Optimal. Leaf size=277 \[ \frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\log \left (a+b x^4\right )}{4 b}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.20, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1876, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 205, 260} \[ \frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\log \left (a+b x^4\right )}{4 b}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 260
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1876
Rubi steps
\begin {align*} \int \frac {1+x+x^2+x^3}{a+b x^4} \, dx &=\int \left (\frac {1+x^2}{a+b x^4}+\frac {x \left (1+x^2\right )}{a+b x^4}\right ) \, dx\\ &=\int \frac {1+x^2}{a+b x^4} \, dx+\int \frac {x \left (1+x^2\right )}{a+b x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x}{a+b x^2} \, dx,x,x^2\right )-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 b}+\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b}\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )+\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}+\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} b^{3/4}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\log \left (a+b x^4\right )}{4 b}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\log \left (a+b x^4\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 283, normalized size = 1.02 \[ \frac {\sqrt {2} \sqrt [4]{b} \left (a^{3/4}-\sqrt [4]{a} \sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )+\sqrt {2} \sqrt [4]{b} \left (\sqrt [4]{a} \sqrt {b}-a^{3/4}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )+2 a \log \left (a+b x^4\right )-2 \sqrt [4]{a} \sqrt [4]{b} \left (2 \sqrt [4]{a} \sqrt [4]{b}+\sqrt {2} \sqrt {a}+\sqrt {2} \sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{a} \sqrt [4]{b} \left (-2 \sqrt [4]{a} \sqrt [4]{b}+\sqrt {2} \sqrt {a}+\sqrt {2} \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 a b} \]
Antiderivative was successfully verified.
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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 [A] time = 0.21, size = 270, normalized size = 0.97 \[ \frac {\log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \sqrt {2} \sqrt {a b^{3}} b + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} + \sqrt {2} \sqrt {a b^{3}} b + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 286, normalized size = 1.03 \[ \frac {\arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{2 \sqrt {a b}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 a}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {\ln \left (b \,x^{4}+a \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 296, normalized size = 1.07 \[ \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} - \sqrt {a} \sqrt {b} + b\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} + \sqrt {a} \sqrt {b} - b\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{8 \, a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {{\left ({\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {a}\right )} b + {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} + 2 \, a\right )} \sqrt {b} - 2 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {{\left ({\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {a}\right )} b + {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} - 2 \, a\right )} \sqrt {b} + 2 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.04, size = 305, normalized size = 1.10 \[ \sum _{k=1}^4\ln \left (-\mathrm {root}\left (256\,a^3\,b^4\,z^4-256\,a^3\,b^3\,z^3+96\,a^3\,b^2\,z^2+96\,a^2\,b^3\,z^2-16\,a^3\,b\,z-16\,a\,b^3\,z-32\,a^2\,b^2\,z+3\,a^2\,b+3\,a\,b^2+b^3+a^3,z,k\right )\,\left (\mathrm {root}\left (256\,a^3\,b^4\,z^4-256\,a^3\,b^3\,z^3+96\,a^3\,b^2\,z^2+96\,a^2\,b^3\,z^2-16\,a^3\,b\,z-16\,a\,b^3\,z-32\,a^2\,b^2\,z+3\,a^2\,b+3\,a\,b^2+b^3+a^3,z,k\right )\,\left (16\,a\,b^3-16\,a\,b^3\,x\right )+x\,\left (4\,b^3+4\,a\,b^2\right )\right )\right )\,\mathrm {root}\left (256\,a^3\,b^4\,z^4-256\,a^3\,b^3\,z^3+96\,a^3\,b^2\,z^2+96\,a^2\,b^3\,z^2-16\,a^3\,b\,z-16\,a\,b^3\,z-32\,a^2\,b^2\,z+3\,a^2\,b+3\,a\,b^2+b^3+a^3,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.27, size = 187, normalized size = 0.68 \[ \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{4} - 256 t^{3} a^{3} b^{3} + t^{2} \left (96 a^{3} b^{2} + 96 a^{2} b^{3}\right ) + t \left (- 16 a^{3} b - 32 a^{2} b^{2} - 16 a b^{3}\right ) + a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a^{3} b^{3} - 48 t^{2} a^{3} b^{2} + 16 t^{2} a^{2} b^{3} + 12 t a^{3} b + 16 t a^{2} b^{2} + 4 t a b^{3} - a^{3} - 2 a^{2} b - a b^{2}}{a^{2} b + 2 a b^{2} + b^{3}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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