Integrand size = 24, antiderivative size = 329 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {2 e^2 \sqrt {e x}}{b d}-\frac {2 c^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{d^{3/2} \left (b c^2+a d^2\right )}+\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{5/4} \left (b c^2+a d^2\right )}-\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{5/4} \left (b c^2+a d^2\right )}+\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} b^{5/4} \left (b c^2+a d^2\right )} \] Output:
2*e^2*(e*x)^(1/2)/b/d-2*c^(5/2)*e^(5/2)*arctan(d^(1/2)*(e*x)^(1/2)/c^(1/2) /e^(1/2))/d^(3/2)/(a*d^2+b*c^2)+1/2*a^(3/4)*(b^(1/2)*c+a^(1/2)*d)*e^(5/2)* arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(5/4)/(a*d ^2+b*c^2)-1/2*a^(3/4)*(b^(1/2)*c+a^(1/2)*d)*e^(5/2)*arctan(1+2^(1/2)*b^(1/ 4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/b^(5/4)/(a*d^2+b*c^2)+1/2*a^(3/4)* (b^(1/2)*c-a^(1/2)*d)*e^(5/2)*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^(5/4)/(a*d^2+b*c^2)
Time = 0.49 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {(e x)^{5/2} \left (-4 b^{5/4} c^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )+\sqrt {d} \left (4 \sqrt [4]{b} \left (b c^2+a d^2\right ) \sqrt {x}+\sqrt {2} a^{3/4} d \left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )-\sqrt {2} a^{3/4} d^{3/2} \left (-\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{2 b^{5/4} d^{3/2} \left (b c^2+a d^2\right ) x^{5/2}} \] Input:
Integrate[(e*x)^(5/2)/((c + d*x)*(a + b*x^2)),x]
Output:
((e*x)^(5/2)*(-4*b^(5/4)*c^(5/2)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]] + Sqrt[ d]*(4*b^(1/4)*(b*c^2 + a*d^2)*Sqrt[x] + Sqrt[2]*a^(3/4)*d*(Sqrt[b]*c + Sqr t[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])]) - Sqrt[2]*a^(3/4)*d^(3/2)*(-(Sqrt[b]*c) + Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a^(1/ 4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(2*b^(5/4)*d^(3/2)*(b*c^2 + a *d^2)*x^(5/2))
Time = 1.53 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.46, 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.
| \(\displaystyle \int \frac {(e x)^{5/2}}{\left (a+b x^2\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \int \left (\frac {b (e x)^{5/2} (c-d x)}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^2 (e x)^{5/2}}{(c+d x) \left (a d^2+b c^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} e^{5/2} \left (\sqrt {a} d+\sqrt {b} c\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} b^{5/4} \left (a d^2+b c^2\right )}-\frac {a^{3/4} e^{5/2} \left (\sqrt {a} d+\sqrt {b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} b^{5/4} \left (a d^2+b c^2\right )}-\frac {a^{3/4} e^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} b^{5/4} \left (a d^2+b c^2\right )}+\frac {a^{3/4} e^{5/2} \left (\sqrt {b} c-\sqrt {a} d\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}+\sqrt {a} \sqrt {e}+\sqrt {b} \sqrt {e} x\right )}{2 \sqrt {2} b^{5/4} \left (a d^2+b c^2\right )}-\frac {2 c^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{d^{3/2} \left (a d^2+b c^2\right )}+\frac {2 a d e^2 \sqrt {e x}}{b \left (a d^2+b c^2\right )}+\frac {2 c^2 e^2 \sqrt {e x}}{d \left (a d^2+b c^2\right )}\) |
