Optimal. Leaf size=132 \[ -\frac {\sqrt {3 \sqrt {a}-2 \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}-2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}}+\frac {\sqrt {3 \sqrt {a}+2 \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {2+3 x}}{\sqrt {3 \sqrt {a}+2 \sqrt {b}}}\right )}{\sqrt {a} b^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, 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 = {714, 1144, 214,
211} \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}} \text {ArcTan}\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.
[In]
[Out]
Rule 211
Rule 214
Rule 714
Rule 1144
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{a-b x^2} \, dx &=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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 150, normalized size = 1.14 \begin {gather*} \frac {-\sqrt {-3 \sqrt {a} \sqrt {b}-2 b} \tan ^{-1}\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} \tan ^{-1}\left (\frac {\sqrt {3 \sqrt {a} \sqrt {b}-2 b} \sqrt {2+3 x}}{3 \sqrt {a}-2 \sqrt {b}}\right )}{\sqrt {a} b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.50, size = 127, normalized size = 0.96
method | result | size |
derivativedivides | \(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \arctanh \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\) | \(127\) |
default | \(-6 b \left (-\frac {\left (3 \sqrt {a b}+2 b \right ) \arctanh \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}+2 b \right ) b}}+\frac {\left (3 \sqrt {a b}-2 b \right ) \arctan \left (\frac {b \sqrt {2+3 x}}{\sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )}{6 b \sqrt {a b}\, \sqrt {\left (3 \sqrt {a b}-2 b \right ) b}}\right )\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (92) = 184\).
time = 2.64, 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.69, 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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs.
\(2 (92) = 184\).
time = 2.07, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________