Integrand size = 27, antiderivative size = 56 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=-4 x+12 \arcsin \left (\frac {1-x}{2}\right )-24 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {1+x}}{\sqrt {3-x}}\right )+21 \log (x)-9 \log (1+x) \] Output:
-4*x-12*arcsin(-1/2+1/2*x)-24*3^(1/2)*arctanh(3^(1/2)*(1+x)^(1/2)/(3-x)^(1 /2))+21*ln(x)-9*ln(1+x)
Leaf count is larger than twice the leaf count of optimal. \(361\) vs. \(2(56)=112\).
Time = 0.49 (sec) , antiderivative size = 361, normalized size of antiderivative = 6.45 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=-4-4 x-48 \arctan \left (\frac {\sqrt {1+x}}{-2+\sqrt {3-x}}\right )-42 \log \left (-2+\sqrt {3-x}\right )-9 \log (1+x)+21 \log \left (-2+\sqrt {3-x}-2 \sqrt {1+x}-\sqrt {3} \sqrt {1+x}\right )+12 \sqrt {3} \log \left (-2+\sqrt {3-x}-2 \sqrt {1+x}-\sqrt {3} \sqrt {1+x}\right )+21 \log \left (-2+\sqrt {3-x}+2 \sqrt {1+x}-\sqrt {3} \sqrt {1+x}\right )+12 \sqrt {3} \log \left (-2+\sqrt {3-x}+2 \sqrt {1+x}-\sqrt {3} \sqrt {1+x}\right )+21 \log \left (-2+\sqrt {3-x}-2 \sqrt {1+x}+\sqrt {3} \sqrt {1+x}\right )-12 \sqrt {3} \log \left (-2+\sqrt {3-x}-2 \sqrt {1+x}+\sqrt {3} \sqrt {1+x}\right )+21 \log \left (-2+\sqrt {3-x}+2 \sqrt {1+x}+\sqrt {3} \sqrt {1+x}\right )-12 \sqrt {3} \log \left (-2+\sqrt {3-x}+2 \sqrt {1+x}+\sqrt {3} \sqrt {1+x}\right ) \] Input:
Integrate[(2*Sqrt[3 - x] + 3/Sqrt[1 + x])^2/x,x]
Output:
-4 - 4*x - 48*ArcTan[Sqrt[1 + x]/(-2 + Sqrt[3 - x])] - 42*Log[-2 + Sqrt[3 - x]] - 9*Log[1 + x] + 21*Log[-2 + Sqrt[3 - x] - 2*Sqrt[1 + x] - Sqrt[3]*S qrt[1 + x]] + 12*Sqrt[3]*Log[-2 + Sqrt[3 - x] - 2*Sqrt[1 + x] - Sqrt[3]*Sq rt[1 + x]] + 21*Log[-2 + Sqrt[3 - x] + 2*Sqrt[1 + x] - Sqrt[3]*Sqrt[1 + x] ] + 12*Sqrt[3]*Log[-2 + Sqrt[3 - x] + 2*Sqrt[1 + x] - Sqrt[3]*Sqrt[1 + x]] + 21*Log[-2 + Sqrt[3 - x] - 2*Sqrt[1 + x] + Sqrt[3]*Sqrt[1 + x]] - 12*Sqr t[3]*Log[-2 + Sqrt[3 - x] - 2*Sqrt[1 + x] + Sqrt[3]*Sqrt[1 + x]] + 21*Log[ -2 + Sqrt[3 - x] + 2*Sqrt[1 + x] + Sqrt[3]*Sqrt[1 + x]] - 12*Sqrt[3]*Log[- 2 + Sqrt[3 - x] + 2*Sqrt[1 + x] + Sqrt[3]*Sqrt[1 + x]]
Time = 0.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {x+1}}\right )^2}{x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {12 \sqrt {3-x}}{x \sqrt {x+1}}+\frac {12}{x}+\frac {9}{x (x+1)}-4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \arcsin \left (\frac {1-x}{2}\right )-24 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+1}}{\sqrt {3-x}}\right )-4 x+21 \log (x)-9 \log (x+1)\) |
