\(\int \frac {(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [324]

Optimal result

Integrand size = 39, antiderivative size = 196 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {2 e^{3/2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \] Output:

-2/3*(e*x+d)^(3/2)/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+ 
2*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
-2*e^(3/2)*arctan(1/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e^ 
2+c*d^2)^(1/2)*(e*x+d)^(1/2))/(-a*e^2+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} \left (\sqrt {c d^2-a e^2} \left (-4 a e^2+c d (d-3 e x)\right )-3 e^{3/2} (a e+c d x)^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 \left (c d^2-a e^2\right )^{5/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1133, 1135, 1132, 1136, 218}

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.

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

\(\Big \downarrow \) 1133

\(\displaystyle -\frac {2 e \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle -\frac {2 e \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1132

\(\displaystyle -\frac {2 e \left (\frac {3 c d \left (-\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle -\frac {2 e \left (\frac {3 c d \left (-\frac {2 e^2 \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {2 e \left (\frac {3 c d \left (-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 c d}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

Input:

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

Output:

(-2*Sqrt[d + e*x])/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - 
 (2*e*(1/((c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2]) + (3*c*d*((-2*Sqrt[d + e*x])/((c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d 
^2 + a*e^2)*x + c*d*e*x^2]) - (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d 
^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(c*d^2 - 
 a*e^2)^(3/2)))/(2*(c*d^2 - a*e^2))))/(3*c*d)
 

Defintions of rubi rules used

rule 218

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

rule 1132

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 
 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 
 4*a*c)))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free 
Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ 
[0, m, 1] && IntegerQ[2*p]
 

rule 1133

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), 
 x] - Simp[e^2*((m + p)/(c*(p + 1)))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x 
^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e 
^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
 

rule 1135

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e)))   Int 
[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I 
ntegerQ[2*p]
 

rule 1136

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x 
_Symbol] :> Simp[2*e   Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + 
 b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 
- b*d*e + a*e^2, 0]
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.14

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c d \,e^{2} x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,e^{3} \sqrt {c d x +a e}-3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c d e x -4 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,e^{2}+\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c \,d^{2}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(224\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (174) = 348\).

Time = 0.14 (sec) , antiderivative size = 788, normalized size of antiderivative = 4.02 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} + {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} + {\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} + {\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} + {\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a^{2} d e^{3} + {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{2} + {\left (2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x\right )} \sqrt {\frac {e}{c d^{2} - a e^{2}}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x - c d^{2} + 4 \, a e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, {\left (a^{2} c^{2} d^{5} e^{2} - 2 \, a^{3} c d^{3} e^{4} + a^{4} d e^{6} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{3} + {\left (c^{4} d^{7} - 3 \, a^{2} c^{2} d^{3} e^{4} + 2 \, a^{3} c d e^{6}\right )} x^{2} + {\left (2 \, a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + a^{4} e^{7}\right )} x\right )}}\right ] \] Input:

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

Output:

[1/3*(3*(c^2*d^2*e^2*x^3 + a^2*d*e^3 + (c^2*d^3*e + 2*a*c*d*e^3)*x^2 + (2* 
a*c*d^2*e^2 + a^2*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a 
*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) 
*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e* 
x + d^2)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*e*x - c*d 
^2 + 4*a*e^2)*sqrt(e*x + d))/(a^2*c^2*d^5*e^2 - 2*a^3*c*d^3*e^4 + a^4*d*e^ 
6 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^3 + (c^4*d^7 - 3*a^2 
*c^2*d^3*e^4 + 2*a^3*c*d*e^6)*x^2 + (2*a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + a 
^4*e^7)*x), 2/3*(3*(c^2*d^2*e^2*x^3 + a^2*d*e^3 + (c^2*d^3*e + 2*a*c*d*e^3 
)*x^2 + (2*a*c*d^2*e^2 + a^2*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(-sqrt( 
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt( 
e/(c*d^2 - a*e^2))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + sqrt(c 
*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*e*x - c*d^2 + 4*a*e^2)*sqrt(e 
*x + d))/(a^2*c^2*d^5*e^2 - 2*a^3*c*d^3*e^4 + a^4*d*e^6 + (c^4*d^6*e - 2*a 
*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^3 + (c^4*d^7 - 3*a^2*c^2*d^3*e^4 + 2*a^3 
*c*d*e^6)*x^2 + (2*a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + a^4*e^7)*x)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit 
hm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2}{3} \, e^{2} {\left (\frac {3 \, e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} {\left | e \right |} - 2 \, a c d^{2} e^{2} {\left | e \right |} + a^{2} e^{4} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {c d^{2} e^{2} - a e^{4} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} e}{{\left (c^{2} d^{4} {\left | e \right |} - 2 \, a c d^{2} e^{2} {\left | e \right |} + a^{2} e^{4} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}\right )} \] Input:

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

Output:

2/3*e^2*(3*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - 
 a*e^3))/((c^2*d^4*abs(e) - 2*a*c*d^2*e^2*abs(e) + a^2*e^4*abs(e))*sqrt(c* 
d^2*e - a*e^3)) - (c*d^2*e^2 - a*e^4 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^ 
3)*e)/((c^2*d^4*abs(e) - 2*a*c*d^2*e^2*abs(e) + a^2*e^4*abs(e))*((e*x + d) 
*c*d*e - c*d^2*e + a*e^3)^(3/2)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {-2 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) a \,e^{2}-2 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c d e x +\frac {8 a^{2} e^{4}}{3}-\frac {10 a c \,d^{2} e^{2}}{3}+2 a c d \,e^{3} x +\frac {2 c^{2} d^{4}}{3}-2 c^{2} d^{3} e x}{\sqrt {c d x +a e}\, \left (a^{3} c d \,e^{6} x -3 a^{2} c^{2} d^{3} e^{4} x +3 a \,c^{3} d^{5} e^{2} x -c^{4} d^{7} x +a^{4} e^{7}-3 a^{3} c \,d^{2} e^{5}+3 a^{2} c^{2} d^{4} e^{3}-a \,c^{3} d^{6} e \right )} \] Input:

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

Output:

(2*( - 3*sqrt(e)*sqrt(a*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e 
 + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*a*e**2 - 3*sqrt(e)*sqrt(a 
*e + c*d*x)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*s 
qrt( - a*e**2 + c*d**2)))*c*d*e*x + 4*a**2*e**4 - 5*a*c*d**2*e**2 + 3*a*c* 
d*e**3*x + c**2*d**4 - 3*c**2*d**3*e*x))/(3*sqrt(a*e + c*d*x)*(a**4*e**7 - 
 3*a**3*c*d**2*e**5 + a**3*c*d*e**6*x + 3*a**2*c**2*d**4*e**3 - 3*a**2*c** 
2*d**3*e**4*x - a*c**3*d**6*e + 3*a*c**3*d**5*e**2*x - c**4*d**7*x))