Integrand size = 20, antiderivative size = 132 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=-\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \arctan \left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}+\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]
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Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {714, 1144, 214, 211} \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}-\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \arctan \left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]
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Rule 211
Rule 214
Rule 714
Rule 1144
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 ) \\ & = \left (3-\frac {2 \sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-3 \sqrt {a} \sqrt {b}+2 b-b x^2} \, dx,x,\sqrt {2+3 x}\right )+\left (3+\frac {2 \sqrt {b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{3 \sqrt {a} \sqrt {b}+2 b-b x^2} \, dx,x,\sqrt {2+3 x}\right ) \\ & = -\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}+\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {-\sqrt {-3 \sqrt {a} \sqrt {b}-2 b} \arctan \left (\frac {\sqrt {-3 \sqrt {a} \sqrt {b}-2 b} \sqrt {2+3 x}}{3 \sqrt {a}+2 \sqrt {b}}\right )-\sqrt {3 \sqrt {a} \sqrt {b}-2 b} \arctan \left (\frac {\sqrt {3 \sqrt {a} \sqrt {b}-2 b} \sqrt {2+3 x}}{3 \sqrt {a}-2 \sqrt {b}}\right )}{\sqrt {a} b} \]
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Time = 3.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(-\frac {-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}}{\sqrt {a b}}\) | \(114\) |
derivativedivides | \(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\) | \(127\) |
default | \(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \operatorname {arctanh}\left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\) | \(127\) |
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (92) = 184\).
Time = 0.51 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=\frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \log \left (a b^{2} \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \log \left (-a b^{2} \sqrt {\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} + 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) - \frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \log \left (a b^{2} \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) + \frac {1}{2} \, \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \log \left (-a b^{2} \sqrt {-\frac {3 \, a b \sqrt {\frac {1}{a b^{3}}} - 2}{a b}} \sqrt {\frac {1}{a b^{3}}} + \sqrt {3 \, x + 2}\right ) \]
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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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Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (92) = 184\).
Time = 0.40 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=-\frac {{\left (4 \, \sqrt {a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {a b} b} a + 17 \, \sqrt {a b} \sqrt {-2 \, b^{2} - 3 \, \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {3 \, x + 2}}{\sqrt {-\frac {2 \, b + \sqrt {{\left (9 \, a - 4 \, b\right )} b + 4 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + 17 \, a b^{4}} + \frac {{\left (4 \, \sqrt {a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {a b} b} a + 17 \, \sqrt {a b} \sqrt {-2 \, b^{2} + 3 \, \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {\sqrt {3 \, x + 2}}{\sqrt {-\frac {2 \, b - \sqrt {{\left (9 \, a - 4 \, b\right )} b + 4 \, b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + 17 \, a b^{4}} \]
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Time = 0.76 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.93 \[ \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,\left (\left (576\,b^3+1296\,a\,b^2\right )\,\sqrt {3\,x+2}+\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {a^3\,b^3}-2\,a\,b^2\right )}{a}\right )\,\sqrt {-\frac {3\,\sqrt {a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}}{3888\,a\,b-1728\,b^2}\right )\,\sqrt {-\frac {3\,\sqrt {a^3\,b^3}-2\,a\,b^2}{4\,a^2\,b^3}}+2\,\mathrm {atanh}\left (\frac {2\,\left (\left (576\,b^3+1296\,a\,b^2\right )\,\sqrt {3\,x+2}-\frac {288\,b\,\sqrt {3\,x+2}\,\left (3\,\sqrt {a^3\,b^3}+2\,a\,b^2\right )}{a}\right )\,\sqrt {\frac {3\,\sqrt {a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}}}{3888\,a\,b-1728\,b^2}\right )\,\sqrt {\frac {3\,\sqrt {a^3\,b^3}+2\,a\,b^2}{4\,a^2\,b^3}} \]
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