Integrand size = 34, antiderivative size = 98 \[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\frac {\sqrt {3-5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticPi}\left (\frac {6}{11},\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right ),\frac {9}{77}\right )}{\sqrt {77} \sqrt {\frac {3-5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}} \] Output:
1/77*(-5*x^2+3)^(1/2)*((11*x^2+7)/(2*x^2+1))^(1/2)*EllipticPi(1/3*33^(1/2) *x/(2*x^2+1)^(1/2),6/11,3/77*77^(1/2))*77^(1/2)/((-5*x^2+3)/(2*x^2+1))^(1/ 2)/(11*x^2+7)^(1/2)
Time = 1.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\frac {\sqrt {3-5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticPi}\left (\frac {6}{11},\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right ),\frac {9}{77}\right )}{\sqrt {77} \sqrt {\frac {3-5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}} \] Input:
Integrate[Sqrt[1 + 2*x^2]/(Sqrt[3 - 5*x^2]*Sqrt[7 + 11*x^2]),x]
Output:
(Sqrt[3 - 5*x^2]*Sqrt[(7 + 11*x^2)/(1 + 2*x^2)]*EllipticPi[6/11, ArcSin[(S qrt[11/3]*x)/Sqrt[1 + 2*x^2]], 9/77])/(Sqrt[77]*Sqrt[(3 - 5*x^2)/(1 + 2*x^ 2)]*Sqrt[7 + 11*x^2])
Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {428, 27, 412}
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.
\(\displaystyle \int \frac {\sqrt {2 x^2+1}}{\sqrt {3-5 x^2} \sqrt {11 x^2+7}} \, dx\) |
\(\Big \downarrow \) 428 |
\(\displaystyle \frac {\sqrt {3-5 x^2} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \int \frac {\sqrt {21}}{\sqrt {3-\frac {11 x^2}{2 x^2+1}} \sqrt {7-\frac {3 x^2}{2 x^2+1}} \left (1-\frac {2 x^2}{2 x^2+1}\right )}d\frac {x}{\sqrt {2 x^2+1}}}{\sqrt {21} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \sqrt {11 x^2+7}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {3-5 x^2} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \int \frac {1}{\sqrt {3-\frac {11 x^2}{2 x^2+1}} \sqrt {7-\frac {3 x^2}{2 x^2+1}} \left (1-\frac {2 x^2}{2 x^2+1}\right )}d\frac {x}{\sqrt {2 x^2+1}}}{\sqrt {\frac {3-5 x^2}{2 x^2+1}} \sqrt {11 x^2+7}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\sqrt {3-5 x^2} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticPi}\left (\frac {6}{11},\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {2 x^2+1}}\right ),\frac {9}{77}\right )}{\sqrt {77} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \sqrt {11 x^2+7}}\) |
Input:
Int[Sqrt[1 + 2*x^2]/(Sqrt[3 - 5*x^2]*Sqrt[7 + 11*x^2]),x]
Output:
(Sqrt[3 - 5*x^2]*Sqrt[(7 + 11*x^2)/(1 + 2*x^2)]*EllipticPi[6/11, ArcSin[(S qrt[11/3]*x)/Sqrt[1 + 2*x^2]], 9/77])/(Sqrt[77]*Sqrt[(3 - 5*x^2)/(1 + 2*x^ 2)]*Sqrt[7 + 11*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* (x_)^2]), x_Symbol] :> Simp[a*Sqrt[c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + b*x^2)))]/(c*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))])) Subs t[Int[1/((1 - b*x^2)*Sqrt[1 - (b*c - a*d)*(x^2/c)]*Sqrt[1 - (b*e - a*f)*(x^ 2/e)]), x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[{a, b, c, d, e, f}, x]
\[\int \frac {\sqrt {2 x^{2}+1}}{\sqrt {-5 x^{2}+3}\, \sqrt {11 x^{2}+7}}d x\]
Input:
int((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x)
Output:
int((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x)
\[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + 1}}{\sqrt {11 \, x^{2} + 7} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x, algorithm=" fricas")
Output:
integral(-sqrt(11*x^2 + 7)*sqrt(2*x^2 + 1)*sqrt(-5*x^2 + 3)/(55*x^4 + 2*x^ 2 - 21), x)
\[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\int \frac {\sqrt {2 x^{2} + 1}}{\sqrt {3 - 5 x^{2}} \sqrt {11 x^{2} + 7}}\, dx \] Input:
integrate((2*x**2+1)**(1/2)/(-5*x**2+3)**(1/2)/(11*x**2+7)**(1/2),x)
Output:
Integral(sqrt(2*x**2 + 1)/(sqrt(3 - 5*x**2)*sqrt(11*x**2 + 7)), x)
\[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + 1}}{\sqrt {11 \, x^{2} + 7} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x, algorithm=" maxima")
Output:
integrate(sqrt(2*x^2 + 1)/(sqrt(11*x^2 + 7)*sqrt(-5*x^2 + 3)), x)
\[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + 1}}{\sqrt {11 \, x^{2} + 7} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x, algorithm=" giac")
Output:
integrate(sqrt(2*x^2 + 1)/(sqrt(11*x^2 + 7)*sqrt(-5*x^2 + 3)), x)
Timed out. \[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=\int \frac {\sqrt {2\,x^2+1}}{\sqrt {3-5\,x^2}\,\sqrt {11\,x^2+7}} \,d x \] Input:
int((2*x^2 + 1)^(1/2)/((3 - 5*x^2)^(1/2)*(11*x^2 + 7)^(1/2)),x)
Output:
int((2*x^2 + 1)^(1/2)/((3 - 5*x^2)^(1/2)*(11*x^2 + 7)^(1/2)), x)
\[ \int \frac {\sqrt {1+2 x^2}}{\sqrt {3-5 x^2} \sqrt {7+11 x^2}} \, dx=-\left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {11 x^{2}+7}\, \sqrt {-5 x^{2}+3}}{55 x^{4}+2 x^{2}-21}d x \right ) \] Input:
int((2*x^2+1)^(1/2)/(-5*x^2+3)^(1/2)/(11*x^2+7)^(1/2),x)
Output:
- int((sqrt(2*x**2 + 1)*sqrt(11*x**2 + 7)*sqrt( - 5*x**2 + 3))/(55*x**4 + 2*x**2 - 21),x)