Integrand size = 49, antiderivative size = 45 \[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=-\frac {1}{5} \sqrt {14} E\left (\arctan \left (\sqrt {\frac {7}{5}} x\right )|-\frac {1}{14}\right )+\frac {3 \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {7}{5}} x\right ),-\frac {1}{14}\right )}{\sqrt {14}} \] Output:
-1/5*14^(1/2)*EllipticE(35^(1/2)*x/(35*x^2+25)^(1/2),1/14*I*14^(1/2))+3/14 *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 = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=-\frac {7 x \sqrt {2+3 x^2}}{5 \sqrt {5+7 x^2}}-i \sqrt {\frac {3}{5}} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {14}{15}\right ) \] Input:
Integrate[(-7*Sqrt[2 + 3*x^2])/(5 + 7*x^2)^(3/2) + 3/(Sqrt[2 + 3*x^2]*Sqrt [5 + 7*x^2]),x]
Output:
(-7*x*Sqrt[2 + 3*x^2])/(5*Sqrt[5 + 7*x^2]) - I*Sqrt[3/5]*EllipticE[I*ArcSi nh[Sqrt[3/2]*x], 14/15]
Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(45)=90\).
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.91, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, 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.
\(\displaystyle \int \left (\frac {3}{\sqrt {3 x^2+2} \sqrt {7 x^2+5}}-\frac {7 \sqrt {3 x^2+2}}{\left (7 x^2+5\right )^{3/2}}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt {3 x^2+2} \operatorname {EllipticF}\left (\arctan \left (\sqrt {\frac {7}{5}} x\right ),-\frac {1}{14}\right )}{\sqrt {14} \sqrt {\frac {3 x^2+2}{7 x^2+5}} \sqrt {7 x^2+5}}-\frac {\sqrt {14} \sqrt {3 x^2+2} E\left (\arctan \left (\sqrt {\frac {7}{5}} x\right )|-\frac {1}{14}\right )}{5 \sqrt {\frac {3 x^2+2}{7 x^2+5}} \sqrt {7 x^2+5}}\) |
Input:
Int[(-7*Sqrt[2 + 3*x^2])/(5 + 7*x^2)^(3/2) + 3/(Sqrt[2 + 3*x^2]*Sqrt[5 + 7 *x^2]),x]
Output:
-1/5*(Sqrt[14]*Sqrt[2 + 3*x^2]*EllipticE[ArcTan[Sqrt[7/5]*x], -1/14])/(Sqr t[(2 + 3*x^2)/(5 + 7*x^2)]*Sqrt[5 + 7*x^2]) + (3*Sqrt[2 + 3*x^2]*EllipticF [ArcTan[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 = 4.44 (sec) , antiderivative size = 156, normalized size of antiderivative = 3.47
method | result | size |
default | \(-\frac {3 i \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {2}\, \sqrt {7}}{14}+\frac {\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}\, \left (i \sqrt {7}\, \sqrt {2}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}-i \sqrt {7}\, \sqrt {2}\, \operatorname {EllipticE}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {35}\, \sqrt {6}}{14}\right ) \sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}-21 x^{3}-14 x \right )}{105 x^{4}+145 x^{2}+50}\) | \(156\) |
elliptic | \(\frac {\sqrt {\left (3 x^{2}+2\right ) \left (7 x^{2}+5\right )}\, \left (-\frac {3 i \sqrt {35}\, \sqrt {35 x^{2}+25}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )}{70 \sqrt {21 x^{4}+29 x^{2}+10}}+\frac {i \sqrt {35}\, \sqrt {35 x^{2}+25}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {35}}{5}, \frac {\sqrt {210}}{14}\right )\right )}{25 \sqrt {21 x^{4}+29 x^{2}+10}}-\frac {\left (21 x^{2}+14\right ) x}{5 \sqrt {\left (x^{2}+\frac {5}{7}\right ) \left (21 x^{2}+14\right )}}\right )}{\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}}\) | \(181\) |
Input:
int(-7*(3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+3/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2),x ,method=_RETURNVERBOSE)
