Integrand size = 24, antiderivative size = 312 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{b c^2+a d^2}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right )} \] Output:
-2*c^(1/2)*d^(1/2)*e^(1/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/(a* d^2+b*c^2)-1/2*(b^(1/2)*c+a^(1/2)*d)*e^(1/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x )^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(1/4)/b^(1/4)/(a*d^2+b*c^2)+1/2*(b^(1/2 )*c+a^(1/2)*d)*e^(1/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2 ))*2^(1/2)/a^(1/4)/b^(1/4)/(a*d^2+b*c^2)-1/2*(b^(1/2)*c-a^(1/2)*d)*e^(1/2) *arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2)*x))* 2^(1/2)/a^(1/4)/b^(1/4)/(a*d^2+b*c^2)
Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=-\frac {\sqrt {e x} \left (4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {c} \sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+\sqrt {2} \left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (\sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (b c^2+a d^2\right ) \sqrt {x}} \] Input:
Integrate[Sqrt[e*x]/((c + d*x)*(a + b*x^2)),x]
Output:
-1/2*(Sqrt[e*x]*(4*a^(1/4)*b^(1/4)*Sqrt[c]*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[x] )/Sqrt[c]] + Sqrt[2]*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/ (Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + Sqrt[2]*(Sqrt[b]*c - Sqrt[a]*d)*ArcTa nh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(a^(1/4)*b^( 1/4)*(b*c^2 + a*d^2)*Sqrt[x])
Time = 1.16 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 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.
| \(\displaystyle \int \frac {\sqrt {e x}}{\left (a+b x^2\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b \sqrt {e x} (c-d x)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2 \sqrt {e x}}{(c+d x) \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{a d^2+b c^2}-\frac {\sqrt {e} \left (\sqrt {a} d+\sqrt {b} c\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^2+b c^2\right )}+\frac {\sqrt {e} \left (\sqrt {a} d+\sqrt {b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^2+b c^2\right )}+\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^2+b c^2\right )}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \left (a d^2+b c^2\right )}\) |
Input:
Int[Sqrt[e*x]/((c + d*x)*(a + b*x^2)),x]
Output:
(-2*Sqrt[c]*Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])]) /(b*c^2 + a*d^2) - ((Sqrt[b]*c + Sqrt[a]*d)*Sqrt[e]*ArcTan[1 - (Sqrt[2]*b^ (1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d ^2)) + ((Sqrt[b]*c + Sqrt[a]*d)*Sqrt[e]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e *x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2)) + ((Sqr t[b]*c - Sqrt[a]*d)*Sqrt[e]*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e]*x - Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2) ) - ((Sqrt[b]*c - Sqrt[a]*d)*Sqrt[e]*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e] *x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*a^(1/4)*b^(1/4)*(b*c^2 + a*d^2))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(2 e^{2} \left (\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e \left (a \,d^{2}+b \,c^{2}\right )}-\frac {c d \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(340\) |
| default | \(2 e^{2} \left (\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{e \left (a \,d^{2}+b \,c^{2}\right )}-\frac {c d \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{e \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(340\) |
| pseudoelliptic | \(\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) c \sqrt {2}\, e \sqrt {d e c}+\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) d \sqrt {\frac {a \,e^{2}}{b}}\, \sqrt {2}\, \sqrt {d e c}+2 \sqrt {d e c}\, \sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \sqrt {d e c}\, \sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-8 c d \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) e \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\sqrt {d e c}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \left (4 a \,d^{2}+4 b \,c^{2}\right )}\) | \(359\) |
Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
2*e^2*(1/e/(a*d^2+b*c^2)*(1/8*d/e*(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x+(a*e^2/ b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^( 1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2 )+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+1/8*c/(a*e^2/b)^(1/4 )*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e *x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/ (a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2 )-1)))-c/e*d/(a*d^2+b*c^2)/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2 )))
Leaf count of result is larger than twice the leaf count of optimal. 2024 vs. \(2 (229) = 458\).
