Integrand size = 19, antiderivative size = 427 \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\frac {3 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}-\sqrt {2} \sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}+\sqrt {2} \sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}+\frac {3 \log \left (\sqrt {9 a+4 b}-\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \log \left (\sqrt {9 a+4 b}+\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}} \]
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Time = 0.39 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {714, 1143, 648, 632, 212, 642} \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\frac {3 \text {arctanh}\left (\frac {\sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}-\sqrt {2} \sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}+\sqrt {2} \sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}+\frac {3 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt {3 x+2} \sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}+\sqrt {9 a+4 b}+\sqrt {b} (3 x+2)\right )}{2 \sqrt {2} b^{3/4} \sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}}-\frac {3 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt {3 x+2} \sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}+\sqrt {9 a+4 b}+\sqrt {b} (3 x+2)\right )}{2 \sqrt {2} b^{3/4} \sqrt {\sqrt {9 a+4 b}+2 \sqrt {b}}} \]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 714
Rule 1143
Rubi steps \begin{align*} \text {integral}& = 6 \text {Subst}\left (\int \frac {x^2}{9 a+4 b-4 b x^2+b x^4} \, dx,x,\sqrt {2+3 x}\right ) \\ & = \frac {3 \text {Subst}\left (\int \frac {x}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \text {Subst}\left (\int \frac {x}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}} \\ & = \frac {3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 b}+\frac {3 \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}+2 x}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}+2 x}{\frac {\sqrt {9 a+4 b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {2+3 x}\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}} \\ & = \frac {3 \log \left (\sqrt {9 a+4 b}-\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \log \left (\sqrt {9 a+4 b}+\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \text {Subst}\left (\int \frac {1}{2 \left (2-\frac {\sqrt {9 a+4 b}}{\sqrt {b}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}+2 \sqrt {2+3 x}\right )}{b}-\frac {3 \text {Subst}\left (\int \frac {1}{2 \left (2-\frac {\sqrt {9 a+4 b}}{\sqrt {b}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}+2 \sqrt {2+3 x}\right )}{b} \\ & = \frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \left (\frac {\sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}-\sqrt {2} \sqrt {2+3 x}\right )}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{b} \left (\frac {\sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}{\sqrt [4]{b}}+\sqrt {2} \sqrt {2+3 x}\right )}{\sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}\right )}{\sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}-\sqrt {9 a+4 b}}}+\frac {3 \log \left (\sqrt {9 a+4 b}-\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}}-\frac {3 \log \left (\sqrt {9 a+4 b}+\sqrt {2} \sqrt [4]{b} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}} \sqrt {2+3 x}+\sqrt {b} (2+3 x)\right )}{2 \sqrt {2} b^{3/4} \sqrt {2 \sqrt {b}+\sqrt {9 a+4 b}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\frac {\sqrt [4]{-1} \sqrt {3 \sqrt {a} \sqrt {b}-2 i b} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {3 \sqrt {a} \sqrt {b}-2 i b} \sqrt {2+3 x}}{3 \sqrt {a}-2 i \sqrt {b}}\right )+(-1)^{3/4} \sqrt {3 \sqrt {a} \sqrt {b}+2 i b} \arctan \left (\frac {(-1)^{3/4} \sqrt {3 \sqrt {a} \sqrt {b}+2 i b} \sqrt {2+3 x}}{3 \sqrt {a}+2 i \sqrt {b}}\right )}{\sqrt {a} b} \]
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Time = 5.89 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {4 \left (-\frac {\sqrt {b \left (9 a +4 b \right )}}{2}+b \right ) \left (-\frac {\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \left (\ln \left (\left (-2-3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}-\sqrt {9 a +4 b}\right )-\ln \left (\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}+\sqrt {9 a +4 b}\right )\right )}{8}+\left (b +\frac {\sqrt {b \left (9 a +4 b \right )}}{2}\right ) \left (\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )-\arctan \left (\frac {2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )\right )\right )}{3 \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}\, b^{\frac {3}{2}} a}\) | \(364\) |
derivativedivides | \(\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (-\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}-\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}-\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}+\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}\) | \(447\) |
default | \(\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (-\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}-\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {-2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}-\frac {\sqrt {2 \sqrt {9 a b +4 b^{2}}+4 b}\, \left (\sqrt {9 a b +4 b^{2}}-2 b \right ) \left (\frac {\ln \left (\left (2+3 x \right ) \sqrt {b}+\sqrt {2+3 x}\, \sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}+\sqrt {9 a +4 b}\right )}{2 \sqrt {b}}-\frac {\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}\, \arctan \left (\frac {2 \sqrt {b}\, \sqrt {2+3 x}+\sqrt {2 \sqrt {b \left (9 a +4 b \right )}+4 b}}{\sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{\sqrt {b}\, \sqrt {4 \sqrt {9 a +4 b}\, \sqrt {b}-2 \sqrt {b \left (9 a +4 b \right )}-4 b}}\right )}{6 a b}\) | \(447\) |
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Time = 0.29 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=-\frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt {-\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {-\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt {-\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {-\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt {\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {-\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt {\frac {3 \, a b \sqrt {-\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {-\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) \]
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\[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\int \frac {\sqrt {3 x + 2}}{a + b x^{2}}\, dx \]
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\[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{b x^{2} + a} \,d x } \]
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Time = 0.40 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=\frac {{\left (4 \, \sqrt {-a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {-a b} b} a - 17 \, \sqrt {-a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {-a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {3 \, x + 2}}{\sqrt {-\frac {4 \, b + \sqrt {-4 \, {\left (9 \, a + 4 \, b\right )} b + 16 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} - 17 \, a b^{4}} + \frac {{\left (4 \, \sqrt {-a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {-a b} b} a - 17 \, \sqrt {-a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {-a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {3 \, x + 2}}{\sqrt {-\frac {4 \, b - \sqrt {-4 \, {\left (9 \, a + 4 \, b\right )} b + 16 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} - 17 \, a b^{4}} \]
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Time = 10.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {2+3 x}}{a+b x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (1296\,a\,b^2-576\,b^3\right )\,\sqrt {3\,x+2}-\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {-a^3\,b^3}-2\,a\,b^2\right )}{a}\right )\,\sqrt {\frac {3\,\sqrt {-a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}}{1728\,b^2+3888\,a\,b}\right )\,\sqrt {\frac {3\,\sqrt {-a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\left (1296\,a\,b^2-576\,b^3\right )\,\sqrt {3\,x+2}+\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {-a^3\,b^3}+2\,a\,b^2\right )}{a}\right )\,\sqrt {-\frac {3\,\sqrt {-a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}}}{1728\,b^2+3888\,a\,b}\right )\,\sqrt {-\frac {3\,\sqrt {-a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}} \]
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