Input:
Int[(2*Sqrt[3 - x] + 3/Sqrt[1 + x])^2/x,x]
Output:
-4*x + 12*ArcSin[(1 - x)/2] - 24*Sqrt[3]*ArcTanh[(Sqrt[3]*Sqrt[1 + x])/Sqr t[3 - x]] + 21*Log[x] - 9*Log[1 + x]
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36
method | result | size |
default | \(21 \ln \left (x \right )-9 \ln \left (1+x \right )+\frac {12 \sqrt {3-x}\, \sqrt {1+x}\, \left (-\arcsin \left (-\frac {1}{2}+\frac {x}{2}\right )-\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (x +3\right ) \sqrt {3}}{3 \sqrt {-x^{2}+2 x +3}}\right )\right )}{\sqrt {-x^{2}+2 x +3}}-4 x\) | \(76\) |
Input:
int((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x,method=_RETURNVERBOSE)
Output:
21*ln(x)-9*ln(1+x)+12*(3-x)^(1/2)*(1+x)^(1/2)/(-x^2+2*x+3)^(1/2)*(-arcsin( -1/2+1/2*x)-3^(1/2)*arctanh(1/3*(x+3)*3^(1/2)/(-x^2+2*x+3)^(1/2)))-4*x
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=6 \, \sqrt {3} \log \left (-\frac {\sqrt {3} {\left (x + 3\right )} \sqrt {x + 1} \sqrt {-x + 3} + x^{2} - 6 \, x - 9}{x^{2}}\right ) - 4 \, x + 12 \, \arctan \left (\frac {\sqrt {x + 1} {\left (x - 1\right )} \sqrt {-x + 3}}{x^{2} - 2 \, x - 3}\right ) - 9 \, \log \left (x + 1\right ) + 21 \, \log \left (x\right ) \] Input:
integrate((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x, algorithm="fricas")
Output:
6*sqrt(3)*log(-(sqrt(3)*(x + 3)*sqrt(x + 1)*sqrt(-x + 3) + x^2 - 6*x - 9)/ x^2) - 4*x + 12*arctan(sqrt(x + 1)*(x - 1)*sqrt(-x + 3)/(x^2 - 2*x - 3)) - 9*log(x + 1) + 21*log(x)
\[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=\int \frac {\left (2 \sqrt {3 - x} \sqrt {x + 1} + 3\right )^{2}}{x \left (x + 1\right )}\, dx \] Input:
integrate((2*(3-x)**(1/2)+3/(1+x)**(1/2))**2/x,x)
Output:
Integral((2*sqrt(3 - x)*sqrt(x + 1) + 3)**2/(x*(x + 1)), x)
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=-12 \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {-x^{2} + 2 \, x + 3}}{{\left | x \right |}} + \frac {6}{{\left | x \right |}} + 2\right ) - 4 \, x + 12 \, \arcsin \left (-\frac {1}{2} \, x + \frac {1}{2}\right ) - 9 \, \log \left (x + 1\right ) + 21 \, \log \left (x\right ) \] Input:
integrate((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x, algorithm="maxima")
Output:
-12*sqrt(3)*log(2*sqrt(3)*sqrt(-x^2 + 2*x + 3)/abs(x) + 6/abs(x) + 2) - 4* x + 12*arcsin(-1/2*x + 1/2) - 9*log(x + 1) + 21*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (44) = 88\).
Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.70 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=12 \, \pi + 12 \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + \frac {6 \, {\left (\sqrt {x + 1} - 2\right )}}{\sqrt {-x + 3}} - \frac {6 \, \sqrt {-x + 3}}{\sqrt {x + 1} - 2} \right |}}{{\left | 4 \, \sqrt {3} + \frac {6 \, {\left (\sqrt {x + 1} - 2\right )}}{\sqrt {-x + 3}} - \frac {6 \, \sqrt {-x + 3}}{\sqrt {x + 1} - 2} \right |}}\right ) - 4 \, x + 24 \, \arctan \left (\frac {\sqrt {-x + 3} {\left (\frac {{\left (\sqrt {x + 1} - 2\right )}^{2}}{x - 3} + 1\right )}}{2 \, {\left (\sqrt {x + 1} - 2\right )}}\right ) + 21 \, \log \left ({\left | x \right |}\right ) - 9 \, \log \left ({\left | -x - 1 \right |}\right ) + 12 \] Input:
integrate((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x, algorithm="giac")
Output:
12*pi + 12*sqrt(3)*log(abs(-4*sqrt(3) + 6*(sqrt(x + 1) - 2)/sqrt(-x + 3) - 6*sqrt(-x + 3)/(sqrt(x + 1) - 2))/abs(4*sqrt(3) + 6*(sqrt(x + 1) - 2)/sqr t(-x + 3) - 6*sqrt(-x + 3)/(sqrt(x + 1) - 2))) - 4*x + 24*arctan(1/2*sqrt( -x + 3)*((sqrt(x + 1) - 2)^2/(x - 3) + 1)/(sqrt(x + 1) - 2)) + 21*log(abs( x)) - 9*log(abs(-x - 1)) + 12
Time = 26.61 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.82 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=48\,\mathrm {atan}\left (\frac {\sqrt {3-x}-4\,\sqrt {3}+3\,\sqrt {3}\,\sqrt {x+1}}{\sqrt {x+1}-3\,\sqrt {3}\,\sqrt {3-x}+8}\right )-9\,\ln \left (x+1\right )-4\,x+21\,\ln \left (x\right )+12\,\sqrt {3}\,\ln \left (\frac {6\,x-12\,\sqrt {x+1}+4\,\sqrt {3}\,\sqrt {3-x}+2\,\sqrt {3}\,\sqrt {x+1}\,\sqrt {3-x}-6}{3\,x+6\,\sqrt {3}\,\sqrt {3-x}-18}\right )-12\,\sqrt {3}\,\ln \left (\frac {\sqrt {x+1}-1}{\sqrt {3}-\sqrt {3-x}}\right ) \] Input:
int((3/(x + 1)^(1/2) + 2*(3 - x)^(1/2))^2/x,x)
Output:
48*atan(((3 - x)^(1/2) - 4*3^(1/2) + 3*3^(1/2)*(x + 1)^(1/2))/((x + 1)^(1/ 2) - 3*3^(1/2)*(3 - x)^(1/2) + 8)) - 9*log(x + 1) - 4*x + 21*log(x) + 12*3 ^(1/2)*log((6*x - 12*(x + 1)^(1/2) + 4*3^(1/2)*(3 - x)^(1/2) + 2*3^(1/2)*( x + 1)^(1/2)*(3 - x)^(1/2) - 6)/(3*x + 6*3^(1/2)*(3 - x)^(1/2) - 18)) - 12 *3^(1/2)*log(((x + 1)^(1/2) - 1)/(3^(1/2) - (3 - x)^(1/2)))
Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.02 \[ \int \frac {\left (2 \sqrt {3-x}+\frac {3}{\sqrt {1+x}}\right )^2}{x} \, dx=24 \mathit {asin} \left (\frac {\sqrt {-x +3}}{2}\right )-12 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-x +3}}{2}\right )}{2}\right )\right )+12 \sqrt {3}\, \mathrm {log}\left (-\sqrt {3}+3 \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-x +3}}{2}\right )}{2}\right )\right )+12 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-x +3}}{2}\right )}{2}\right )\right )-12 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+3 \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-x +3}}{2}\right )}{2}\right )\right )-9 \,\mathrm {log}\left (x +1\right )+21 \,\mathrm {log}\left (x \right )-4 x \] Input:
int((2*(3-x)^(1/2)+3/(1+x)^(1/2))^2/x,x)
Output:
24*asin(sqrt( - x + 3)/2) - 12*sqrt(3)*log( - sqrt(3) + tan(asin(sqrt( - x + 3)/2)/2)) + 12*sqrt(3)*log( - sqrt(3) + 3*tan(asin(sqrt( - x + 3)/2)/2) ) + 12*sqrt(3)*log(sqrt(3) + tan(asin(sqrt( - x + 3)/2)/2)) - 12*sqrt(3)*l og(sqrt(3) + 3*tan(asin(sqrt( - x + 3)/2)/2)) - 9*log(x + 1) + 21*log(x) - 4*x