Optimal. Leaf size=135 \[ -\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt {x^2+1}+1}{x \sqrt [4]{x^2+1}}\right )-\frac {1}{2} a \tanh ^{-1}\left (\frac {1-\sqrt {x^2+1}}{x \sqrt [4]{x^2+1}}\right )-\frac {b \tan ^{-1}\left (\frac {1-\sqrt {x^2+1}}{\sqrt {2} \sqrt [4]{x^2+1}}\right )}{\sqrt {2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {x^2+1}+1}{\sqrt {2} \sqrt [4]{x^2+1}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1010, 397, 439} \[ -\frac {1}{2} a \tan ^{-1}\left (\frac {\sqrt {x^2+1}+1}{x \sqrt [4]{x^2+1}}\right )-\frac {1}{2} a \tanh ^{-1}\left (\frac {1-\sqrt {x^2+1}}{x \sqrt [4]{x^2+1}}\right )-\frac {b \tan ^{-1}\left (\frac {1-\sqrt {x^2+1}}{\sqrt {2} \sqrt [4]{x^2+1}}\right )}{\sqrt {2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {x^2+1}+1}{\sqrt {2} \sqrt [4]{x^2+1}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 397
Rule 439
Rule 1010
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt [4]{1+x^2} \left (2+x^2\right )} \, dx &=a \int \frac {1}{\sqrt [4]{1+x^2} \left (2+x^2\right )} \, dx+b \int \frac {x}{\sqrt [4]{1+x^2} \left (2+x^2\right )} \, dx\\ &=-\frac {b \tan ^{-1}\left (\frac {1-\sqrt {1+x^2}}{\sqrt {2} \sqrt [4]{1+x^2}}\right )}{\sqrt {2}}-\frac {1}{2} a \tan ^{-1}\left (\frac {1+\sqrt {1+x^2}}{x \sqrt [4]{1+x^2}}\right )-\frac {1}{2} a \tanh ^{-1}\left (\frac {1-\sqrt {1+x^2}}{x \sqrt [4]{1+x^2}}\right )-\frac {b \tanh ^{-1}\left (\frac {1+\sqrt {1+x^2}}{\sqrt {2} \sqrt [4]{1+x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 152, normalized size = 1.13 \[ \frac {1}{4} b x^2 F_1\left (1;\frac {1}{4},1;2;-x^2,-\frac {x^2}{2}\right )-\frac {6 a x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-x^2,-\frac {x^2}{2}\right )}{\sqrt [4]{x^2+1} \left (x^2+2\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-x^2,-\frac {x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-x^2,-\frac {x^2}{2}\right )\right )-6 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-x^2,-\frac {x^2}{2}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {b x + a}{{\left (x^{2} + 2\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {b x +a}{\left (x^{2}+1\right )^{\frac {1}{4}} \left (x^{2}+2\right )}\, dx \]
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 \[ \int \frac {b x + a}{{\left (x^{2} + 2\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {a+b\,x}{{\left (x^2+1\right )}^{1/4}\,\left (x^2+2\right )} \,d x \]
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 \[ \int \frac {a + b x}{\sqrt [4]{x^{2} + 1} \left (x^{2} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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