\(\int \frac {1}{\sqrt {x} (a+b x^2) (c+d x^2)} \, dx\) [790]

Optimal result

Integrand size = 24, antiderivative size = 332 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {b^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)}+\frac {b^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)}+\frac {d^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} (b c-a d)}-\frac {d^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{3/4} (b c-a d)}+\frac {b^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{3/4} (b c-a d)}-\frac {d^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{3/4} (b c-a d)} \] Output:

-1/2*b^(3/4)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(3/4)/(-a 
*d+b*c)+1/2*b^(3/4)*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(3 
/4)/(-a*d+b*c)+1/2*d^(3/4)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/ 
2)/c^(3/4)/(-a*d+b*c)-1/2*d^(3/4)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4) 
)*2^(1/2)/c^(3/4)/(-a*d+b*c)+1/2*b^(3/4)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x 
^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(3/4)/(-a*d+b*c)-1/2*d^(3/4)*arctanh 
(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(3/4)/(-a* 
d+b*c)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-b^{3/4} c^{3/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+a^{3/4} d^{3/4} \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+b^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )-a^{3/4} d^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} a^{3/4} c^{3/4} (b c-a d)} \] Input:

Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

(-(b^(3/4)*c^(3/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*S 
qrt[x])]) + a^(3/4)*d^(3/4)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)* 
d^(1/4)*Sqrt[x])] + b^(3/4)*c^(3/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[ 
x])/(Sqrt[a] + Sqrt[b]*x)] - a^(3/4)*d^(3/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1 
/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(Sqrt[2]*a^(3/4)*c^(3/4)*(b*c - a*d))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {368, 917, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle 2 \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}\)

\(\Big \downarrow \) 917

\(\displaystyle 2 \left (\frac {b \int \frac {1}{b x^2+a}d\sqrt {x}}{b c-a d}-\frac {d \int \frac {1}{d x^2+c}d\sqrt {x}}{b c-a d}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle 2 \left (\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle 2 \left (\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 2 \left (\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\frac {b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle 2 \left (\frac {b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b c-a d}-\frac {d \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{b c-a d}\right )\)

Input:

Int[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)),x]
 

Output:

2*((b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b 
^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)* 
b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x 
] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b 
^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/(b* 
c - a*d) - (d*((-(ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]*c 
^(1/4)*d^(1/4))) + ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)]/(Sqrt[2]* 
c^(1/4)*d^(1/4)))/(2*Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4 
)*Sqrt[x] + Sqrt[d]*x]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c 
^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2*Sqrt[c 
])))/(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 368

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[x^(k*(m + 1) 
 - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], 
 x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m 
] && IntegerQ[p]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 917

Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Sim 
p[b/(b*c - a*d)   Int[1/(a + b*x^n), x], x] - Simp[d/(b*c - a*d)   Int[1/(c 
 + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) c}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) a}\)	\(234\)
default	\(\frac {d \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) c}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) a}\)	\(234\)

Input:

int(1/x^(1/2)/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

1/4*d/(a*d-b*c)*(c/d)^(1/4)/c*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+( 
c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/ 
(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1))-1/4*b/(a*d 
-b*c)*(a/b)^(1/4)/a*2^(1/2)*(ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2) 
)/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4 
)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 1187, normalized size of antiderivative = 3.58 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^(1/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")
 

Output:

