\(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [326]

Optimal result

Integrand size = 39, antiderivative size = 332 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{2 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {7 c d \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {35 c^2 d^2 (d+e x)^{3/2}}{12 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {35 c^2 d^2 e \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^2 d^2 e^{3/2} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )^{9/2}} \] Output:

1/2/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+7 
/4*c*d*(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)^(3 
/2)-35/12*c^2*d^2*(e*x+d)^(3/2)/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(3/2)+35/4*c^2*d^2*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2 
+c*d^2)*x+c*d*e*x^2)^(1/2)-35/4*c^2*d^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)^(9/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^2 d^2 (d+e x)^{5/2} \left (-\frac {(a e+c d x) \left (6 a^3 e^6-3 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (40 d^2+119 d e x+70 e^2 x^2\right )+c^3 d^3 \left (8 d^3-56 d^2 e x-175 d e^2 x^2-105 e^3 x^3\right )\right )}{c^2 d^2 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {105 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}\right )}{12 ((a e+c d x) (d+e x))^{5/2}} \] Input:

Integrate[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)), 
x]
 

Output:

(c^2*d^2*(d + e*x)^(5/2)*(-(((a*e + c*d*x)*(6*a^3*e^6 - 3*a^2*c*d*e^4*(13* 
d + 7*e*x) - 2*a*c^2*d^2*e^2*(40*d^2 + 119*d*e*x + 70*e^2*x^2) + c^3*d^3*( 
8*d^3 - 56*d^2*e*x - 175*d*e^2*x^2 - 105*e^3*x^3)))/(c^2*d^2*(c*d^2 - a*e^ 
2)^4*(d + e*x)^2)) + (105*e^(3/2)*(a*e + c*d*x)^(5/2)*ArcTan[(Sqrt[e]*Sqrt 
[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(9/2)))/(12*((a*e + c 
*d*x)*(d + e*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1135, 1132, 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.

\(\displaystyle \int \frac {1}{\sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {7 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1132

\(\displaystyle \frac {7 c d \left (-\frac {5 e \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1132

\(\displaystyle \frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 e^2 \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

Input:

Int[1/(Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
 

Output:

1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) 
^(3/2)) + (7*c*d*((-2*Sqrt[d + e*x])/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + 
a*e^2)*x + c*d*e*x^2)^(3/2)) - (5*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]*Arc 
Tan[(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^2 - a*e^2))))/(4*(c*d^2 - a*e^2))
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(657\) vs. \(2(294)=588\).

Time = 1.08 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.98

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{4} x^{3} \sqrt {c d x +a e}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{2} e^{5} x^{2} \sqrt {c d x +a e}+210 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{3} x^{2} \sqrt {c d x +a e}+210 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{3} e^{4} x \sqrt {c d x +a e}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e^{2} x \sqrt {c d x +a e}-105 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a \,c^{2} d^{4} e^{3} \sqrt {c d x +a e}-140 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}-175 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}-21 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -238 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -56 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{5} e x +6 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} e^{6}-39 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-80 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}+8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{12 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(658\)

Input:

int(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETU 
RNVERBOSE)
 

Output:

-1/12*((e*x+d)*(c*d*x+a*e))^(1/2)*(105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2 
-c*d^2)*e)^(1/2))*c^3*d^3*e^4*x^3*(c*d*x+a*e)^(1/2)+105*arctanh(e*(c*d*x+a 
*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^2*d^2*e^5*x^2*(c*d*x+a*e)^(1/2)+210 
*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^3*d^4*e^3*x^2*(c*d 
*x+a*e)^(1/2)+210*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c 
^2*d^3*e^4*x*(c*d*x+a*e)^(1/2)+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d 
^2)*e)^(1/2))*c^3*d^5*e^2*x*(c*d*x+a*e)^(1/2)-105*((a*e^2-c*d^2)*e)^(1/2)* 
c^3*d^3*e^3*x^3+105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a 
*c^2*d^4*e^3*(c*d*x+a*e)^(1/2)-140*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^2*e^4*x 
^2-175*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^4*e^2*x^2-21*((a*e^2-c*d^2)*e)^(1/2)* 
a^2*c*d*e^5*x-238*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^3*e^3*x-56*((a*e^2-c*d^2 
)*e)^(1/2)*c^3*d^5*e*x+6*((a*e^2-c*d^2)*e)^(1/2)*a^3*e^6-39*((a*e^2-c*d^2) 
*e)^(1/2)*a^2*c*d^2*e^4-80*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4*e^2+8*((a*e^2 
-c*d^2)*e)^(1/2)*c^3*d^6)/(e*x+d)^(5/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^4/((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. 878 vs. \(2 (294) = 588\).

