Integrand size = 28, antiderivative size = 550 \[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\frac {\sqrt {\frac {5}{3}} (4+5 x) \sqrt {\frac {1+x^2}{(4+5 x)^2}} \text {arctanh}\left (\frac {2 \sqrt {2+3 x}}{\sqrt {15} \sqrt {4+5 x} \sqrt {13+\frac {41 (2+3 x)^2}{(4+5 x)^2}-\frac {46 (2+3 x)}{4+5 x}}}\right )}{\sqrt {1+x^2}}+\frac {2\ 41^{3/4} (4+5 x) \sqrt {\frac {1+x^2}{(4+5 x)^2}} \sqrt {\frac {13+\frac {41 (2+3 x)^2}{(4+5 x)^2}-\frac {46 (2+3 x)}{4+5 x}}{\left (13+\frac {\sqrt {533} (2+3 x)}{4+5 x}\right )^2}} \left (13+\frac {\sqrt {533} (2+3 x)}{4+5 x}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {41}{13}} \sqrt {2+3 x}}{\sqrt {4+5 x}}\right ),\frac {533+23 \sqrt {533}}{1066}\right )}{\sqrt [4]{13} \left (123+5 \sqrt {533}\right ) \sqrt {1+x^2} \sqrt {13+\frac {41 (2+3 x)^2}{(4+5 x)^2}-\frac {46 (2+3 x)}{4+5 x}}}+\frac {\sqrt [4]{41} \left (65-3 \sqrt {533}\right ) (4+5 x) \sqrt {\frac {1+x^2}{(4+5 x)^2}} \sqrt {\frac {13+\frac {41 (2+3 x)^2}{(4+5 x)^2}-\frac {46 (2+3 x)}{4+5 x}}{\left (13+\frac {\sqrt {533} (2+3 x)}{4+5 x}\right )^2}} \left (13+\frac {\sqrt {533} (2+3 x)}{4+5 x}\right ) \operatorname {EllipticPi}\left (\frac {7995+347 \sqrt {533}}{15990},2 \arctan \left (\frac {\sqrt [4]{\frac {41}{13}} \sqrt {2+3 x}}{\sqrt {4+5 x}}\right ),\frac {533+23 \sqrt {533}}{1066}\right )}{3\ 13^{3/4} \left (123+5 \sqrt {533}\right ) \sqrt {1+x^2} \sqrt {13+\frac {41 (2+3 x)^2}{(4+5 x)^2}-\frac {46 (2+3 x)}{4+5 x}}} \] Output:
1/3*15^(1/2)*(4+5*x)*((x^2+1)/(4+5*x)^2)^(1/2)*arctanh(2/15*(2+3*x)^(1/2)* 15^(1/2)/(4+5*x)^(1/2)/(13+41*(2+3*x)^2/(4+5*x)^2-46*(2+3*x)/(4+5*x))^(1/2 ))/(x^2+1)^(1/2)+2/13*41^(3/4)*(4+5*x)*((x^2+1)/(4+5*x)^2)^(1/2)*((13+41*( 2+3*x)^2/(4+5*x)^2-46*(2+3*x)/(4+5*x))/(13+533^(1/2)*(2+3*x)/(4+5*x))^2)^( 1/2)*(13+533^(1/2)*(2+3*x)/(4+5*x))*InverseJacobiAM(2*arctan(1/13*13^(3/4) *41^(1/4)/(4+5*x)^(1/2)*(2+3*x)^(1/2)),1/1066*(568178+24518*533^(1/2))^(1/ 2))*13^(3/4)/(5*533^(1/2)+123)/(x^2+1)^(1/2)/(13+41*(2+3*x)^2/(4+5*x)^2-46 *(2+3*x)/(4+5*x))^(1/2)+1/39*41^(1/4)*(65-3*533^(1/2))*(4+5*x)*((x^2+1)/(4 +5*x)^2)^(1/2)*((13+41*(2+3*x)^2/(4+5*x)^2-46*(2+3*x)/(4+5*x))/(13+533^(1/ 2)*(2+3*x)/(4+5*x))^2)^(1/2)*(13+533^(1/2)*(2+3*x)/(4+5*x))*EllipticPi(sin (2*arctan(1/13*13^(3/4)*41^(1/4)/(4+5*x)^(1/2)*(2+3*x)^(1/2))),1/2+347/159 90*533^(1/2),1/1066*(568178+24518*533^(1/2))^(1/2))*13^(1/4)/(5*533^(1/2)+ 123)/(x^2+1)^(1/2)/(13+41*(2+3*x)^2/(4+5*x)^2-46*(2+3*x)/(4+5*x))^(1/2)
Result contains complex when optimal does not.
