Integrand size = 47, antiderivative size = 46 \[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\frac {1}{5} \sqrt {\frac {2}{7}} E\left (\arctan \left (\sqrt {\frac {7}{5}} x\right )|-\frac {1}{14}\right )+\frac {\operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {7}{5}} x\right ),-\frac {1}{14}\right )}{\sqrt {14}} \] Output:
1/35*14^(1/2)*EllipticE(35^(1/2)*x/(35*x^2+25)^(1/2),1/14*I*14^(1/2))+1/14 *InverseJacobiAM(arctan(1/5*35^(1/2)*x),1/14*I*14^(1/2))*14^(1/2)
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.83 \[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\frac {1}{70} \left (\frac {14 x \sqrt {2+3 x^2}}{\sqrt {5+7 x^2}}+2 i \sqrt {14} E\left (i \text {arcsinh}\left (\sqrt {\frac {7}{5}} x\right )|\frac {15}{14}\right )-7 i \sqrt {14} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {7}{5}} x\right ),\frac {15}{14}\right )\right ) \] Input:
Integrate[Sqrt[2 + 3*x^2]/(5 + 7*x^2)^(3/2) + 1/(Sqrt[2 + 3*x^2]*Sqrt[5 + 7*x^2]),x]
Output:
((14*x*Sqrt[2 + 3*x^2])/Sqrt[5 + 7*x^2] + (2*I)*Sqrt[14]*EllipticE[I*ArcSi nh[Sqrt[7/5]*x], 15/14] - (7*I)*Sqrt[14]*EllipticF[I*ArcSinh[Sqrt[7/5]*x], 15/14])/70
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(46)=92\).
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.87, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {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.
\(\displaystyle \int \left (\frac {\sqrt {3 x^2+2}}{\left (7 x^2+5\right )^{3/2}}+\frac {1}{\sqrt {7 x^2+5} \sqrt {3 x^2+2}}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {7}{5}} x\right ),-\frac {1}{14}\right )}{\sqrt {14} \sqrt {\frac {3 x^2+2}{7 x^2+5}} \sqrt {7 x^2+5}}+\frac {\sqrt {\frac {2}{7}} \sqrt {3 x^2+2} E\left (\arctan \left (\sqrt {\frac {7}{5}} x\right )|-\frac {1}{14}\right )}{5 \sqrt {\frac {3 x^2+2}{7 x^2+5}} \sqrt {7 x^2+5}}\) |
Input:
Int[Sqrt[2 + 3*x^2]/(5 + 7*x^2)^(3/2) + 1/(Sqrt[2 + 3*x^2]*Sqrt[5 + 7*x^2] ),x]
Output:
(Sqrt[2/7]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[Sqrt[7/5]*x], -1/14])/(5*Sqrt[ (2 + 3*x^2)/(5 + 7*x^2)]*Sqrt[5 + 7*x^2]) + (Sqrt[2 + 3*x^2]*EllipticF[Arc Tan[Sqrt[7/5]*x], -1/14])/(Sqrt[14]*Sqrt[(2 + 3*x^2)/(5 + 7*x^2)]*Sqrt[5 + 7*x^2])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (46 ) = 92\).
