\(\int \frac {(e x)^{5/2}}{(c+d x)^3 (a+b x^2)} \, dx\) [397]

Optimal result

Integrand size = 24, antiderivative size = 505 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=-\frac {c^2 e^2 \sqrt {e x}}{2 d \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {c \left (b c^2+9 a d^2\right ) e^2 \sqrt {e x}}{4 d \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {\sqrt {c} \left (b^2 c^4+18 a b c^2 d^2-15 a^2 d^4\right ) e^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{4 d^{3/2} \left (b c^2+a d^2\right )^3}+\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} c d+a d^2\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} c d+a d^2\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} c d+a d^2\right ) e^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3} \] Output:

-1/2*c^2*e^2*(e*x)^(1/2)/d/(a*d^2+b*c^2)/(d*x+c)^2+1/4*c*(9*a*d^2+b*c^2)*e 
^2*(e*x)^(1/2)/d/(a*d^2+b*c^2)^2/(d*x+c)+1/4*c^(1/2)*(-15*a^2*d^4+18*a*b*c 
^2*d^2+b^2*c^4)*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)^3+1/2*a^(3/4)*(b^(1/2)*c-a^(1/2)*d)*(b*c^2+4*a^(1/2)*b^(1/ 
2)*c*d+a*d^2)*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^(1/4)/(a*d^2+b*c^2)^3-1/2*a^(3/4)*(b^(1/2)*c-a^(1/2)*d)*(b*c^2 
+4*a^(1/2)*b^(1/2)*c*d+a*d^2)*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^(1/4)/(a*d^2+b*c^2)^3+1/2*a^(3/4)*(b^(1/2)*c+a 
^(1/2)*d)*(b*c^2-4*a^(1/2)*b^(1/2)*c*d+a*d^2)*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^(1/4)/(a*d^ 
2+b*c^2)^3
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.66 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\frac {(e x)^{5/2} \left (\frac {\left (b c^2+a d^2\right ) \sqrt {x} \left (b c^3 (-c+d x)+a c d^2 (7 c+9 d x)\right )}{d (c+d x)^2}+\frac {\sqrt {c} \left (b^2 c^4+18 a b c^2 d^2-15 a^2 d^4\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{d^{3/2}}-\frac {2 \sqrt {2} a^{3/4} \left (-b^{3/2} c^3-3 \sqrt {a} b c^2 d+3 a \sqrt {b} c d^2+a^{3/2} d^3\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {2 \sqrt {2} a^{3/4} \left (b^{3/2} c^3-3 \sqrt {a} b c^2 d-3 a \sqrt {b} c d^2+a^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{4 \left (b c^2+a d^2\right )^3 x^{5/2}} \] Input:

Integrate[(e*x)^(5/2)/((c + d*x)^3*(a + b*x^2)),x]
 

Output:

((e*x)^(5/2)*(((b*c^2 + a*d^2)*Sqrt[x]*(b*c^3*(-c + d*x) + a*c*d^2*(7*c + 
9*d*x)))/(d*(c + d*x)^2) + (Sqrt[c]*(b^2*c^4 + 18*a*b*c^2*d^2 - 15*a^2*d^4 
)*ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/d^(3/2) - (2*Sqrt[2]*a^(3/4)*(-(b^(3/ 
2)*c^3) - 3*Sqrt[a]*b*c^2*d + 3*a*Sqrt[b]*c*d^2 + a^(3/2)*d^3)*ArcTan[(Sqr 
t[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (2*Sqrt[2] 
*a^(3/4)*(b^(3/2)*c^3 - 3*Sqrt[a]*b*c^2*d - 3*a*Sqrt[b]*c*d^2 + a^(3/2)*d^ 
3)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(1/ 
4)))/(4*(b*c^2 + a*d^2)^3*x^(5/2))
 

Rubi [A] (verified)

Time = 2.51 (sec) , antiderivative size = 994, normalized size of antiderivative = 1.97, 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.