Time = 15.49 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=-\frac {4 \sqrt {\frac {(4+5 i) (i+x)}{2+3 x}} \sqrt {4+5 x} \left ((23+2 i) \sqrt {\frac {(4-5 i) (-i+x)}{2+3 x}} (4+5 x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(23+2 i) (4+5 x)}{2+3 x}}}{\sqrt {41}}\right ),\frac {525}{533}-\frac {92 i}{533}\right )-(2+3 x) \sqrt {\frac {(23+2 i) (4+5 x)}{2+3 x}} \sqrt {\frac {(102-107 i) \left (-4 i+(4-5 i) x+5 x^2\right )}{(2+3 x)^2}} \operatorname {EllipticPi}\left (\frac {69}{65}-\frac {6 i}{65},\arcsin \left (\frac {\sqrt {\frac {(23+2 i) (4+5 x)}{2+3 x}}}{\sqrt {41}}\right ),\frac {525}{533}-\frac {92 i}{533}\right )\right )}{3 \sqrt {41} \sqrt {2+3 x} \left (\frac {(23+2 i) (4+5 x)}{2+3 x}\right )^{3/2} \sqrt {1+x^2}} \] Input:
Integrate[Sqrt[4 + 5*x]/(Sqrt[2 + 3*x]*Sqrt[1 + x^2]),x]
Output:
(-4*Sqrt[((4 + 5*I)*(I + x))/(2 + 3*x)]*Sqrt[4 + 5*x]*((23 + 2*I)*Sqrt[((4 - 5*I)*(-I + x))/(2 + 3*x)]*(4 + 5*x)*EllipticF[ArcSin[Sqrt[((23 + 2*I)*( 4 + 5*x))/(2 + 3*x)]/Sqrt[41]], 525/533 - (92*I)/533] - (2 + 3*x)*Sqrt[((2 3 + 2*I)*(4 + 5*x))/(2 + 3*x)]*Sqrt[((102 - 107*I)*(-4*I + (4 - 5*I)*x + 5 *x^2))/(2 + 3*x)^2]*EllipticPi[69/65 - (6*I)/65, ArcSin[Sqrt[((23 + 2*I)*( 4 + 5*x))/(2 + 3*x)]/Sqrt[41]], 525/533 - (92*I)/533]))/(3*Sqrt[41]*Sqrt[2 + 3*x]*(((23 + 2*I)*(4 + 5*x))/(2 + 3*x))^(3/2)*Sqrt[1 + x^2])
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.22, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {726}
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.
