Integrand size = 24, antiderivative size = 313 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {2 c^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {d} \left (b c^2+a d^2\right )}+\frac {\sqrt [4]{a} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{3/4} \left (b c^2+a d^2\right )}-\frac {\sqrt [4]{a} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{3/4} \left (b c^2+a d^2\right )}-\frac {\sqrt [4]{a} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} b^{3/4} \left (b c^2+a d^2\right )} \] Output:
2*c^(3/2)*e^(3/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2)/e^(1/2))/d^(1/2)/(a*d ^2+b*c^2)+1/2*a^(1/4)*(b^(1/2)*c-a^(1/2)*d)*e^(3/2)*arctan(1-2^(1/2)*b^(1/ 4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(3/4)/(a*d^2+b*c^2)-1/2*a^(1/4)* (b^(1/2)*c-a^(1/2)*d)*e^(3/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4) /e^(1/2))*2^(1/2)/b^(3/4)/(a*d^2+b*c^2)-1/2*a^(1/4)*(b^(1/2)*c+a^(1/2)*d)* e^(3/2)*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)/b^(3/4)/(a*d^2+b*c^2)
Time = 0.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.66 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {(e x)^{3/2} \left (4 b^{3/4} c^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+\sqrt {2} \sqrt [4]{a} \sqrt {d} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\sqrt {2} \sqrt [4]{a} \sqrt {d} \left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{2 b^{3/4} \sqrt {d} \left (b c^2+a d^2\right ) x^{3/2}} \] Input:
Integrate[(e*x)^(3/2)/((c + d*x)*(a + b*x^2)),x]
Output:
((e*x)^(3/2)*(4*b^(3/4)*c^(3/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] + Sqrt[2 ]*a^(1/4)*Sqrt[d]*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - Sqrt[2]*a^(1/4)*Sqrt[d]*(Sqrt[b]*c + Sqr t[a]*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])) /(2*b^(3/4)*Sqrt[d]*(b*c^2 + a*d^2)*x^(3/2))
Time = 1.31 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.34, 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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\left (a+b x^2\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b (e x)^{3/2} (c-d x)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2 (e x)^{3/2}}{(c+d x) \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{a} e^{3/2} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{3/4} \left (a d^2+b c^2\right )}-\frac {\sqrt [4]{a} e^{3/2} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} b^{3/4} \left (a d^2+b c^2\right )}+\frac {2 c^{3/2} e^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{\sqrt {d} \left (a d^2+b c^2\right )}+\frac {\sqrt [4]{a} e^{3/2} \left (\sqrt {a} d+\sqrt {b} c\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} b^{3/4} \left (a d^2+b c^2\right )}-\frac {\sqrt [4]{a} e^{3/2} \left (\sqrt {a} d+\sqrt {b} c\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} b^{3/4} \left (a d^2+b c^2\right )}\) |
Input:
Int[(e*x)^(3/2)/((c + d*x)*(a + b*x^2)),x]
Output:
(2*c^(3/2)*e^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(Sqrt[d] *(b*c^2 + a*d^2)) + (a^(1/4)*(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)*ArcTan[1 - (S qrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(3/4)*(b*c^2 + a* d^2)) - (a^(1/4)*(Sqrt[b]*c - Sqrt[a]*d)*e^(3/2)*ArcTan[1 + (Sqrt[2]*b^(1/ 4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2)) + (a^( 1/4)*(Sqrt[b]*c + Sqrt[a]*d)*e^(3/2)*Log[Sqrt[a]*Sqrt[e] + Sqrt[b]*Sqrt[e] *x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*b^(3/4)*(b*c^2 + a*d^2 )) - (a^(1/4)*(Sqrt[b]*c + Sqrt[a]*d)*e^(3/2)*Log[Sqrt[a]*Sqrt[e] + Sqrt[b ]*Sqrt[e]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*b^(3/4)*(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 = 0.43 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {e \left (-\sqrt {d e c}\, b \sqrt {2}\, \left (\frac {\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )\right ) c \sqrt {\frac {a \,e^{2}}{b}}+e \left (4 c^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+d \sqrt {d e c}\, \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}\right ) \sqrt {2}\, a \right )\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right ) b}\) | \(341\) |
| derivativedivides | \(2 e^{2} \left (-\frac {a \left (\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}-\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {c^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(342\) |
| default | \(2 e^{2} \left (-\frac {a \left (\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}-\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {c^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(342\) |
Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*e/(a*e^2/b)^(1/4)*(-(d*e*c)^(1/2)*b*2^(1/2)*(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)))+arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1)+arc tan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1))*c*(a*e^2/b)^(1/2)+e*(4*c^2*arc tan(d*(e*x)^(1/2)/(d*e*c)^(1/2))*b*(a*e^2/b)^(1/4)+d*(d*e*c)^(1/2)*(arctan (2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1)+arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e* x)^(1/2)+1)+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^(1/2)*a)) /(d*e*c)^(1/2)/(a*d^2+b*c^2)/b
Leaf count of result is larger than twice the leaf count of optimal. 2128 vs. \(2 (230) = 460\).