Time = 0.37 (sec) , antiderivative size = 1778, normalized size of antiderivative = 5.36 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor 
ithm="fricas")
 

Output:

[1/24*(105*(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5*e^3 + 2*a*c^3*d 
^3*e^5)*x^4 + (3*c^4*d^6*e^2 + 6*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^3 + (c 
^4*d^7*e + 6*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5)*x^2 + (2*a*c^3*d^6*e^2 + 3 
*a^2*c^2*d^4*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*(105*c^3*d^3*e^3*x^3 - 8*c^3*d^6 + 80*a*c^2*d^4*e^2 + 39*a^2*c*d^ 
2*e^4 - 6*a^3*e^6 + 35*(5*c^3*d^4*e^2 + 4*a*c^2*d^2*e^4)*x^2 + 7*(8*c^3*d^ 
5*e + 34*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 
 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^4*d^11*e^2 - 4*a^3*c^3*d^9*e^4 + 6*a^4* 
c^2*d^7*e^6 - 4*a^5*c*d^5*e^8 + a^6*d^3*e^10 + (c^6*d^10*e^3 - 4*a*c^5*d^8 
*e^5 + 6*a^2*c^4*d^6*e^7 - 4*a^3*c^3*d^4*e^9 + a^4*c^2*d^2*e^11)*x^5 + (3* 
c^6*d^11*e^2 - 10*a*c^5*d^9*e^4 + 10*a^2*c^4*d^7*e^6 - 5*a^4*c^2*d^3*e^10 
+ 2*a^5*c*d*e^12)*x^4 + (3*c^6*d^12*e - 6*a*c^5*d^10*e^3 - 5*a^2*c^4*d^8*e 
^5 + 20*a^3*c^3*d^6*e^7 - 15*a^4*c^2*d^4*e^9 + 2*a^5*c*d^2*e^11 + a^6*e^13 
)*x^3 + (c^6*d^13 + 2*a*c^5*d^11*e^2 - 15*a^2*c^4*d^9*e^4 + 20*a^3*c^3*d^7 
*e^6 - 5*a^4*c^2*d^5*e^8 - 6*a^5*c*d^3*e^10 + 3*a^6*d*e^12)*x^2 + (2*a*c^5 
*d^12*e - 5*a^2*c^4*d^10*e^3 + 10*a^4*c^2*d^6*e^7 - 10*a^5*c*d^4*e^9 + 3*a 
^6*d^2*e^11)*x), 1/12*(105*(c^4*d^4*e^4*x^5 + a^2*c^2*d^5*e^3 + (3*c^4*d^5 
*e^3 + 2*a*c^3*d^3*e^5)*x^4 + (3*c^4*d^6*e^2 + 6*a*c^3*d^4*e^4 + a^2*c^...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \] Input:

integrate(1/(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
 

Output:

Integral(1/(((d + e*x)*(a*e + c*d*x))**(5/2)*sqrt(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \] Input:

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor 
ithm="maxima")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*sqrt(e*x + d)), 
 x)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{12} \, {\left (\frac {105 \, c^{2} d^{2} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {8 \, {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4} - 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{2} d^{2} e\right )}}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {3 \, {\left (13 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 13 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e\right )}}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} {\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}\right )} e^{2} \] Input:

integrate(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algor 
ithm="giac")
 

Output:

1/12*(105*c^2*d^2*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^4*d^8*abs(e) - 4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4* 
e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*sqrt(c*d^2*e - a*e^3 
)) - 8*(c^3*d^4*e^2 - a*c^2*d^2*e^4 - 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3 
)*c^2*d^2*e)/((c^4*d^8*abs(e) - 4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4 
*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*((e*x + d)*c*d*e - c*d^ 
2*e + a*e^3)^(3/2)) + 3*(13*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^ 
4*e^2 - 13*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e^4 + 11*((e* 
x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^2*d^2*e)/((c^4*d^8*abs(e) - 4*a*c^ 
3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4 
*e^8*abs(e))*(e*x + d)^2*c^2*d^2*e^2))*e^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {d+e\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int(1/((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
 

Output:

int(1/((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 930, normalized size of antiderivative = 2.80 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

( - 105*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*c**2*d**4*e**2 - 210*sqr 
t(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*c**2*d**3*e**3*x - 105*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*c**2*d**2*e**4*x**2 - 105*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)))*c**3*d**5*e*x - 210*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)))*c**3*d**4*e**2*x**2 - 105*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)))* 
c**3*d**3*e**3*x**3 - 6*a**4*e**8 + 45*a**3*c*d**2*e**6 + 21*a**3*c*d*e**7 
*x + 41*a**2*c**2*d**4*e**4 + 217*a**2*c**2*d**3*e**5*x + 140*a**2*c**2*d* 
*2*e**6*x**2 - 88*a*c**3*d**6*e**2 - 182*a*c**3*d**5*e**3*x + 35*a*c**3*d* 
*4*e**4*x**2 + 105*a*c**3*d**3*e**5*x**3 + 8*c**4*d**8 - 56*c**4*d**7*e*x 
- 175*c**4*d**6*e**2*x**2 - 105*c**4*d**5*e**3*x**3)/(12*sqrt(a*e + c*d*x) 
*(a**6*d**2*e**11 + 2*a**6*d*e**12*x + a**6*e**13*x**2 - 5*a**5*c*d**4*e** 
9 - 9*a**5*c*d**3*e**10*x - 3*a**5*c*d**2*e**11*x**2 + a**5*c*d*e**12*x**3 
 + 10*a**4*c**2*d**6*e**7 + 15*a**4*c**2*d**5*e**8*x - 5*a**4*c**2*d**3*e* 
*10*x**3 - 10*a**3*c**3*d**8*e**5 - 10*a**3*c**3*d**7*e**6*x + 10*a**3*...