Time = 4.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.39
method | result | size |
default | \(-\frac {i \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {2}\, \sqrt {7}}{14}+\frac {\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}\, \left (i \sqrt {7}\, \sqrt {2}\, \operatorname {EllipticE}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}-i \sqrt {7}\, \sqrt {2}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}+21 x^{3}+14 x \right )}{735 x^{4}+1015 x^{2}+350}\) | \(156\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}+2\right ) \left (7 x^{2}+5\right )}\, \left (\frac {\left (21 x^{2}+14\right ) x}{35 \sqrt {\left (x^{2}+\frac {5}{7}\right ) \left (21 x^{2}+14\right )}}-\frac {i \sqrt {35}\, \sqrt {35 x^{2}+25}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )\right )}{175 \sqrt {21 x^{4}+29 x^{2}+10}}-\frac {i \sqrt {35}\, \sqrt {35 x^{2}+25}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )}{70 \sqrt {21 x^{4}+29 x^{2}+10}}\right )}{\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}}\) | \(181\) |
Input:
int((3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+1/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2),x,me thod=_RETURNVERBOSE)
Output:
-1/14*I*EllipticF(1/5*I*x*35^(1/2),1/14*35^(1/2)*6^(1/2))*2^(1/2)*7^(1/2)+ 1/35*(3*x^2+2)^(1/2)*(7*x^2+5)^(1/2)*(I*EllipticE(1/5*I*x*35^(1/2),1/14*35 ^(1/2)*6^(1/2))*7^(1/2)*2^(1/2)*(3*x^2+2)^(1/2)*(7*x^2+5)^(1/2)-I*Elliptic F(1/5*I*x*35^(1/2),1/14*35^(1/2)*6^(1/2))*7^(1/2)*2^(1/2)*(3*x^2+2)^(1/2)* (7*x^2+5)^(1/2)+21*x^3+14*x)/(21*x^4+29*x^2+10)
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.72 \[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=-\frac {14 \, \sqrt {10} \sqrt {-\frac {7}{5}} {\left (7 \, x^{2} + 5\right )} E(\arcsin \left (\sqrt {-\frac {7}{5}} x\right )\,|\,\frac {15}{14}) + 11 \, \sqrt {10} \sqrt {-\frac {7}{5}} {\left (7 \, x^{2} + 5\right )} F(\arcsin \left (\sqrt {-\frac {7}{5}} x\right )\,|\,\frac {15}{14}) - 70 \, \sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2} x}{350 \, {\left (7 \, x^{2} + 5\right )}} \] Input:
integrate((3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+1/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2 ),x, algorithm="fricas")
Output:
-1/350*(14*sqrt(10)*sqrt(-7/5)*(7*x^2 + 5)*elliptic_e(arcsin(sqrt(-7/5)*x) , 15/14) + 11*sqrt(10)*sqrt(-7/5)*(7*x^2 + 5)*elliptic_f(arcsin(sqrt(-7/5) *x), 15/14) - 70*sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)*x)/(7*x^2 + 5)
\[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int \frac {10 x^{2} + 7}{\sqrt {3 x^{2} + 2} \left (7 x^{2} + 5\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((3*x**2+2)**(1/2)/(7*x**2+5)**(3/2)+1/(3*x**2+2)**(1/2)/(7*x**2+ 5)**(1/2),x)
Output:
Integral((10*x**2 + 7)/(sqrt(3*x**2 + 2)*(7*x**2 + 5)**(3/2)), x)
\[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int { \frac {1}{\sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2}} + \frac {\sqrt {3 \, x^{2} + 2}}{{\left (7 \, x^{2} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+1/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2 ),x, algorithm="maxima")
Output:
integrate(1/(sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)) + sqrt(3*x^2 + 2)/(7*x^2 + 5 )^(3/2), x)
\[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int { \frac {1}{\sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2}} + \frac {\sqrt {3 \, x^{2} + 2}}{{\left (7 \, x^{2} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+1/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2 ),x, algorithm="giac")
Output:
integrate(1/(sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)) + sqrt(3*x^2 + 2)/(7*x^2 + 5 )^(3/2), x)
Timed out. \[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int \frac {1}{\sqrt {3\,x^2+2}\,\sqrt {7\,x^2+5}}+\frac {\sqrt {3\,x^2+2}}{{\left (7\,x^2+5\right )}^{3/2}} \,d x \] Input:
int(1/((3*x^2 + 2)^(1/2)*(7*x^2 + 5)^(1/2)) + (3*x^2 + 2)^(1/2)/(7*x^2 + 5 )^(3/2),x)
Output:
int(1/((3*x^2 + 2)^(1/2)*(7*x^2 + 5)^(1/2)) + (3*x^2 + 2)^(1/2)/(7*x^2 + 5 )^(3/2), x)
\[ \int \left (\frac {\sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {1}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=10 \left (\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}\, x^{2}}{147 x^{6}+308 x^{4}+215 x^{2}+50}d x \right )+7 \left (\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}}{147 x^{6}+308 x^{4}+215 x^{2}+50}d x \right ) \] Input:
int((3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+1/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2),x)
Output:
10*int((sqrt(3*x**2 + 2)*sqrt(7*x**2 + 5)*x**2)/(147*x**6 + 308*x**4 + 215 *x**2 + 50),x) + 7*int((sqrt(3*x**2 + 2)*sqrt(7*x**2 + 5))/(147*x**6 + 308 *x**4 + 215*x**2 + 50),x)