Time = 0.50 (sec) , antiderivative size = 4271, normalized size of antiderivative = 13.65 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 9.65 (sec) , antiderivative size = 2492, normalized size of antiderivative = 7.96 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)**(3/2)/(d*x+c)/(b*x**2+a),x)
Output:
3*a**(3/4)*b**(1/4)*d**(3/2)*e**(3/2)*x**2*exp(3*I*pi/4)*log(-a**(1/4)*exp _polar(I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-3/4)*gamma(3/4)*gamma(5/4)*g amma(7/4)/(8*a*b*d**(5/2)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*gamma(7/4) + 8*b**2*c**2*sqrt(d)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*gamma(7/4)) - 3*I*a**(3/4)*b**(1/4)*d**(3/2)*e**(3/2)*x**2*exp(3*I*pi/4)*log(-a**(1/4)* exp_polar(3*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-3/4)*gamma(3/4)*gamma(5 /4)*gamma(7/4)/(8*a*b*d**(5/2)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*gamma (7/4) + 8*b**2*c**2*sqrt(d)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*gamma(7/ 4)) - 3*a**(3/4)*b**(1/4)*d**(3/2)*e**(3/2)*x**2*exp(3*I*pi/4)*log(-a**(1/ 4)*exp_polar(5*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-3/4)*gamma(3/4)*gamm a(5/4)*gamma(7/4)/(8*a*b*d**(5/2)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*ga mma(7/4) + 8*b**2*c**2*sqrt(d)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)*gamma (7/4)) + 3*I*a**(3/4)*b**(1/4)*d**(3/2)*e**(3/2)*x**2*exp(3*I*pi/4)*log(-a **(1/4)*exp_polar(7*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-3/4)*gamma(3/4) *gamma(5/4)*gamma(7/4)/(8*a*b*d**(5/2)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/ 4)*gamma(7/4) + 8*b**2*c**2*sqrt(d)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)* gamma(7/4)) - a**(1/4)*b**(3/4)*c*sqrt(d)*e**(3/2)*x**2*exp(I*pi/4)*log(-a **(1/4)*exp_polar(I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-1/4)*gamma(1/4)*g amma(5/4)*gamma(7/4)/(8*a*b*d**(5/2)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4) *gamma(7/4) + 8*b**2*c**2*sqrt(d)*x**2*gamma(1/4)*gamma(3/4)*gamma(5/4)...