Input:
Int[(e*x)^(5/2)/((c + d*x)*(a + b*x^2)),x]
Output:
(2*c^2*e^2*Sqrt[e*x])/(d*(b*c^2 + a*d^2)) + (2*a*d*e^2*Sqrt[e*x])/(b*(b*c^ 2 + a*d^2)) - (2*c^(5/2)*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[ e])])/(d^(3/2)*(b*c^2 + a*d^2)) + (a^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*e^(5/2) *ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(5/ 4)*(b*c^2 + a*d^2)) - (a^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*e^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(5/4)*(b*c^2 + a*d^2)) - (a^(3/4)*(Sqrt[b]*c - Sqrt[a]*d)*e^(5/2)*Log[Sqrt[a]*Sqrt[e] + S qrt[b]*Sqrt[e]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2]*b^(5/4)* (b*c^2 + a*d^2)) + (a^(3/4)*(Sqrt[b]*c - Sqrt[a]*d)*e^(5/2)*Log[Sqrt[a]*Sq rt[e] + Sqrt[b]*Sqrt[e]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[e*x]])/(2*Sqrt[2] *b^(5/4)*(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 = 0.70 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(2 e^{2} \left (\frac {\sqrt {e x}}{d b}-\frac {e a \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) b}-\frac {c^{3} e \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{d \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(357\) |
| default | \(2 e^{2} \left (\frac {\sqrt {e x}}{d b}-\frac {e a \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) b}-\frac {c^{3} e \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{d \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right )\) | \(357\) |
| risch | \(\frac {2 x \,e^{3}}{d b \sqrt {e x}}-\frac {\left (\frac {2 d a \left (\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {2 c^{3} b \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}}\right ) e^{3}}{b d}\) | \(363\) |
| pseudoelliptic | \(-\frac {2 \left (\frac {d^{2} \sqrt {2}\, \sqrt {d e c}\, a \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\frac {a \,e^{2}}{b}}}{8}+\left (-\sqrt {e x}\, \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}+\arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) b \,c^{3} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}+\frac {d e \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}\right ) \sqrt {2}\, \sqrt {d e c}\, a c}{4}\right ) e^{2}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, b d \left (a \,d^{2}+b \,c^{2}\right )}\) | \(376\) |
Input:
int((e*x)^(5/2)/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
2*e^2*(1/d/b*(e*x)^(1/2)-e*a/(a*d^2+b*c^2)/b*(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)))-1/d*c^3*e/(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. 2294 vs. \(2 (245) = 490\).
Time = 1.07 (sec) , antiderivative size = 4603, normalized size of antiderivative = 13.99 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)^(5/2)/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
Too large to include
Result contains complex when optimal does not.
Time = 22.71 (sec) , antiderivative size = 3074, normalized size of antiderivative = 9.34 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x)**(5/2)/(d*x+c)/(b*x**2+a),x)
Output:
-5*a**(5/4)*b**(7/4)*sqrt(c)*d**3*e**(5/2)*x**(13/2)*exp(I*pi/4)*log(-a**( 1/4)*exp_polar(I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-5/4)*gamma(1/4)*gamm a(3/4)*gamma(5/4)/(8*a*b**3*sqrt(c)*d**4*x**(13/2)*gamma(-1/4)*gamma(1/4)* gamma(3/4)*gamma(5/4) + 8*b**4*c**(5/2)*d**2*x**(13/2)*gamma(-1/4)*gamma(1 /4)*gamma(3/4)*gamma(5/4)) - 5*I*a**(5/4)*b**(7/4)*sqrt(c)*d**3*e**(5/2)*x **(13/2)*exp(I*pi/4)*log(-a**(1/4)*exp_polar(3*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-5/4)*gamma(1/4)*gamma(3/4)*gamma(5/4)/(8*a*b**3*sqrt(c)*d**4*x **(13/2)*gamma(-1/4)*gamma(1/4)*gamma(3/4)*gamma(5/4) + 8*b**4*c**(5/2)*d* *2*x**(13/2)*gamma(-1/4)*gamma(1/4)*gamma(3/4)*gamma(5/4)) + 5*a**(5/4)*b* *(7/4)*sqrt(c)*d**3*e**(5/2)*x**(13/2)*exp(I*pi/4)*log(-a**(1/4)*exp_polar (5*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(-5/4)*gamma(1/4)*gamma(3/4)*gamma (5/4)/(8*a*b**3*sqrt(c)*d**4*x**(13/2)*gamma(-1/4)*gamma(1/4)*gamma(3/4)*g amma(5/4) + 8*b**4*c**(5/2)*d**2*x**(13/2)*gamma(-1/4)*gamma(1/4)*gamma(3/ 4)*gamma(5/4)) + 5*I*a**(5/4)*b**(7/4)*sqrt(c)*d**3*e**(5/2)*x**(13/2)*exp (I*pi/4)*log(-a**(1/4)*exp_polar(7*I*pi/4)/(b**(1/4)*sqrt(x)) + 1)*gamma(- 5/4)*gamma(1/4)*gamma(3/4)*gamma(5/4)/(8*a*b**3*sqrt(c)*d**4*x**(13/2)*gam ma(-1/4)*gamma(1/4)*gamma(3/4)*gamma(5/4) + 8*b**4*c**(5/2)*d**2*x**(13/2) *gamma(-1/4)*gamma(1/4)*gamma(3/4)*gamma(5/4)) - 3*a**(3/4)*b**(9/4)*c**(3 /2)*d**2*e**(5/2)*x**(13/2)*exp(3*I*pi/4)*log(-a**(1/4)*exp_polar(I*pi/4)/ (b**(1/4)*sqrt(x)) + 1)*gamma(-3/4)*gamma(-1/4)*gamma(3/4)*gamma(5/4)/(...
Exception generated. \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(5/2)/(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 = 0.15 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.30 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=-\frac {1}{2} \, {\left (\frac {4 \, c^{3} e \arctan \left (\frac {\sqrt {e x} d}{\sqrt {c d e}}\right )}{{\left (b c^{2} d + a d^{3}\right )} \sqrt {c d e}} + \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} e + \sqrt {2} a b^{3} d^{2} e} + \frac {2 \, {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{4} c^{2} e + \sqrt {2} a b^{3} d^{2} e} + \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} b^{4} c^{2} e + \sqrt {2} a b^{3} d^{2} e} - \frac {{\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{\sqrt {2} b^{4} c^{2} e + \sqrt {2} a b^{3} d^{2} e} - \frac {4 \, \sqrt {e x}}{b d}\right )} e^{2} \] Input:
integrate((e*x)^(5/2)/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
-1/2*(4*c^3*e*arctan(sqrt(e*x)*d/sqrt(c*d*e))/((b*c^2*d + a*d^3)*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)*b^4*c ^2*e + sqrt(2)*a*b^3*d^2*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)*b^4*c^2*e + sqrt(2)*a*b^3*d^2*e) + ((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)*b^4*c^2*e + sqrt(2)*a*b^3*d^2*e) - ((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)*b^4*c^2*e + sqrt(2)*a*b^3*d^2*e) - 4*sqr t(e*x)/(b*d))*e^2
Time = 9.61 (sec) , antiderivative size = 8313, normalized size of antiderivative = 25.27 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:
int((e*x)^(5/2)/((a + b*x^2)*(c + d*x)),x)