1/2*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d 
^3 + a^7*d^4))^(1/4)*log(b*sqrt(x) + (a*b*c - a^2*d)*(-b^3/(a^3*b^4*c^4 - 
4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)) - 1 
/2*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^ 
3 + a^7*d^4))^(1/4)*log(b*sqrt(x) - (a*b*c - a^2*d)*(-b^3/(a^3*b^4*c^4 - 4 
*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)) - 1/ 
2*I*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d 
^3 + a^7*d^4))^(1/4)*log(b*sqrt(x) - (I*a*b*c - I*a^2*d)*(-b^3/(a^3*b^4*c^ 
4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^(1/4)) 
 + 1/2*I*(-b^3/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6* 
b*c*d^3 + a^7*d^4))^(1/4)*log(b*sqrt(x) - (-I*a*b*c + I*a^2*d)*(-b^3/(a^3* 
b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4))^ 
(1/4)) - 1/2*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b* 
c^4*d^3 + a^4*c^3*d^4))^(1/4)*log(d*sqrt(x) + (b*c^2 - a*c*d)*(-d^3/(b^4*c 
^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4))^( 
1/4)) + 1/2*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c 
^4*d^3 + a^4*c^3*d^4))^(1/4)*log(d*sqrt(x) - (b*c^2 - a*c*d)*(-d^3/(b^4*c^ 
7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4))^(1 
/4)) + 1/2*I*(-d^3/(b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b* 
c^4*d^3 + a^4*c^3*d^4))^(1/4)*log(d*sqrt(x) - (I*b*c^2 - I*a*c*d)*(-d^3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**(1/2)/(b*x**2+a)/(d*x**2+c),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\frac {2 \, \sqrt {2} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (b c - a d\right )}} - \frac {\frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} d^{\frac {3}{4}} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}}} - \frac {\sqrt {2} d^{\frac {3}{4}} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}}}}{4 \, {\left (b c - a d\right )}} \] Input:

integrate(1/x^(1/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")
 

Output:

1/4*(2*sqrt(2)*b*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*s 
qrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2) 
*b*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt( 
sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(3/4)*log(sq 
rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^( 
3/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3/4))/ 
(b*c - a*d) - 1/4*(2*sqrt(2)*d*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) 
 + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d) 
)) + 2*sqrt(2)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)* 
sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)* 
d^(3/4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/c^(3/4) 
 - sqrt(2)*d^(3/4)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt 
(c))/c^(3/4))/(b*c - a*d)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b c - \sqrt {2} a^{2} d} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b c - \sqrt {2} a^{2} d} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{2} - \sqrt {2} a c d} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{2} - \sqrt {2} a c d} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b c - \sqrt {2} a^{2} d\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b c - \sqrt {2} a^{2} d\right )}} - \frac {\left (c d^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{2} - \sqrt {2} a c d\right )}} + \frac {\left (c d^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{2} - \sqrt {2} a c d\right )}} \] Input:

integrate(1/x^(1/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")
 

Output:

(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^( 
1/4))/(sqrt(2)*a*b*c - sqrt(2)*a^2*d) + (a*b^3)^(1/4)*arctan(-1/2*sqrt(2)* 
(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b*c - sqrt(2)*a^ 
2*d) - (c*d^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/ 
(c/d)^(1/4))/(sqrt(2)*b*c^2 - sqrt(2)*a*c*d) - (c*d^3)^(1/4)*arctan(-1/2*s 
qrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b*c^2 - sqr 
t(2)*a*c*d) + 1/2*(a*b^3)^(1/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt 
(a/b))/(sqrt(2)*a*b*c - sqrt(2)*a^2*d) - 1/2*(a*b^3)^(1/4)*log(-sqrt(2)*sq 
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b*c - sqrt(2)*a^2*d) - 1/2*( 
c*d^3)^(1/4)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c 
^2 - sqrt(2)*a*c*d) + 1/2*(c*d^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + 
 x + sqrt(c/d))/(sqrt(2)*b*c^2 - sqrt(2)*a*c*d)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 2.12 (sec) , antiderivative size = 8785, normalized size of antiderivative = 26.46 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \] Input:

int(1/(x^(1/2)*(a + b*x^2)*(c + d*x^2)),x)
 

Output:

