Optimal. Leaf size=198 \[ -\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {a \tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {3 b \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1024, 401, 455,
57, 631, 210, 31} \begin {gather*} \frac {a \tan ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 401
Rule 455
Rule 631
Rule 1024
Rubi steps
\begin {align*} \int \frac {a+b x}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx &=a \int \frac {1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx+b \int \frac {x}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx\\ &=-\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {a \tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {1}{2} b \text {Subst}\left (\int \frac {1}{(3-x) \sqrt [3]{1+x}} \, dx,x,x^2\right )\\ &=-\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {a \tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {1}{4} (3 b) \text {Subst}\left (\int \frac {1}{2 \sqrt [3]{2}+2^{2/3} x+x^2} \, dx,x,\sqrt [3]{1+x^2}\right )+\frac {(3 b) \text {Subst}\left (\int \frac {1}{2^{2/3}-x} \, dx,x,\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}\\ &=-\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {a \tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {3 b \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{2\ 2^{2/3}}\\ &=-\frac {a \tan ^{-1}(x)}{6\ 2^{2/3}}+\frac {a \tan ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} b \tan ^{-1}\left (\frac {1+\sqrt [3]{2} \sqrt [3]{1+x^2}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac {3 b \log \left (2^{2/3}-\sqrt [3]{1+x^2}\right )}{4\ 2^{2/3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1072\) vs. \(2(198)=396\).
time = 8.61, size = 1072, normalized size = 5.41 \begin {gather*} \frac {\frac {2 \left (\sqrt {3} a^4-12 a^3 b+18 \sqrt {3} a^2 b^2-36 a b^3+9 \sqrt {3} b^4\right ) \tan ^{-1}\left (\frac {3 b+a x+6 \sqrt [3]{2} b \sqrt [3]{1+x^2}-\sqrt {3} \left (a+b x+2 \sqrt [3]{2} a \sqrt [3]{1+x^2}\right )}{3 b \left (\sqrt {3}-x\right )+a \left (-3+\sqrt {3} x\right )}\right )}{\sqrt {3} a^3-9 a^2 b+9 \sqrt {3} a b^2-9 b^3}-\frac {2 \left (\sqrt {3} a^4+12 a^3 b+18 \sqrt {3} a^2 b^2+36 a b^3+9 \sqrt {3} b^4\right ) \tan ^{-1}\left (\frac {3 b+a x+6 \sqrt [3]{2} b \sqrt [3]{1+x^2}+\sqrt {3} \left (a+b x+2 \sqrt [3]{2} a \sqrt [3]{1+x^2}\right )}{3 b \left (\sqrt {3}+x\right )+a \left (3+\sqrt {3} x\right )}\right )}{\sqrt {3} a^3+9 a^2 b+9 \sqrt {3} a b^2+9 b^3}+\frac {2 \left (a^4-4 \sqrt {3} a^3 b+18 a^2 b^2-12 \sqrt {3} a b^3+9 b^4\right ) \log \left (3 b+a x-3 \sqrt [3]{2} b \sqrt [3]{1+x^2}-\sqrt {3} \left (a+b x-\sqrt [3]{2} a \sqrt [3]{1+x^2}\right )\right )}{\sqrt {3} a^3-9 a^2 b+9 \sqrt {3} a b^2-9 b^3}-\frac {2 \left (a^4+4 \sqrt {3} a^3 b+18 a^2 b^2+12 \sqrt {3} a b^3+9 b^4\right ) \log \left (-3 b-a x+3 \sqrt [3]{2} b \sqrt [3]{1+x^2}-\sqrt {3} \left (a+b x-\sqrt [3]{2} a \sqrt [3]{1+x^2}\right )\right )}{\sqrt {3} a^3+9 a^2 b+9 \sqrt {3} a b^2+9 b^3}-\frac {\left (a^4-4 \sqrt {3} a^3 b+18 a^2 b^2-12 \sqrt {3} a b^3+9 b^4\right ) \log \left (3 a^2+9 b^2+12 a b x+a^2 x^2+3 b^2 x^2+3 \sqrt [3]{2} \left (a^2+3 b^2+2 a b x\right ) \sqrt [3]{1+x^2}+3\ 2^{2/3} \left (a^2+3 b^2\right ) \left (1+x^2\right )^{2/3}+\sqrt {3} \left (-6 a b-2 a^2 x-6 b^2 x-2 a b x^2-\sqrt [3]{2} \left (6 a b+a^2 x+3 b^2 x\right ) \sqrt [3]{1+x^2}-6\ 2^{2/3} a b \left (1+x^2\right )^{2/3}\right )\right )}{\sqrt {3} a^3-9 a^2 b+9 \sqrt {3} a b^2-9 b^3}+\frac {\left (a^4+4 \sqrt {3} a^3 b+18 a^2 b^2+12 \sqrt {3} a b^3+9 b^4\right ) \log \left (3 a^2+9 b^2+12 a b x+a^2 x^2+3 b^2 x^2+3 \sqrt [3]{2} \left (a^2+3 b^2+2 a b x\right ) \sqrt [3]{1+x^2}+3\ 2^{2/3} \left (a^2+3 b^2\right ) \left (1+x^2\right )^{2/3}+\sqrt {3} \left (6 a b+2 a^2 x+6 b^2 x+2 a b x^2+\sqrt [3]{2} \left (6 a b+a^2 x+3 b^2 x\right ) \sqrt [3]{1+x^2}+6\ 2^{2/3} a b \left (1+x^2\right )^{2/3}\right )\right )}{\sqrt {3} a^3+9 a^2 b+9 \sqrt {3} a b^2+9 b^3}}{12\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {b x +a}{\left (-x^{2}+3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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} \sqrt [3]{x^{2} + 1} - 3 \sqrt [3]{x^{2} + 1}}\, dx - \int \frac {b x}{x^{2} \sqrt [3]{x^{2} + 1} - 3 \sqrt [3]{x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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+b\,x}{{\left (x^2+1\right )}^{1/3}\,\left (x^2-3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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