Output:
-3/14*I*EllipticF(1/5*I*x*35^(1/2),1/14*35^(1/2)*6^(1/2))*2^(1/2)*7^(1/2)+ 1/5*(3*x^2+2)^(1/2)*(7*x^2+5)^(1/2)*(I*7^(1/2)*2^(1/2)*EllipticF(1/5*I*x*3 5^(1/2),1/14*35^(1/2)*6^(1/2))*(3*x^2+2)^(1/2)*(7*x^2+5)^(1/2)-I*7^(1/2)*2 ^(1/2)*EllipticE(1/5*I*x*35^(1/2),1/14*35^(1/2)*6^(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 = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.76 \[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\frac {98 \, \sqrt {10} \sqrt {-\frac {7}{5}} {\left (7 \, x^{2} + 5\right )} E(\arcsin \left (\sqrt {-\frac {7}{5}} x\right )\,|\,\frac {15}{14}) - 173 \, \sqrt {10} \sqrt {-\frac {7}{5}} {\left (7 \, x^{2} + 5\right )} F(\arcsin \left (\sqrt {-\frac {7}{5}} x\right )\,|\,\frac {15}{14}) - 490 \, \sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2} x}{350 \, {\left (7 \, x^{2} + 5\right )}} \] Input:
integrate(-7*(3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+3/(3*x^2+2)^(1/2)/(7*x^2+5)^( 1/2),x, algorithm="fricas")
Output:
1/350*(98*sqrt(10)*sqrt(-7/5)*(7*x^2 + 5)*elliptic_e(arcsin(sqrt(-7/5)*x), 15/14) - 173*sqrt(10)*sqrt(-7/5)*(7*x^2 + 5)*elliptic_f(arcsin(sqrt(-7/5) *x), 15/14) - 490*sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)*x)/(7*x^2 + 5)
\[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int \frac {1}{\sqrt {3 x^{2} + 2} \left (7 x^{2} + 5\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(-7*(3*x**2+2)**(1/2)/(7*x**2+5)**(3/2)+3/(3*x**2+2)**(1/2)/(7*x* *2+5)**(1/2),x)
Output:
Integral(1/(sqrt(3*x**2 + 2)*(7*x**2 + 5)**(3/2)), x)
\[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int { \frac {3}{\sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2}} - \frac {7 \, \sqrt {3 \, x^{2} + 2}}{{\left (7 \, x^{2} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(-7*(3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+3/(3*x^2+2)^(1/2)/(7*x^2+5)^( 1/2),x, algorithm="maxima")
Output:
integrate(3/(sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)) - 7*sqrt(3*x^2 + 2)/(7*x^2 + 5)^(3/2), x)
\[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int { \frac {3}{\sqrt {7 \, x^{2} + 5} \sqrt {3 \, x^{2} + 2}} - \frac {7 \, \sqrt {3 \, x^{2} + 2}}{{\left (7 \, x^{2} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(-7*(3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+3/(3*x^2+2)^(1/2)/(7*x^2+5)^( 1/2),x, algorithm="giac")
Output:
integrate(3/(sqrt(7*x^2 + 5)*sqrt(3*x^2 + 2)) - 7*sqrt(3*x^2 + 2)/(7*x^2 + 5)^(3/2), x)
Timed out. \[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int \frac {3}{\sqrt {3\,x^2+2}\,\sqrt {7\,x^2+5}}-\frac {7\,\sqrt {3\,x^2+2}}{{\left (7\,x^2+5\right )}^{3/2}} \,d x \] Input:
int(3/((3*x^2 + 2)^(1/2)*(7*x^2 + 5)^(1/2)) - (7*(3*x^2 + 2)^(1/2))/(7*x^2 + 5)^(3/2),x)
Output:
int(3/((3*x^2 + 2)^(1/2)*(7*x^2 + 5)^(1/2)) - (7*(3*x^2 + 2)^(1/2))/(7*x^2 + 5)^(3/2), x)
\[ \int \left (-\frac {7 \sqrt {2+3 x^2}}{\left (5+7 x^2\right )^{3/2}}+\frac {3}{\sqrt {2+3 x^2} \sqrt {5+7 x^2}}\right ) \, dx=\int \frac {\sqrt {3 x^{2}+2}\, \sqrt {7 x^{2}+5}}{147 x^{6}+308 x^{4}+215 x^{2}+50}d x \] Input:
int(-7*(3*x^2+2)^(1/2)/(7*x^2+5)^(3/2)+3/(3*x^2+2)^(1/2)/(7*x^2+5)^(1/2),x )
Output:
int((sqrt(3*x**2 + 2)*sqrt(7*x**2 + 5))/(147*x**6 + 308*x**4 + 215*x**2 + 50),x)