- atan(((-d^3/(16*b^4*c^7 + 16*a^4*c^3*d^4 - 64*a^3*b*c^4*d^3 + 96*a^2*b^2 
*c^5*d^2 - 64*a*b^3*c^6*d))^(1/4)*((-d^3/(16*b^4*c^7 + 16*a^4*c^3*d^4 - 64 
*a^3*b*c^4*d^3 + 96*a^2*b^2*c^5*d^2 - 64*a*b^3*c^6*d))^(1/4)*(((-d^3/(16*b 
^4*c^7 + 16*a^4*c^3*d^4 - 64*a^3*b*c^4*d^3 + 96*a^2*b^2*c^5*d^2 - 64*a*b^3 
*c^6*d))^(1/4)*(8192*a*b^11*c^8*d^4 + 8192*a^8*b^4*c*d^11 - 40960*a^2*b^10 
*c^7*d^5 + 73728*a^3*b^9*c^6*d^6 - 40960*a^4*b^8*c^5*d^7 - 40960*a^5*b^7*c 
^4*d^8 + 73728*a^6*b^6*c^3*d^9 - 40960*a^7*b^5*c^2*d^10) + x^(1/2)*(4096*a 
^7*b^4*d^11 + 4096*b^11*c^7*d^4 - 16384*a*b^10*c^6*d^5 - 16384*a^6*b^5*c*d 
^10 + 24576*a^2*b^9*c^5*d^6 - 12288*a^3*b^8*c^4*d^7 - 12288*a^4*b^7*c^3*d^ 
8 + 24576*a^5*b^6*c^2*d^9))*(-d^3/(16*b^4*c^7 + 16*a^4*c^3*d^4 - 64*a^3*b* 
c^4*d^3 + 96*a^2*b^2*c^5*d^2 - 64*a*b^3*c^6*d))^(3/4) - 512*a^2*b^6*d^8 - 
512*b^8*c^2*d^6 + 1024*a*b^7*c*d^7) + 512*b^7*d^7*x^(1/2))*1i - (-d^3/(16* 
b^4*c^7 + 16*a^4*c^3*d^4 - 64*a^3*b*c^4*d^3 + 96*a^2*b^2*c^5*d^2 - 64*a*b^ 
3*c^6*d))^(1/4)*((-d^3/(16*b^4*c^7 + 16*a^4*c^3*d^4 - 64*a^3*b*c^4*d^3 + 9 
6*a^2*b^2*c^5*d^2 - 64*a*b^3*c^6*d))^(1/4)*(((-d^3/(16*b^4*c^7 + 16*a^4*c^ 
3*d^4 - 64*a^3*b*c^4*d^3 + 96*a^2*b^2*c^5*d^2 - 64*a*b^3*c^6*d))^(1/4)*(81 
92*a*b^11*c^8*d^4 + 8192*a^8*b^4*c*d^11 - 40960*a^2*b^10*c^7*d^5 + 73728*a 
^3*b^9*c^6*d^6 - 40960*a^4*b^8*c^5*d^7 - 40960*a^5*b^7*c^4*d^8 + 73728*a^6 
*b^6*c^3*d^9 - 40960*a^7*b^5*c^2*d^10) - x^(1/2)*(4096*a^7*b^4*d^11 + 4096 
*b^11*c^7*d^4 - 16384*a*b^10*c^6*d^5 - 16384*a^6*b^5*c*d^10 + 24576*a^2...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\sqrt {2}\, \left (2 b^{\frac {3}{4}} a^{\frac {1}{4}} \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -2 b^{\frac {3}{4}} a^{\frac {1}{4}} \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -2 d^{\frac {3}{4}} c^{\frac {1}{4}} \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a +2 d^{\frac {3}{4}} c^{\frac {1}{4}} \mathit {atan} \left (\frac {d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {d}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a +b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -d^{\frac {3}{4}} c^{\frac {1}{4}} \mathrm {log}\left (-\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a +d^{\frac {3}{4}} c^{\frac {1}{4}} \mathrm {log}\left (\sqrt {x}\, d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}+\sqrt {d}\, x \right ) a \right )}{4 a c \left (a d -b c \right )} \] Input:

int(1/x^(1/2)/(b*x^2+a)/(d*x^2+c),x)
 

Output:

(sqrt(2)*(2*b**(3/4)*a**(1/4)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)* 
sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*c - 2*b**(3/4)*a**(1/4)*atan((b**(1/ 
4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*c - 
2*d**(3/4)*c**(1/4)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/( 
d**(1/4)*c**(1/4)*sqrt(2)))*a + 2*d**(3/4)*c**(1/4)*atan((d**(1/4)*c**(1/4 
)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a + b**(3/4)*a 
**(1/4)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*c 
- b**(3/4)*a**(1/4)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt 
(b)*x)*c - d**(3/4)*c**(1/4)*log( - sqrt(x)*d**(1/4)*c**(1/4)*sqrt(2) + sq 
rt(c) + sqrt(d)*x)*a + d**(3/4)*c**(1/4)*log(sqrt(x)*d**(1/4)*c**(1/4)*sqr 
t(2) + sqrt(c) + sqrt(d)*x)*a))/(4*a*c*(a*d - b*c))