2 a^{3/4}}-\frac {3 \sqrt [4]{2} b \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{a^{3/4}}+x \sqrt [4]{a x^4+b x^2} \]
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Rubi [A] time = 0.42, antiderivative size = 290, normalized size of antiderivative = 1.75, number of steps used = 14, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2056, 581, 584, 329, 331, 298, 203, 206, 466, 494} \begin {gather*} -\frac {7 b \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {3 \sqrt [4]{2} b \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {7 b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {3 \sqrt [4]{2} b \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4} \sqrt {x} \sqrt [4]{a x^2+b}}+x \sqrt [4]{a x^4+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 331
Rule 466
Rule 494
Rule 581
Rule 584
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (b+2 a x^2\right )}{-b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \left (5 a b^2+7 a^2 b x^2\right )}{\left (-b+a x^2\right ) \left (b+a x^2\right )^{3/4}} \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\sqrt [4]{b x^2+a x^4} \int \left (\frac {7 a b \sqrt {x}}{\left (b+a x^2\right )^{3/4}}+\frac {12 a b^2 \sqrt {x}}{\left (-b+a x^2\right ) \left (b+a x^2\right )^{3/4}}\right ) \, dx}{2 a \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (7 b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (6 b^2 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (-b+a x^2\right ) \left (b+a x^2\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (7 b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (12 b^2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+a x^4\right ) \left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (7 b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (12 b^2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-b+2 a b x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}+\frac {\left (7 b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (7 b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (3 \sqrt {2} b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (3 \sqrt {2} b \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=x \sqrt [4]{b x^2+a x^4}-\frac {7 b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {3 \sqrt [4]{2} b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {7 b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {3 \sqrt [4]{2} b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{a^{3/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 156, normalized size = 0.94 \begin {gather*} \frac {x^3 \left (-3 a x^2 \left (1-\frac {a^2 x^4}{b^2}\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )-5 b \left (\frac {a x^2}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {2 a x^2}{b-a x^2}\right )+3 \left (a x^2+b\right ) \left (1-\frac {a x^2}{b}\right )^{3/4}\right )}{3 \left (x^2 \left (a x^2+b\right )\right )^{3/4} \left (1-\frac {a x^2}{b}\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.60, size = 166, normalized size = 1.00 \begin {gather*} x \sqrt [4]{b x^2+a x^4}-\frac {7 b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}+\frac {3 \sqrt [4]{2} b \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{a^{3/4}}+\frac {7 b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}-\frac {3 \sqrt [4]{2} b \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 394, normalized size = 2.37 \begin {gather*} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \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 )}{2 \, a} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \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 )}{2 \, a} + \frac {3 \cdot 2^{\frac {3}{4}} b \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 )}{4 \, \left (-a\right )^{\frac {3}{4}}} + \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \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 )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \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 )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \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 )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{8 \, a} + \frac {7 \, \sqrt {2} b \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} \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 \[\int \frac {\left (2 a \,x^{2}+b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}{a \,x^{2}-b}\, dx\]
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 {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (2 \, a x^{2} + b\right )}}{a x^{2} - b}\,{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 {\left (2\,a\,x^2+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{b-a\,x^2} \,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 {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (2 a x^{2} + b\right )}{a x^{2} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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