Integrand size = 25, antiderivative size = 339 \[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=-\frac {\sqrt {7+x} \sqrt {3+2 x+5 x^2}}{3 x}+\frac {\sqrt {130} \sqrt {7+x} \sqrt {3+2 x+5 x^2}}{3 \left (78+\sqrt {130} (7+x)\right )}+\frac {11 \text {arctanh}\left (\frac {\sqrt {\frac {3}{7}} \sqrt {7+x}}{\sqrt {3+2 x+5 x^2}}\right )}{6 \sqrt {21}}-\frac {\sqrt [4]{130} \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right )|\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{\sqrt {3} \sqrt {3+2 x+5 x^2}}-\frac {\left (479-42 \sqrt {130}\right ) \sqrt {\frac {3+2 x+5 x^2}{\left (78+\sqrt {130} (7+x)\right )^2}} \left (78+\sqrt {130} (7+x)\right ) \operatorname {EllipticPi}\left (\frac {5460+479 \sqrt {130}}{10920},2 \arctan \left (\frac {\sqrt [4]{\frac {5}{26}} \sqrt {7+x}}{\sqrt {3}}\right ),\frac {1}{390} \left (195+17 \sqrt {130}\right )\right )}{84 \sqrt {3} \sqrt [4]{130} \sqrt {3+2 x+5 x^2}} \] Output:
-1/3*(7+x)^(1/2)*(5*x^2+2*x+3)^(1/2)/x+130^(1/2)*(7+x)^(1/2)*(5*x^2+2*x+3) ^(1/2)/(234+3*130^(1/2)*(7+x))+11/126*21^(1/2)*arctanh(1/7*21^(1/2)*(7+x)^ (1/2)/(5*x^2+2*x+3)^(1/2))-1/3*130^(1/4)*((5*x^2+2*x+3)/(78+130^(1/2)*(7+x ))^2)^(1/2)*(78+130^(1/2)*(7+x))*EllipticE(sin(2*arctan(1/78*5^(1/4)*26^(3 /4)*(7+x)^(1/2)*3^(1/2))),1/390*(76050+6630*130^(1/2))^(1/2))*3^(1/2)/(5*x ^2+2*x+3)^(1/2)-1/32760*(479-42*130^(1/2))*((5*x^2+2*x+3)/(78+130^(1/2)*(7 +x))^2)^(1/2)*(78+130^(1/2)*(7+x))*EllipticPi(sin(2*arctan(1/78*5^(1/4)*26 ^(3/4)*(7+x)^(1/2)*3^(1/2))),1/2+479/10920*130^(1/2),1/390*(76050+6630*130 ^(1/2))^(1/2))*3^(1/2)*130^(3/4)/(5*x^2+2*x+3)^(1/2)
Result contains complex when optimal does not.
Time = 24.45 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=-\frac {\sqrt {7+x} \sqrt {3+2 x+5 x^2}}{3 x}+\frac {(7+x)^{3/2} \left (2730 \sqrt {-\frac {i}{34 i+\sqrt {14}}}+\frac {127764 \sqrt {-\frac {i}{34 i+\sqrt {14}}}}{(7+x)^2}-\frac {37128 \sqrt {-\frac {i}{34 i+\sqrt {14}}}}{7+x}+\frac {14 \sqrt {13} \left (-17 i \sqrt {2}+\sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} E\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right )|\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}+\frac {2 i \sqrt {13} \left (2 \sqrt {2}+7 i \sqrt {7}\right ) \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}-\frac {11 i \sqrt {26} \sqrt {\frac {34 i+\sqrt {14}-\frac {234 i}{7+x}}{34 i+\sqrt {14}}} \sqrt {\frac {-34 i+\sqrt {14}+\frac {234 i}{7+x}}{-34 i+\sqrt {14}}} \operatorname {EllipticPi}\left (\frac {7}{234} \left (34-i \sqrt {14}\right ),i \text {arcsinh}\left (\frac {3 \sqrt {-\frac {26 i}{34 i+\sqrt {14}}}}{\sqrt {7+x}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{\sqrt {7+x}}\right )}{1638 \sqrt {-\frac {i}{34 i+\sqrt {14}}} \sqrt {3+2 x+5 x^2}} \] Input:
Integrate[Sqrt[7 + x]/(x^2*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
-1/3*(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/x + ((7 + x)^(3/2)*(2730*Sqrt[(-I )/(34*I + Sqrt[14])] + (127764*Sqrt[(-I)/(34*I + Sqrt[14])])/(7 + x)^2 - ( 37128*Sqrt[(-I)/(34*I + Sqrt[14])])/(7 + x) + (14*Sqrt[13]*((-17*I)*Sqrt[2 ] + Sqrt[7])*Sqrt[(34*I + Sqrt[14] - (234*I)/(7 + x))/(34*I + Sqrt[14])]*S qrt[(-34*I + Sqrt[14] + (234*I)/(7 + x))/(-34*I + Sqrt[14])]*EllipticE[I*A rcSinh[(3*Sqrt[(-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14]) /(34*I - Sqrt[14])])/Sqrt[7 + x] + ((2*I)*Sqrt[13]*(2*Sqrt[2] + (7*I)*Sqrt [7])*Sqrt[(34*I + Sqrt[14] - (234*I)/(7 + x))/(34*I + Sqrt[14])]*Sqrt[(-34 *I + Sqrt[14] + (234*I)/(7 + x))/(-34*I + Sqrt[14])]*EllipticF[I*ArcSinh[( 3*Sqrt[(-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/Sqrt[7 + x] - ((11*I)*Sqrt[26]*Sqrt[(34*I + Sqrt[14] - (234*I )/(7 + x))/(34*I + Sqrt[14])]*Sqrt[(-34*I + Sqrt[14] + (234*I)/(7 + x))/(- 34*I + Sqrt[14])]*EllipticPi[(7*(34 - I*Sqrt[14]))/234, I*ArcSinh[(3*Sqrt[ (-26*I)/(34*I + Sqrt[14])])/Sqrt[7 + x]], (34*I + Sqrt[14])/(34*I - Sqrt[1 4])])/Sqrt[7 + x]))/(1638*Sqrt[(-I)/(34*I + Sqrt[14])]*Sqrt[3 + 2*x + 5*x^ 2])
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {1285, 2154, 27, 1269, 1172, 321, 327, 1279, 27, 187, 413, 413, 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.
| \(\displaystyle \int \frac {\sqrt {x+7}}{x^2 \sqrt {5 x^2+2 x+3}} \, dx\) |
| \(\Big \downarrow \) 1285 |
| \(\displaystyle -\frac {1}{6} \int \frac {11-5 x^2}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle \frac {1}{6} \left (-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-\int -\frac {5 x}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (5 \int \frac {x}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{6} \left (5 \left (\int \frac {\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}dx-7 \int \frac {1}{\sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx\right )-\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+5 \left (\frac {2 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \int \frac {1}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1} \sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+5 \left (\frac {2 i \sqrt {x+7} \int \frac {\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{34 i+\sqrt {14}}+1}}{\sqrt {\frac {i \left (5 x+i \sqrt {14}+1\right )}{2 \sqrt {14}}+1}}d\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (-11 \int \frac {1}{x \sqrt {x+7} \sqrt {5 x^2+2 x+3}}dx+5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 1279 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )-\frac {22 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{2 x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )-\frac {11 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int \frac {1}{x \sqrt {x+7} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1}}dx}{\sqrt {5 x^2+2 x+3}}\right )\) |
| \(\Big \downarrow \) 187 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (\frac {22 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \int -\frac {1}{x \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3}}+5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (\frac {22 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {5 (x+7)+i \sqrt {14}-34} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34}}+5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (\frac {22 \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \int -\frac {1}{x \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}}}d\sqrt {x+7}}{\sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}+5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {\sqrt {x+7} \sqrt {5 x^2+2 x+3}}{3 x}+\frac {1}{6} \left (\frac {22 \sqrt {\frac {1}{5} \left (34-i \sqrt {14}\right )} \sqrt {5 x-i \sqrt {14}+1} \sqrt {5 x+i \sqrt {14}+1} \sqrt {1-\frac {5 (x+7)}{34-i \sqrt {14}}} \sqrt {1-\frac {5 (x+7)}{34+i \sqrt {14}}} \operatorname {EllipticPi}\left (\frac {1}{35} \left (34-i \sqrt {14}\right ),\arcsin \left (\frac {\sqrt {5} \sqrt {x+7}}{\sqrt {34-i \sqrt {14}}}\right ),\frac {34 i+\sqrt {14}}{34 i-\sqrt {14}}\right )}{7 \sqrt {5 x^2+2 x+3} \sqrt {5 (x+7)-i \sqrt {14}-34} \sqrt {5 (x+7)+i \sqrt {14}-34}}+5 \left (\frac {2 i \sqrt {x+7} E\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right )|\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{5 \sqrt {\frac {x+7}{34-i \sqrt {14}}}}-\frac {14 i \sqrt {\frac {x+7}{34-i \sqrt {14}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-i \left (5 x+i \sqrt {14}+1\right )}}{2^{3/4} \sqrt [4]{7}}\right ),\frac {2 \sqrt {14}}{34 i+\sqrt {14}}\right )}{\sqrt {x+7}}\right )\right )\) |
Input:
Int[Sqrt[7 + x]/(x^2*Sqrt[3 + 2*x + 5*x^2]),x]
Output:
-1/3*(Sqrt[7 + x]*Sqrt[3 + 2*x + 5*x^2])/x + (5*((((2*I)/5)*Sqrt[7 + x]*El lipticE[ArcSin[Sqrt[(-I)*(1 + I*Sqrt[14] + 5*x)]/(2^(3/4)*7^(1/4))], (2*Sq rt[14])/(34*I + Sqrt[14])])/Sqrt[(7 + x)/(34 - I*Sqrt[14])] - ((14*I)*Sqrt [(7 + x)/(34 - I*Sqrt[14])]*EllipticF[ArcSin[Sqrt[(-I)*(1 + I*Sqrt[14] + 5 *x)]/(2^(3/4)*7^(1/4))], (2*Sqrt[14])/(34*I + Sqrt[14])])/Sqrt[7 + x]) + ( 22*Sqrt[(34 - I*Sqrt[14])/5]*Sqrt[1 - I*Sqrt[14] + 5*x]*Sqrt[1 + I*Sqrt[14 ] + 5*x]*Sqrt[1 - (5*(7 + x))/(34 - I*Sqrt[14])]*Sqrt[1 - (5*(7 + x))/(34 + I*Sqrt[14])]*EllipticPi[(34 - I*Sqrt[14])/35, ArcSin[(Sqrt[5]*Sqrt[7 + x ])/Sqrt[34 - I*Sqrt[14]]], (34*I + Sqrt[14])/(34*I - Sqrt[14])])/(7*Sqrt[3 + 2*x + 5*x^2]*Sqrt[-34 - I*Sqrt[14] + 5*(7 + x)]*Sqrt[-34 + I*Sqrt[14] + 5*(7 + x)]))/6
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_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && !SimplerQ[e + f*x, c + d*x] && !SimplerQ[g + h*x, c + d*x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2]) Int[1/((d + e*x )*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ {a, b, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_.) + (b_.) *(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]* (Sqrt[a + b*x + c*x^2]/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*( m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqr t[a + b*x + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g + b*f*(2*m + 3)) - 2*(b* e*g*(2 + m) - c*(d*g*(m + 1) - e*f*(m + 2)))*x - c*e*g*(2*m + 5)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 3.56 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.10
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}\, \left (-\frac {\sqrt {5 x^{3}+37 x^{2}+17 x +21}}{3 x}+\frac {5 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{3 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {11 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{21 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right )}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(373\) |
| risch | \(-\frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}{3 x}+\frac {\left (\frac {11 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )}{21 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}+\frac {5 \left (\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}-\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\, \sqrt {\frac {x +\frac {1}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}+\frac {i \sqrt {14}}{5}}}\, \left (\left (-\frac {34}{5}-\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )+\left (-\frac {1}{5}+\frac {i \sqrt {14}}{5}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +7}{\frac {34}{5}-\frac {i \sqrt {14}}{5}}}, \sqrt {\frac {-\frac {34}{5}+\frac {i \sqrt {14}}{5}}{-\frac {34}{5}-\frac {i \sqrt {14}}{5}}}\right )\right )}{3 \sqrt {5 x^{3}+37 x^{2}+17 x +21}}\right ) \sqrt {\left (x +7\right ) \left (5 x^{2}+2 x +3\right )}}{\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}}\) | \(374\) |