Time = 0.41 (sec) , antiderivative size = 4060, normalized size of antiderivative = 13.01 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 7.42 (sec) , antiderivative size = 2516, normalized size of antiderivative = 8.06 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)**(1/2)/(d*x+c)/(b*x**2+a),x)
Output:
-4*pi*a**(1/4)*b**(1/4)*c**(3/2)*sqrt(d)*sqrt(e)*sqrt(x)*gamma(1/4)*gamma( 3/4)/(8*pi*a**(5/4)*b**(1/4)*sqrt(c)*d**(5/2)*x*gamma(3/4)*gamma(5/4) + 8* pi*a**(1/4)*b**(5/4)*c**(5/2)*sqrt(d)*x*gamma(3/4)*gamma(5/4)) + 4*pi*a**( 1/4)*b**(1/4)*sqrt(c)*d**(3/2)*sqrt(e)*x**(3/2)*gamma(-1/4)*gamma(5/4)/(8* pi*a**(5/4)*b**(1/4)*sqrt(c)*d**(5/2)*x*gamma(3/4)*gamma(5/4) + 8*pi*a**(1 /4)*b**(5/4)*c**(5/2)*sqrt(d)*x*gamma(3/4)*gamma(5/4)) - pi*a**(1/4)*b**(1 /4)*c*d*sqrt(e)*x*log(1 - sqrt(d)*sqrt(x)/sqrt(c))*gamma(1/4)*gamma(3/4)/( 8*pi*a**(5/4)*b**(1/4)*sqrt(c)*d**(5/2)*x*gamma(3/4)*gamma(5/4) + 8*pi*a** (1/4)*b**(5/4)*c**(5/2)*sqrt(d)*x*gamma(3/4)*gamma(5/4)) - pi*a**(1/4)*b** (1/4)*c*d*sqrt(e)*x*log(1 - sqrt(d)*sqrt(x)/sqrt(c))*gamma(-1/4)*gamma(5/4 )/(8*pi*a**(5/4)*b**(1/4)*sqrt(c)*d**(5/2)*x*gamma(3/4)*gamma(5/4) + 8*pi* a**(1/4)*b**(5/4)*c**(5/2)*sqrt(d)*x*gamma(3/4)*gamma(5/4)) + I*pi*a**(1/4 )*b**(1/4)*c*d*sqrt(e)*x*log(1 - sqrt(d)*sqrt(x)*exp_polar(I*pi/2)/sqrt(c) )*gamma(-1/4)*gamma(5/4)/(8*pi*a**(5/4)*b**(1/4)*sqrt(c)*d**(5/2)*x*gamma( 3/4)*gamma(5/4) + 8*pi*a**(1/4)*b**(5/4)*c**(5/2)*sqrt(d)*x*gamma(3/4)*gam ma(5/4)) - I*pi*a**(1/4)*b**(1/4)*c*d*sqrt(e)*x*log(1 - sqrt(d)*sqrt(x)*ex p_polar(I*pi/2)/sqrt(c))*gamma(1/4)*gamma(3/4)/(8*pi*a**(5/4)*b**(1/4)*sqr t(c)*d**(5/2)*x*gamma(3/4)*gamma(5/4) + 8*pi*a**(1/4)*b**(5/4)*c**(5/2)*sq rt(d)*x*gamma(3/4)*gamma(5/4)) + pi*a**(1/4)*b**(1/4)*c*d*sqrt(e)*x*log(1 - sqrt(d)*sqrt(x)*exp_polar(I*pi)/sqrt(c))*gamma(-1/4)*gamma(5/4)/(8*pi...
Exception generated. \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.14 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=-\frac {\frac {4 \, c d e^{2} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {c d e}} - \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c^{2} + \sqrt {2} a^{2} b^{2} d^{2}} - \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c^{2} + \sqrt {2} a^{2} b^{2} d^{2}} - \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} a b^{3} c^{2} + \sqrt {2} a^{2} b^{2} d^{2}} + \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} a b^{3} c^{2} + \sqrt {2} a^{2} b^{2} d^{2}}}{2 \, e} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*(4*c*d*e^2*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b*c^2 + a*d^2)*sqrt(c*d* e)) - 2*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^(3/4)*c)*arctan(1/2*sqrt( 2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a*b^3 *c^2 + sqrt(2)*a^2*b^2*d^2) - 2*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^( 3/4)*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2 /b)^(1/4))/(sqrt(2)*a*b^3*c^2 + sqrt(2)*a^2*b^2*d^2) - ((a*b^3*e^2)^(1/4)* a*b*d*e - (a*b^3*e^2)^(3/4)*c)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a*b^3*c^2 + sqrt(2)*a^2*b^2*d^2) + ((a*b^3*e^2) ^(1/4)*a*b*d*e - (a*b^3*e^2)^(3/4)*c)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sq rt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a*b^3*c^2 + sqrt(2)*a^2*b^2*d^2))/e