Exception generated. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(3/2)/(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 = 425, normalized size of antiderivative = 1.36 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {\frac {4 \, c^{2} e^{3} \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {c d e}} - \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{2} - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} + \sqrt {2} a b^{3} d^{2}} - \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{2} - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} + \sqrt {2} a b^{3} d^{2}} - \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{2} + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} b^{4} c^{2} + \sqrt {2} a b^{3} d^{2}} + \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e^{2} + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d e\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} b^{4} c^{2} + \sqrt {2} a b^{3} d^{2}}}{2 \, e} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
1/2*(4*c^2*e^3*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)*b^2*c*e^2 - (a*b^3*e^2)^(3/4)*d*e)*arctan(1/2*sq rt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*b^ 4*c^2 + sqrt(2)*a*b^3*d^2) - 2*((a*b^3*e^2)^(1/4)*b^2*c*e^2 - (a*b^3*e^2)^ (3/4)*d*e)*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)*b^4*c^2 + sqrt(2)*a*b^3*d^2) - ((a*b^3*e^2)^(1/4)*b ^2*c*e^2 + (a*b^3*e^2)^(3/4)*d*e)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e *x) + sqrt(a*e^2/b))/(sqrt(2)*b^4*c^2 + sqrt(2)*a*b^3*d^2) + ((a*b^3*e^2)^ (1/4)*b^2*c*e^2 + (a*b^3*e^2)^(3/4)*d*e)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4) *sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*b^4*c^2 + sqrt(2)*a*b^3*d^2))/e
Time = 9.87 (sec) , antiderivative size = 7335, normalized size of antiderivative = 23.43 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((e*x)^(3/2)/((a + b*x^2)*(c + d*x)),x)
Output:
atan((((32*a^4*b*d^5*e^16 + 64*a^2*b^3*c^4*d*e^16)*(e*x)^(1/2) + ((a*d^2*e ^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3)/(4*(b^5*c^ 4 + a^2*b^3*d^4 + 2*a*b^4*c^2*d^2)))^(1/2)*((((a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3)/(4*(b^5*c^4 + a^2*b^3*d^4 + 2* a*b^4*c^2*d^2)))^(1/2)*((e*x)^(1/2)*((a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3 *(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3)/(4*(b^5*c^4 + a^2*b^3*d^4 + 2*a*b^4*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) + 128*a^2*b^6*c^6*d^2*e^12 + 2 56*a^3*b^5*c^4*d^4*e^12 + 128*a^4*b^4*c^2*d^6*e^12) - (e*x)^(1/2)*(448*a^2 *b^5*c^5*d^2*e^13 - 128*a^3*b^4*c^3*d^4*e^13 + 448*a^4*b^3*c*d^6*e^13))*(( a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3)/(4* (b^5*c^4 + a^2*b^3*d^4 + 2*a*b^4*c^2*d^2)))^(1/2) - 480*a^3*b^3*c^3*d^3*e^ 15 + 128*a^2*b^4*c^5*d*e^15 + 32*a^4*b^2*c*d^5*e^15))*((a*d^2*e^3*(-a*b^3) ^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3)/(4*(b^5*c^4 + a^2*b^3 *d^4 + 2*a*b^4*c^2*d^2)))^(1/2)*1i + ((32*a^4*b*d^5*e^16 + 64*a^2*b^3*c^4* d*e^16)*(e*x)^(1/2) - ((a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2 ) + 2*a*b^2*c*d*e^3)/(4*(b^5*c^4 + a^2*b^3*d^4 + 2*a*b^4*c^2*d^2)))^(1/2)* ((((a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/2) + 2*a*b^2*c*d*e^3) /(4*(b^5*c^4 + a^2*b^3*d^4 + 2*a*b^4*c^2*d^2)))^(1/2)*(128*a^2*b^6*c^6*d^2 *e^12 - (e*x)^(1/2)*((a*d^2*e^3*(-a*b^3)^(1/2) - b*c^2*e^3*(-a*b^3)^(1/...
Time = 0.27 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.07 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, e \left (-2 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{2}+2 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c d +2 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{2}-2 b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c d +8 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b c +b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d^{2}-b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d^{2}+b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c d -b^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c d \right )}{4 b d \left (a \,d^{2}+b \,c^{2}\right )} \] Input:
int((e*x)^(3/2)/(d*x+c)/(b*x^2+a),x)
Output:
(sqrt(e)*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)))*d**2 + 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)))*c*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)))*d**2 - 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)))*c*d + 8*sqrt(d)*sqrt(c)*atan((sqrt(x)*d)/(s qrt(d)*sqrt(c)))*b*c + b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a **(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d**2 - b**(1/4)*a**(3/4)*sqrt(2)*lo g(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d**2 + b**(3/4) *a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqr t(b)*x)*c*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)*c*d))/(4*b*d*(a*d**2 + b*c**2))