| default | \(\frac {\sqrt {x +7}\, \sqrt {5 x^{2}+2 x +3}\, \left (245 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) x -11 i \sqrt {14}\, \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) x -8190 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) x -140 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) x +374 \sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}\, \sqrt {\frac {i \sqrt {14}-5 x -1}{i \sqrt {14}+34}}\, \sqrt {\frac {i \sqrt {14}+5 x +1}{-34+i \sqrt {14}}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {5 \left (x +7\right )}{-34+i \sqrt {14}}}, \frac {34}{35}-\frac {i \sqrt {14}}{35}, \sqrt {-\frac {-34+i \sqrt {14}}{i \sqrt {14}+34}}\right ) x -175 x^{3}-1295 x^{2}-595 x -735\right )}{105 x \left (5 x^{3}+37 x^{2}+17 x +21\right )}\) | \(619\) |
Input:
int((x+7)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
((x+7)*(5*x^2+2*x+3))^(1/2)/(x+7)^(1/2)/(5*x^2+2*x+3)^(1/2)*(-1/3/x*(5*x^3 +37*x^2+17*x+21)^(1/2)+5/3*(34/5-1/5*I*14^(1/2))*((x+7)/(34/5-1/5*I*14^(1/ 2)))^(1/2)*((x+1/5-1/5*I*14^(1/2))/(-34/5-1/5*I*14^(1/2)))^(1/2)*((x+1/5+1 /5*I*14^(1/2))/(-34/5+1/5*I*14^(1/2)))^(1/2)/(5*x^3+37*x^2+17*x+21)^(1/2)* ((-34/5-1/5*I*14^(1/2))*EllipticE(((x+7)/(34/5-1/5*I*14^(1/2)))^(1/2),((-3 4/5+1/5*I*14^(1/2))/(-34/5-1/5*I*14^(1/2)))^(1/2))+(-1/5+1/5*I*14^(1/2))*E llipticF(((x+7)/(34/5-1/5*I*14^(1/2)))^(1/2),((-34/5+1/5*I*14^(1/2))/(-34/ 5-1/5*I*14^(1/2)))^(1/2)))+11/21*(34/5-1/5*I*14^(1/2))*((x+7)/(34/5-1/5*I* 14^(1/2)))^(1/2)*((x+1/5-1/5*I*14^(1/2))/(-34/5-1/5*I*14^(1/2)))^(1/2)*((x +1/5+1/5*I*14^(1/2))/(-34/5+1/5*I*14^(1/2)))^(1/2)/(5*x^3+37*x^2+17*x+21)^ (1/2)*EllipticPi(((x+7)/(34/5-1/5*I*14^(1/2)))^(1/2),34/35-1/35*I*14^(1/2) ,((-34/5+1/5*I*14^(1/2))/(-34/5-1/5*I*14^(1/2)))^(1/2)))
\[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7}}{\sqrt {5 \, x^{2} + 2 \, x + 3} x^{2}} \,d x } \] Input:
integrate((7+x)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(5*x^2 + 2*x + 3)*sqrt(x + 7)/(5*x^4 + 2*x^3 + 3*x^2), x)
\[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x + 7}}{x^{2} \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:
integrate((7+x)**(1/2)/x**2/(5*x**2+2*x+3)**(1/2),x)
Output:
Integral(sqrt(x + 7)/(x**2*sqrt(5*x**2 + 2*x + 3)), x)
\[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7}}{\sqrt {5 \, x^{2} + 2 \, x + 3} x^{2}} \,d x } \] Input:
integrate((7+x)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + 7)/(sqrt(5*x^2 + 2*x + 3)*x^2), x)
\[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int { \frac {\sqrt {x + 7}}{\sqrt {5 \, x^{2} + 2 \, x + 3} x^{2}} \,d x } \] Input:
integrate((7+x)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x + 7)/(sqrt(5*x^2 + 2*x + 3)*x^2), x)
Timed out. \[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x+7}}{x^2\,\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:
int((x + 7)^(1/2)/(x^2*(2*x + 5*x^2 + 3)^(1/2)),x)
Output:
int((x + 7)^(1/2)/(x^2*(2*x + 5*x^2 + 3)^(1/2)), x)
\[ \int \frac {\sqrt {7+x}}{x^2 \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\sqrt {x +7}}{x^{2} \sqrt {5 x^{2}+2 x +3}}d x \] Input:
int((7+x)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x)
Output:
int((7+x)^(1/2)/x^2/(5*x^2+2*x+3)^(1/2),x)