Time = 8.88 (sec) , antiderivative size = 6558, normalized size of antiderivative = 21.02 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((e*x)^(1/2)/((a + b*x^2)*(c + d*x)),x)
Output:
atan(-(((((-(a*d^2*e*(-a*b)^(1/2) - b*c^2*e*(-a*b)^(1/2) + 2*a*b*c*d*e)/(4 *(a*b^3*c^4 + a^3*b*d^4 + 2*a^2*b^2*c^2*d^2)))^(1/2)*((e*x)^(1/2)*(-(a*d^2 *e*(-a*b)^(1/2) - b*c^2*e*(-a*b)^(1/2) + 2*a*b*c*d*e)/(4*(a*b^3*c^4 + a^3* b*d^4 + 2*a^2*b^2*c^2*d^2)))^(1/2)*(512*a^5*b^4*d^9*e^10 - 512*a^2*b^7*c^6 *d^3*e^10 - 512*a^3*b^6*c^4*d^5*e^10 + 512*a^4*b^5*c^2*d^7*e^10) + 384*a^2 *b^6*c^5*d^3*e^11 + 768*a^3*b^5*c^3*d^5*e^11 + 384*a^4*b^4*c*d^7*e^11) - ( e*x)^(1/2)*(640*a^2*b^5*c^3*d^4*e^11 + 64*a*b^6*c^5*d^2*e^11 - 448*a^3*b^4 *c*d^6*e^11))*(-(a*d^2*e*(-a*b)^(1/2) - b*c^2*e*(-a*b)^(1/2) + 2*a*b*c*d*e )/(4*(a*b^3*c^4 + a^3*b*d^4 + 2*a^2*b^2*c^2*d^2)))^(1/2) + 416*a^2*b^4*c^2 *d^4*e^12 + 32*a*b^5*c^4*d^2*e^12)*(-(a*d^2*e*(-a*b)^(1/2) - b*c^2*e*(-a*b )^(1/2) + 2*a*b*c*d*e)/(4*(a*b^3*c^4 + a^3*b*d^4 + 2*a^2*b^2*c^2*d^2)))^(1 /2) + (e*x)^(1/2)*(32*a^2*b^3*d^5*e^12 - 64*a*b^4*c^2*d^3*e^12))*(-(a*d^2* e*(-a*b)^(1/2) - b*c^2*e*(-a*b)^(1/2) + 2*a*b*c*d*e)/(4*(a*b^3*c^4 + a^3*b *d^4 + 2*a^2*b^2*c^2*d^2)))^(1/2)*1i - ((((-(a*d^2*e*(-a*b)^(1/2) - b*c^2* e*(-a*b)^(1/2) + 2*a*b*c*d*e)/(4*(a*b^3*c^4 + a^3*b*d^4 + 2*a^2*b^2*c^2*d^ 2)))^(1/2)*(384*a^2*b^6*c^5*d^3*e^11 - (e*x)^(1/2)*(-(a*d^2*e*(-a*b)^(1/2) - b*c^2*e*(-a*b)^(1/2) + 2*a*b*c*d*e)/(4*(a*b^3*c^4 + a^3*b*d^4 + 2*a^2*b ^2*c^2*d^2)))^(1/2)*(512*a^5*b^4*d^9*e^10 - 512*a^2*b^7*c^6*d^3*e^10 - 512 *a^3*b^6*c^4*d^5*e^10 + 512*a^4*b^5*c^2*d^7*e^10) + 768*a^3*b^5*c^3*d^5*e^ 11 + 384*a^4*b^4*c*d^7*e^11) + (e*x)^(1/2)*(640*a^2*b^5*c^3*d^4*e^11 + ...
Time = 0.26 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (-2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c +2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -8 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) a b +b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d +b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d \right )}{4 a b \left (a \,d^{2}+b \,c^{2}\right )} \] Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a),x)
Output:
(sqrt(e)*( - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c - 2*b**(3/4)*a**(1/4) *sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a* *(1/4)*sqrt(2)))*a*d + 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) *sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*c + 2*b**(3/4 )*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b **(1/4)*a**(1/4)*sqrt(2)))*a*d - 8*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(sqrt( d)*sqrt(c)))*a*b + b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1 /4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c - b**(1/4)*a**(3/4)*sqrt(2)*log(sqr t(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*c - b**(3/4)*a**(1 /4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x )*a*d + b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*d))/(4*a*b*(a*d**2 + b*c**2))