\(\displaystyle \int \frac {(e x)^{5/2}}{\left (a+b x^2\right ) (c+d x)^3} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (\frac {b^2 (e x)^{5/2} \left (c \left (b c^2-3 a d^2\right )-d x \left (3 b c^2-a d^2\right )\right )}{\left (a+b x^2\right ) \left (a d^2+b c^2\right )^3}+\frac {b d^2 (e x)^{5/2} \left (3 b c^2-a d^2\right )}{(c+d x) \left (a d^2+b c^2\right )^3}+\frac {2 b c d^2 (e x)^{5/2}}{(c+d x)^2 \left (a d^2+b c^2\right )^2}+\frac {d^2 (e x)^{5/2}}{(c+d x)^3 \left (a d^2+b c^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {15 \sqrt {c} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) e^{5/2}}{4 d^{3/2} \left (b c^2+a d^2\right )}+\frac {10 b c^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) e^{5/2}}{d^{3/2} \left (b c^2+a d^2\right )^2}-\frac {2 b c^{5/2} \left (3 b c^2-a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) e^{5/2}}{d^{3/2} \left (b c^2+a d^2\right )^3}+\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} d c+a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) e^{5/2}}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {a^{3/4} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+4 \sqrt {a} \sqrt {b} d c+a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) e^{5/2}}{\sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}-\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} d c+a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) e^{5/2}}{2 \sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {a^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2-4 \sqrt {a} \sqrt {b} d c+a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) e^{5/2}}{2 \sqrt {2} \sqrt [4]{b} \left (b c^2+a d^2\right )^3}+\frac {15 \sqrt {e x} e^2}{4 d \left (b c^2+a d^2\right )}-\frac {10 b c^2 \sqrt {e x} e^2}{d \left (b c^2+a d^2\right )^2}+\frac {2 a d \left (3 b c^2-a d^2\right ) \sqrt {e x} e^2}{\left (b c^2+a d^2\right )^3}+\frac {2 b c^2 \left (3 b c^2-a d^2\right ) \sqrt {e x} e^2}{d \left (b c^2+a d^2\right )^3}+\frac {10 b c (e x)^{3/2} e}{3 \left (b c^2+a d^2\right )^2}+\frac {2 b c \left (b c^2-3 a d^2\right ) (e x)^{3/2} e}{3 \left (b c^2+a d^2\right )^3}-\frac {2 b c \left (3 b c^2-a d^2\right ) (e x)^{3/2} e}{3 \left (b c^2+a d^2\right )^3}-\frac {5 (e x)^{3/2} e}{4 \left (b c^2+a d^2\right ) (c+d x)}-\frac {2 b c d (e x)^{5/2}}{\left (b c^2+a d^2\right )^2 (c+d x)}-\frac {d (e x)^{5/2}}{2 \left (b c^2+a d^2\right ) (c+d x)^2}\)

Input:

Int[(e*x)^(5/2)/((c + d*x)^3*(a + b*x^2)),x]
 

Output:

(2*b*c^2*(3*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(d*(b*c^2 + a*d^2)^3) + (2*a*d*( 
3*b*c^2 - a*d^2)*e^2*Sqrt[e*x])/(b*c^2 + a*d^2)^3 - (10*b*c^2*e^2*Sqrt[e*x 
])/(d*(b*c^2 + a*d^2)^2) + (15*e^2*Sqrt[e*x])/(4*d*(b*c^2 + a*d^2)) + (2*b 
*c*(b*c^2 - 3*a*d^2)*e*(e*x)^(3/2))/(3*(b*c^2 + a*d^2)^3) - (2*b*c*(3*b*c^ 
2 - a*d^2)*e*(e*x)^(3/2))/(3*(b*c^2 + a*d^2)^3) + (10*b*c*e*(e*x)^(3/2))/( 
3*(b*c^2 + a*d^2)^2) - (d*(e*x)^(5/2))/(2*(b*c^2 + a*d^2)*(c + d*x)^2) - ( 
5*e*(e*x)^(3/2))/(4*(b*c^2 + a*d^2)*(c + d*x)) - (2*b*c*d*(e*x)^(5/2))/((b 
*c^2 + a*d^2)^2*(c + d*x)) - (2*b*c^(5/2)*(3*b*c^2 - a*d^2)*e^(5/2)*ArcTan 
[(Sqrt[d]*Sqrt[e*x])/(Sqrt[c]*Sqrt[e])])/(d^(3/2)*(b*c^2 + a*d^2)^3) + (10 
*b*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)^2) - (15*Sqrt[c]*e^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt 
[c]*Sqrt[e])])/(4*d^(3/2)*(b*c^2 + a*d^2)) + (a^(3/4)*(Sqrt[b]*c - Sqrt[a] 
*d)*(b*c^2 + 4*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*e^(5/2)*ArcTan[1 - (Sqrt[2]*b^ 
(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2]*b^(1/4)*(b*c^2 + a*d^2)^3) - 
 (a^(3/4)*(Sqrt[b]*c - Sqrt[a]*d)*(b*c^2 + 4*Sqrt[a]*Sqrt[b]*c*d + a*d^2)* 
e^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(Sqrt[2 
]*b^(1/4)*(b*c^2 + a*d^2)^3) - (a^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*(b*c^2 - 4 
*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*e^(5/2)*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^(1/4)*(b*c^2 + a*d^ 
2)^3) + (a^(3/4)*(Sqrt[b]*c + Sqrt[a]*d)*(b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d...
 

Defintions of rubi rules used

rule 615

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 496, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {a \left (\frac {\left (a^{2} d^{3} e -3 a b \,c^{2} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (3 a b c \,d^{2}-c^{3} b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{e \left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c \left (\frac {\left (-\frac {9}{8} a^{2} d^{4}-\frac {5}{4} b \,c^{2} d^{2} a -\frac {1}{8} b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \sqrt {e x}}{8 d}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}-18 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 d \sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} e}\right )\)	\(496\)
default	\(2 e^{4} \left (\frac {a \left (\frac {\left (a^{2} d^{3} e -3 a b \,c^{2} d e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {\left (3 a b c \,d^{2}-c^{3} b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{e \left (a \,d^{2}+b \,c^{2}\right )^{3}}-\frac {c \left (\frac {\left (-\frac {9}{8} a^{2} d^{4}-\frac {5}{4} b \,c^{2} d^{2} a -\frac {1}{8} b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}-\frac {c e \left (7 a^{2} d^{4}+6 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \sqrt {e x}}{8 d}}{\left (d e x +c e \right )^{2}}+\frac {\left (15 a^{2} d^{4}-18 b \,c^{2} d^{2} a -b^{2} c^{4}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{8 d \sqrt {d e c}}\right )}{\left (a \,d^{2}+b \,c^{2}\right )^{3} e}\right )\)	\(496\)
pseudoelliptic	\(\frac {e \left (d^{2} \left (d x +c \right )^{2} e \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {d e c}\, a \left (a \,d^{2}-3 b \,c^{2}\right ) \sqrt {\frac {a \,e^{2}}{b}}+6 \left (\frac {\left (3 d \left (a \,d^{2}+b \,c^{2}\right ) \left (a \,d^{2}+\frac {b \,c^{2}}{9}\right ) \sqrt {d e c}\, \left (e x \right )^{\frac {3}{2}}-5 \left (-\frac {7 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, c \left (a \,d^{2}-\frac {b \,c^{2}}{7}\right ) \sqrt {e x}}{15}+\left (d x +c \right )^{2} \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right ) e \left (a^{2} d^{4}-\frac {6}{5} b \,c^{2} d^{2} a -\frac {1}{15} b^{2} c^{4}\right )\right ) e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{2}+d \left (d x +c \right )^{2} e^{2} \left (a \,d^{2}-\frac {b \,c^{2}}{3}\right ) \left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}\right ) \sqrt {2}\, \sqrt {d e c}\, a \right ) c \right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, \left (a \,d^{2}+b \,c^{2}\right )^{3} \left (d x +c \right )^{2} d}\)	\(504\)

