Optimal. Leaf size=132 \[ \frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}-\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]
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Rubi [A] time = 0.13, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {700, 1130, 208, 205} \begin {gather*} \frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \tanh ^{-1}\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}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 700
Rule 1130
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx &=6 \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 0.09, size = 123, normalized size = 0.93 \begin {gather*} \frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )-\sqrt {3 \sqrt {a}-2 \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {3 x+2}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 200, normalized size = 1.52 \begin {gather*} \frac {\left (3 \sqrt {a}+2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {3 x+2} \sqrt {-3 \sqrt {a} \sqrt {b}-2 b}}{3 \sqrt {a}+2 \sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \sqrt {-\left (\sqrt {b} \left (3 \sqrt {a}+2 \sqrt {b}\right )\right )}}+\frac {\left (2 \sqrt {b}-3 \sqrt {a}\right ) \tan ^{-1}\left (\frac {\sqrt {3 x+2} \sqrt {3 \sqrt {a} \sqrt {b}-2 b}}{3 \sqrt {a}-2 \sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\sqrt {b} \left (3 \sqrt {a}-2 \sqrt {b}\right )}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 299, normalized size = 2.27 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 216, normalized size = 1.64 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 182, normalized size = 1.38 \begin {gather*} \frac {2 b \arctanh \left (\frac {\sqrt {3 x +2}\, b}{\sqrt {\left (2 b +3 \sqrt {a b}\right ) b}}\right )}{\sqrt {a b}\, \sqrt {\left (2 b +3 \sqrt {a b}\right ) b}}+\frac {2 b \arctan \left (\frac {\sqrt {3 x +2}\, b}{\sqrt {\left (-2 b +3 \sqrt {a b}\right ) b}}\right )}{\sqrt {a b}\, \sqrt {\left (-2 b +3 \sqrt {a b}\right ) b}}+\frac {3 \arctanh \left (\frac {\sqrt {3 x +2}\, b}{\sqrt {\left (2 b +3 \sqrt {a b}\right ) b}}\right )}{\sqrt {\left (2 b +3 \sqrt {a b}\right ) b}}-\frac {3 \arctan \left (\frac {\sqrt {3 x +2}\, b}{\sqrt {\left (-2 b +3 \sqrt {a b}\right ) b}}\right )}{\sqrt {\left (-2 b +3 \sqrt {a b}\right ) b}} \end {gather*}
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*} -\int \frac {\sqrt {3 \, x + 2}}{b x^{2} - a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 255, normalized size = 1.93 \begin {gather*} 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.80, size = 58, normalized size = 0.44 \begin {gather*} - 6 \operatorname {RootSum} {\left (20736 t^{4} a^{2} b^{3} - 576 t^{2} a b^{2} - 9 a + 4 b, \left (t \mapsto t \log {\left (- 576 t^{3} a b^{2} + 8 t b + \sqrt {3 x + 2} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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