Optimal. Leaf size=101 \[ \frac {\left (-3 b+4 a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{21 b^2 x^5}-\frac {a \text {RootSum}\left [2 a^2+a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ]}{16 b} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(538\) vs. \(2(101)=202\).
time = 0.91, antiderivative size = 538, normalized size of antiderivative = 5.33, number of steps
used = 19, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2081, 6857,
277, 270, 1284, 1543, 1443, 385, 218, 212, 209} \begin {gather*} \frac {a \sqrt {x} \sqrt [4]{a x^2+b} \text {ArcTan}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \text {ArcTan}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a} \sqrt [4]{a x^4+b x^2}}+\frac {4 a \left (a x^2+b\right )}{21 b^2 x \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a} \sqrt [4]{a x^4+b x^2}}-\frac {a x^2+b}{7 b x^3 \sqrt [4]{a x^4+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 270
Rule 277
Rule 385
Rule 1284
Rule 1443
Rule 1543
Rule 2081
Rule 6857
Rubi steps
\begin {align*} \int \frac {b+a x^4}{x^4 \left (2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {b+a x^4}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \left (\frac {1}{x^{9/2} \sqrt [4]{b+a x^2}}-\frac {b}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )}\right ) \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {\left (4 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{5/2} \sqrt [4]{b+a x^2}} \, dx}{7 b \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4} \left (2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 b x^8 \sqrt [4]{b+a x^4}}-\frac {a}{2 b \sqrt [4]{b+a x^4} \left (2 b+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (4 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{7 b \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 129, normalized size = 1.28 \begin {gather*} \frac {16 \left (-3 b^2+a b x^2+4 a^2 x^4\right )-21 a b x^{7/2} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2+a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{336 b^2 x^3 \sqrt [4]{x^2 \left (b+a x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{x^{4} \left (a \,x^{4}+2 b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{x^{4} \sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} + 2 b\right )}\, dx \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 {a\,x^4+b}{x^4\,{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (a\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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