Input:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

2*e^4*(1/e*a/(a*d^2+b*c^2)^3*(1/8*(a^2*d^3*e-3*a*b*c^2*d*e)*(a*e^2/b)^(1/4 
)/a/e^2*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*(3*a*b*c*d^2-b^2*c^3)/b/(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/(a*d^2+b*c^ 
2)^3/e*(((-9/8*a^2*d^4-5/4*b*c^2*d^2*a-1/8*b^2*c^4)*(e*x)^(3/2)-1/8*c*e*(7 
*a^2*d^4+6*a*b*c^2*d^2-b^2*c^4)/d*(e*x)^(1/2))/(d*e*x+c*e)^2+1/8*(15*a^2*d 
^4-18*a*b*c^2*d^2-b^2*c^4)/d/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1 
/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6629 vs. \(2 (401) = 802\).

Time = 38.99 (sec) , antiderivative size = 13275, normalized size of antiderivative = 26.29 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((e*x)**(5/2)/(d*x+c)**3/(b*x**2+a),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(5/2)/(d*x+c)^3/(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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (401) = 802\).

Time = 0.20 (sec) , antiderivative size = 902, normalized size of antiderivative = 1.79 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a),x, algorithm="giac")
 

Output:

-1/4*e^2*(4*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e - (a*b^3*e^2)^(1/4)*a^2*b*d 
^3*e + (a*b^3*e^2)^(3/4)*b*c^3 - 3*(a*b^3*e^2)^(3/4)*a*c*d^2)*arctan(1/2*s 
qrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*b 
^5*c^6*e + 3*sqrt(2)*a*b^4*c^4*d^2*e + 3*sqrt(2)*a^2*b^3*c^2*d^4*e + sqrt( 
2)*a^3*b^2*d^6*e) + 4*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e - (a*b^3*e^2)^(1/ 
4)*a^2*b*d^3*e + (a*b^3*e^2)^(3/4)*b*c^3 - 3*(a*b^3*e^2)^(3/4)*a*c*d^2)*ar 
ctan(-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^5*c^6*e + 3*sqrt(2)*a*b^4*c^4*d^2*e + 3*sqrt(2)*a^2*b^3*c^2*d^ 
4*e + sqrt(2)*a^3*b^2*d^6*e) + 2*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e - (a*b 
^3*e^2)^(1/4)*a^2*b*d^3*e - (a*b^3*e^2)^(3/4)*b*c^3 + 3*(a*b^3*e^2)^(3/4)* 
a*c*d^2)*log(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqr 
t(2)*b^5*c^6*e + 3*sqrt(2)*a*b^4*c^4*d^2*e + 3*sqrt(2)*a^2*b^3*c^2*d^4*e + 
 sqrt(2)*a^3*b^2*d^6*e) - 2*(3*(a*b^3*e^2)^(1/4)*a*b^2*c^2*d*e - (a*b^3*e^ 
2)^(1/4)*a^2*b*d^3*e - (a*b^3*e^2)^(3/4)*b*c^3 + 3*(a*b^3*e^2)^(3/4)*a*c*d 
^2)*log(e*x - sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)* 
b^5*c^6*e + 3*sqrt(2)*a*b^4*c^4*d^2*e + 3*sqrt(2)*a^2*b^3*c^2*d^4*e + sqrt 
(2)*a^3*b^2*d^6*e) - (b^2*c^5*e + 18*a*b*c^3*d^2*e - 15*a^2*c*d^4*e)*arcta 
n(sqrt(e*x)*d/sqrt(c*d*e))/((b^3*c^6*d + 3*a*b^2*c^4*d^3 + 3*a^2*b*c^2*d^5 
 + a^3*d^7)*sqrt(c*d*e)) - (sqrt(e*x)*b*c^3*d*e^2*x + 9*sqrt(e*x)*a*c*d^3* 
e^2*x - sqrt(e*x)*b*c^4*e^2 + 7*sqrt(e*x)*a*c^2*d^2*e^2)/((b^2*c^4*d + ...
 