| \(\displaystyle \int \frac {\sqrt {5 x+4}}{\sqrt {3 x+2} \sqrt {x^2+1}} \, dx\) |
| \(\Big \downarrow \) 726 |
| \(\displaystyle \frac {4 \sqrt {\frac {(3+2 i) (1+i x)}{5 x+4}} \sqrt {-\frac {(2+3 i) (x+i)}{5 x+4}} (5 x+4) \operatorname {EllipticPi}\left (\frac {115}{123}-\frac {10 i}{123},\arcsin \left (\frac {\sqrt {41} \sqrt {3 x+2}}{\sqrt {23-2 i} \sqrt {5 x+4}}\right ),\frac {525}{533}-\frac {92 i}{533}\right )}{3 \sqrt {13} \sqrt {23+2 i} \sqrt {x^2+1}}\) |
Input:
Int[Sqrt[4 + 5*x]/(Sqrt[2 + 3*x]*Sqrt[1 + x^2]),x]
Output:
(4*Sqrt[((3 + 2*I)*(1 + I*x))/(4 + 5*x)]*Sqrt[((-2 - 3*I)*(I + x))/(4 + 5* x)]*(4 + 5*x)*EllipticPi[115/123 - (10*I)/123, ArcSin[(Sqrt[41]*Sqrt[2 + 3 *x])/(Sqrt[23 - 2*I]*Sqrt[4 + 5*x])], 525/533 - (92*I)/533])/(3*Sqrt[13]*S qrt[23 + 2*I]*Sqrt[1 + x^2])
Int[Sqrt[(d_.) + (e_.)*(x_)]/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x _)^2]), x_Symbol] :> With[{q = Rt[-4*a*c, 2]}, Simp[Sqrt[2]*Sqrt[2*c*f - g* q]*Sqrt[-q + 2*c*x]*(d + e*x)*Sqrt[(e*f - d*g)*((q + 2*c*x)/((2*c*f - g*q)* (d + e*x)))]*(Sqrt[(e*f - d*g)*((2*a + q*x)/((q*f - 2*a*g)*(d + e*x)))]/(g* Sqrt[2*c*d - e*q]*Sqrt[2*a*(c/q) + c*x]*Sqrt[a + c*x^2]))*EllipticPi[e*((2* c*f - g*q)/(g*(2*c*d - e*q))), ArcSin[Sqrt[2*c*d - e*q]*(Sqrt[f + g*x]/(Sqr t[2*c*f - g*q]*Sqrt[d + e*x]))], (q*d - 2*a*e)*((2*c*f - g*q)/((q*f - 2*a*g )*(2*c*d - e*q)))], x]] /; FreeQ[{a, c, d, e, f, g}, x]
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.29
| method | result | size |
| default | \(\frac {\left (\frac {46}{123}-\frac {4 i}{123}\right ) \left (5 x +4\right )^{\frac {5}{2}} \sqrt {3 x +2}\, \sqrt {x^{2}+1}\, \sqrt {\frac {\left (\frac {23}{13}+\frac {2 i}{13}\right ) \left (3 x +2\right )}{5 x +4}}\, \sqrt {\frac {\left (\frac {4}{13}-\frac {6 i}{13}\right ) \left (-x +i\right )}{5 x +4}}\, \sqrt {\frac {\left (-\frac {4}{13}-\frac {6 i}{13}\right ) \left (x +i\right )}{5 x +4}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {23}{13}+\frac {2 i}{13}\right ) \left (3 x +2\right )}{5 x +4}}, \frac {115}{123}-\frac {10 i}{123}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{\sqrt {15 x^{4}+22 x^{3}+23 x^{2}+22 x +8}\, \sqrt {-\left (3 x +2\right ) \left (5 x +4\right ) \left (-x +i\right ) \left (x +i\right )}}\) | \(160\) |
| elliptic | \(\frac {\sqrt {\left (3 x +2\right ) \left (5 x +4\right ) \left (x^{2}+1\right )}\, \left (\frac {\left (\frac {460}{123}-\frac {40 i}{123}\right ) \sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}\, \left (x +\frac {4}{5}\right )^{2} \sqrt {\frac {\left (-\frac {4}{65}+\frac {6 i}{65}\right ) \left (x -i\right )}{x +\frac {4}{5}}}\, \sqrt {\frac {\left (-\frac {4}{65}-\frac {6 i}{65}\right ) \left (x +i\right )}{x +\frac {4}{5}}}\, \sqrt {15}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{\sqrt {\left (x +\frac {2}{3}\right ) \left (x +\frac {4}{5}\right ) \left (x -i\right ) \left (x +i\right )}}+\frac {\left (\frac {575}{123}-\frac {50 i}{123}\right ) \sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}\, \left (x +\frac {4}{5}\right )^{2} \sqrt {\frac {\left (-\frac {4}{65}+\frac {6 i}{65}\right ) \left (x -i\right )}{x +\frac {4}{5}}}\, \sqrt {\frac {\left (-\frac {4}{65}-\frac {6 i}{65}\right ) \left (x +i\right )}{x +\frac {4}{5}}}\, \sqrt {15}\, \left (-\frac {4 \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{5}+\frac {2 \operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}, \frac {115}{123}-\frac {10 i}{123}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{15}\right )}{\sqrt {\left (x +\frac {2}{3}\right ) \left (x +\frac {4}{5}\right ) \left (x -i\right ) \left (x +i\right )}}\right )}{\sqrt {3 x +2}\, \sqrt {5 x +4}\, \sqrt {x^{2}+1}}\) | \(277\) |
Input:
int((5*x+4)^(1/2)/(3*x+2)^(1/2)/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
(46/123-4/123*I)*(5*x+4)^(5/2)*(3*x+2)^(1/2)*(x^2+1)^(1/2)*((23/13+2/13*I) *(3*x+2)/(5*x+4))^(1/2)*((4/13-6/13*I)*(-x+I)/(5*x+4))^(1/2)*((-4/13-6/13* I)*(x+I)/(5*x+4))^(1/2)*EllipticPi(((23/13+2/13*I)*(3*x+2)/(5*x+4))^(1/2), 115/123-10/123*I,23/533*533^(1/2)-2/533*I*533^(1/2))/(15*x^4+22*x^3+23*x^2 +22*x+8)^(1/2)/(-(3*x+2)*(5*x+4)*(-x+I)*(x+I))^(1/2)
\[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 4}}{\sqrt {x^{2} + 1} \sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((4+5*x)^(1/2)/(2+3*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(x^2 + 1)*sqrt(5*x + 4)*sqrt(3*x + 2)/(3*x^3 + 2*x^2 + 3*x + 2), x)
\[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {5 x + 4}}{\sqrt {3 x + 2} \sqrt {x^{2} + 1}}\, dx \] Input:
integrate((4+5*x)**(1/2)/(2+3*x)**(1/2)/(x**2+1)**(1/2),x)
Output:
Integral(sqrt(5*x + 4)/(sqrt(3*x + 2)*sqrt(x**2 + 1)), x)
\[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 4}}{\sqrt {x^{2} + 1} \sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((4+5*x)^(1/2)/(2+3*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(5*x + 4)/(sqrt(x^2 + 1)*sqrt(3*x + 2)), x)
\[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int { \frac {\sqrt {5 \, x + 4}}{\sqrt {x^{2} + 1} \sqrt {3 \, x + 2}} \,d x } \] Input:
integrate((4+5*x)^(1/2)/(2+3*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(5*x + 4)/(sqrt(x^2 + 1)*sqrt(3*x + 2)), x)
Timed out. \[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {5\,x+4}}{\sqrt {3\,x+2}\,\sqrt {x^2+1}} \,d x \] Input:
int((5*x + 4)^(1/2)/((3*x + 2)^(1/2)*(x^2 + 1)^(1/2)),x)
Output:
int((5*x + 4)^(1/2)/((3*x + 2)^(1/2)*(x^2 + 1)^(1/2)), x)
\[ \int \frac {\sqrt {4+5 x}}{\sqrt {2+3 x} \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +4}\, \sqrt {x^{2}+1}}{3 x^{3}+2 x^{2}+3 x +2}d x \] Input:
int((4+5*x)^(1/2)/(2+3*x)^(1/2)/(x^2+1)^(1/2),x)
Output:
int((sqrt(3*x + 2)*sqrt(5*x + 4)*sqrt(x**2 + 1))/(3*x**3 + 2*x**2 + 3*x + 2),x)