Output:
atan(((((((32*(4*a^3*b^6*c^5*d^4*e^13 + 8*a^4*b^5*c^3*d^6*e^13 + 4*a^5*b^4 *c*d^8*e^13))/(b*d) - (32*(e*x)^(1/2)*(-(a*d^2*e^5*(-a^3*b^5)^(1/2) - b*c^ 2*e^5*(-a^3*b^5)^(1/2) + 2*a^2*b^3*c*d*e^5)/(4*(b^7*c^4 + a^2*b^5*d^4 + 2* a*b^6*c^2*d^2)))^(1/2)*(16*a^5*b^5*d^10*e^10 - 16*a^2*b^8*c^6*d^4*e^10 - 1 6*a^3*b^7*c^4*d^6*e^10 + 16*a^4*b^6*c^2*d^8*e^10))/(b*d))*(-(a*d^2*e^5*(-a ^3*b^5)^(1/2) - b*c^2*e^5*(-a^3*b^5)^(1/2) + 2*a^2*b^3*c*d*e^5)/(4*(b^7*c^ 4 + a^2*b^5*d^4 + 2*a*b^6*c^2*d^2)))^(1/2) + (32*(e*x)^(1/2)*(2*a^3*b^5*c^ 5*d^3*e^15 + 4*a^4*b^4*c^3*d^5*e^15 + 16*a^2*b^6*c^7*d*e^15 - 14*a^5*b^3*c *d^7*e^15))/(b*d))*(-(a*d^2*e^5*(-a^3*b^5)^(1/2) - b*c^2*e^5*(-a^3*b^5)^(1 /2) + 2*a^2*b^3*c*d*e^5)/(4*(b^7*c^4 + a^2*b^5*d^4 + 2*a*b^6*c^2*d^2)))^(1 /2) - (32*(a^5*b^2*c^2*d^5*e^18 - 15*a^4*b^3*c^4*d^3*e^18 + 12*a^3*b^4*c^6 *d*e^18))/(b*d))*(-(a*d^2*e^5*(-a^3*b^5)^(1/2) - b*c^2*e^5*(-a^3*b^5)^(1/2 ) + 2*a^2*b^3*c*d*e^5)/(4*(b^7*c^4 + a^2*b^5*d^4 + 2*a*b^6*c^2*d^2)))^(1/2 ) - (32*(e*x)^(1/2)*(a^6*d^6*e^20 - 2*a^3*b^3*c^6*e^20))/(b*d))*(-(a*d^2*e ^5*(-a^3*b^5)^(1/2) - b*c^2*e^5*(-a^3*b^5)^(1/2) + 2*a^2*b^3*c*d*e^5)/(4*( b^7*c^4 + a^2*b^5*d^4 + 2*a*b^6*c^2*d^2)))^(1/2)*1i - (((((32*(4*a^3*b^6*c ^5*d^4*e^13 + 8*a^4*b^5*c^3*d^6*e^13 + 4*a^5*b^4*c*d^8*e^13))/(b*d) + (32* (e*x)^(1/2)*(-(a*d^2*e^5*(-a^3*b^5)^(1/2) - b*c^2*e^5*(-a^3*b^5)^(1/2) + 2 *a^2*b^3*c*d*e^5)/(4*(b^7*c^4 + a^2*b^5*d^4 + 2*a*b^6*c^2*d^2)))^(1/2)*(16 *a^5*b^5*d^10*e^10 - 16*a^2*b^8*c^6*d^4*e^10 - 16*a^3*b^7*c^4*d^6*e^10 ...
Time = 0.26 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.12 \[ \int \frac {(e x)^{5/2}}{(c+d x) \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, e^{2} \left (2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,d^{2}+2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{3}-2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c \,d^{2}-2 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{3}-8 \sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {\sqrt {x}\, d}{\sqrt {d}\, \sqrt {c}}\right ) b^{2} c^{2}-b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c \,d^{2}+b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c \,d^{2}+b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d^{3}-b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d^{3}+8 \sqrt {x}\, a b \,d^{3}+8 \sqrt {x}\, b^{2} c^{2} d \right )}{4 b^{2} d^{2} \left (a \,d^{2}+b \,c^{2}\right )} \] Input:
int((e*x)^(5/2)/(d*x+c)/(b*x^2+a),x)
Output:
(sqrt(e)*e**2*(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*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)))*a*d**3 - 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*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)))*a*d**3 - 8*sqrt(d)*sqrt(c)*atan( (sqrt(x)*d)/(sqrt(d)*sqrt(c)))*b**2*c**2 - 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*d**2 + b**( 1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sq rt(b)*x)*b*c*d**2 + 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**3 - 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**3 + 8*sqrt( x)*a*b*d**3 + 8*sqrt(x)*b**2*c**2*d))/(4*b**2*d**2*(a*d**2 + b*c**2))