Mupad [B] (verification not implemented)

Time = 11.74 (sec) , antiderivative size = 28575, normalized size of antiderivative = 56.58 \[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\text {Too large to display} \] Input:

int((e*x)^(5/2)/((a + b*x^2)*(c + d*x)^3),x)
 

Output:

- atan(((((764*a^4*b^10*c^14*d^3*e^18 + 5892*a^5*b^9*c^12*d^5*e^18 - 10996 
*a^6*b^8*c^10*d^7*e^18 - 16644*a^7*b^7*c^8*d^9*e^18 + 14772*a^8*b^6*c^6*d^ 
11*e^18 + 10476*a^9*b^5*c^4*d^13*e^18 - 4796*a^10*b^4*c^2*d^15*e^18 + 20*a 
^3*b^11*c^16*d*e^18)/(2*(a^8*d^17 + b^8*c^16*d + 8*a*b^7*c^14*d^3 + 8*a^7* 
b*c^2*d^15 + 28*a^2*b^6*c^12*d^5 + 56*a^3*b^5*c^10*d^7 + 70*a^4*b^4*c^8*d^ 
9 + 56*a^5*b^3*c^6*d^11 + 28*a^6*b^2*c^4*d^13)) + (((1408*a^3*b^13*c^19*d^ 
4*e^13 + 9600*a^4*b^12*c^17*d^6*e^13 + 26112*a^5*b^11*c^15*d^8*e^13 + 3225 
6*a^6*b^10*c^13*d^10*e^13 + 5376*a^7*b^9*c^11*d^12*e^13 - 37632*a^8*b^8*c^ 
9*d^14*e^13 - 53760*a^9*b^7*c^7*d^16*e^13 - 35328*a^10*b^6*c^5*d^18*e^13 - 
 11904*a^11*b^5*c^3*d^20*e^13 - 1664*a^12*b^4*c*d^22*e^13)/(2*(a^8*d^17 + 
b^8*c^16*d + 8*a*b^7*c^14*d^3 + 8*a^7*b*c^2*d^15 + 28*a^2*b^6*c^12*d^5 + 5 
6*a^3*b^5*c^10*d^7 + 70*a^4*b^4*c^8*d^9 + 56*a^5*b^3*c^6*d^11 + 28*a^6*b^2 
*c^4*d^13)) - ((e*x)^(1/2)*(-(a^3*d^6*e^5*(-a^3*b)^(1/2) - b^3*c^6*e^5*(-a 
^3*b)^(1/2) - 20*a^3*b^2*c^3*d^3*e^5 + 6*a^4*b*c*d^5*e^5 + 6*a^2*b^3*c^5*d 
*e^5 + 15*a*b^2*c^4*d^2*e^5*(-a^3*b)^(1/2) - 15*a^2*b*c^2*d^4*e^5*(-a^3*b) 
^(1/2))/(4*(b^7*c^12 + a^6*b*d^12 + 6*a*b^6*c^10*d^2 + 15*a^2*b^5*c^8*d^4 
+ 20*a^3*b^4*c^6*d^6 + 15*a^4*b^3*c^4*d^8 + 6*a^5*b^2*c^2*d^10)))^(1/2)*(5 
12*a^13*b^4*d^26*e^10 - 512*a^2*b^15*c^22*d^4*e^10 - 4608*a^3*b^14*c^20*d^ 
6*e^10 - 17920*a^4*b^13*c^18*d^8*e^10 - 38400*a^5*b^12*c^16*d^10*e^10 - 46 
080*a^6*b^11*c^14*d^12*e^10 - 21504*a^7*b^10*c^12*d^14*e^10 + 21504*a^8...
 

Reduce [F]

\[ \int \frac {(e x)^{5/2}}{(c+d x)^3 \left (a+b x^2\right )} \, dx=\int \frac {\left (e x \right )^{\frac {5}{2}}}{\left (d x +c \right )^{3} \left (b \,x^{2}+a \right )}d x \] Input:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a),x)
 

Output:

int((e*x)^(5/2)/(d*x+c)^3/(b*x^2+a),x)