\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^{9/2}} \, dx\) [286]

Optimal result

Integrand size = 39, antiderivative size = 264 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}} \] Output:

-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(e*x+d)^(7/2)+1/12*c*d*(a*d 
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(-a*e^2+c*d^2)/(e*x+d)^(5/2)+1/8*c^2 
*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e/(-a*e^2+c*d^2)^2/(e*x+d)^(3 
/2)+1/8*c^3*d^3*arctan(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))/e^(3/2)/(-a*e^2+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {a e+c d x} \left (-8 a^2 e^4+2 a c d e^2 (7 d-e x)+c^2 d^2 \left (-3 d^2+8 d e x+3 e^2 x^2\right )\right )+3 c^3 d^3 (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{24 e^{3/2} \left (c d^2-a e^2\right )^{5/2} \sqrt {a e+c d x} (d+e x)^{7/2}} \] Input:

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d 
*x]*(-8*a^2*e^4 + 2*a*c*d*e^2*(7*d - e*x) + c^2*d^2*(-3*d^2 + 8*d*e*x + 3* 
e^2*x^2)) + 3*c^3*d^3*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[ 
c*d^2 - a*e^2]]))/(24*e^(3/2)*(c*d^2 - a*e^2)^(5/2)*Sqrt[a*e + c*d*x]*(d + 
 e*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1130, 1135, 1135, 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.

\(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{9/2}} \, dx\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {c d \int \frac {1}{(d+e x)^{5/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {c d \left (\frac {3 c d \int \frac {1}{(d+e x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d e \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\)

Input:

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d + e*x)^(9/2),x]
 

Output:

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

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 1130

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] 
- Simp[c*(p/(e^2*(m + 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] 
 && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & 
& 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 = 1.08 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.69

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+2 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-8 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}-14 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}+3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, e \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(447\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^(9/2),x,method=_RETURN 
VERBOSE)
 

Output:

-1/24*((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^3*d^3*e^3*x^3+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^ 
2)*e)^(1/2))*c^3*d^4*e^2*x^2+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)* 
e)^(1/2))*c^3*d^5*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2 
))*c^3*d^6-3*c^2*d^2*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*a 
*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-8*c^2*d^3*e*x*(c*d*x+ 
a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^( 
1/2)*a^2*e^4-14*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e^2+3*(( 
a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/((a*e^2-c*d 
^2)*e)^(1/2)/e/(a*e^2-c*d^2)^2/(c*d*x+a*e)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (232) = 464\).

Time = 0.13 (sec) , antiderivative size = 1114, normalized size of antiderivative = 4.22 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorit 
hm="fricas")
 

Output:

[-1/48*(3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3 
*d^6*e*x + c^3*d^7)*sqrt(-c*d^2*e + a*e^3)*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)*sqrt(-c 
*d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^3*d^6*e 
 - 17*a*c^2*d^4*e^3 + 22*a^2*c*d^2*e^5 - 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^ 
2*d^2*e^5)*x^2 - 2*(4*c^3*d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt 
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^10*e^2 - 3*a 
*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8 + (c^3*d^6*e^6 - 3*a*c^2*d^4* 
e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12)*x^4 + 4*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 
+ 3*a^2*c*d^3*e^9 - a^3*d*e^11)*x^3 + 6*(c^3*d^8*e^4 - 3*a*c^2*d^6*e^6 + 3 
*a^2*c*d^4*e^8 - a^3*d^2*e^10)*x^2 + 4*(c^3*d^9*e^3 - 3*a*c^2*d^7*e^5 + 3* 
a^2*c*d^5*e^7 - a^3*d^3*e^9)*x), -1/24*(3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3 
*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7)*sqrt(c*d^2*e - a*e^3)* 
arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)* 
sqrt(e*x + d)/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)) + (3*c^3*d^6*e - 17 
*a*c^2*d^4*e^3 + 22*a^2*c*d^2*e^5 - 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^2*d^2 
*e^5)*x^2 - 2*(4*c^3*d^5*e^2 - 5*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x)*sqrt(c*d* 
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^10*e^2 - 3*a*c^2* 
d^8*e^4 + 3*a^2*c*d^6*e^6 - a^3*d^4*e^8 + (c^3*d^6*e^6 - 3*a*c^2*d^4*e^8 + 
 3*a^2*c*d^2*e^10 - a^3*e^12)*x^4 + 4*(c^3*d^7*e^5 - 3*a*c^2*d^5*e^7 + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorit 
hm="maxima")
 

Output:

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {1}{24} \, c^{3} d^{3} e {\left (\frac {3 \, \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{4} e^{2} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c d^{2} e^{4} + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} e^{6} - 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} e + 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}}{{\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} {\left (e x + d\right )}^{3} c^{3} d^{3} e^{3}}\right )} {\left | e \right |} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x, algorit 
hm="giac")
 

Output:

1/24*c^3*d^3*e*(3*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*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*sqrt(c*d^2*e - a*e^ 
3)) - (3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^4*e^2 - 6*sqrt((e*x 
 + d)*c*d*e - c*d^2*e + a*e^3)*a*c*d^2*e^4 + 3*sqrt((e*x + d)*c*d*e - c*d^ 
2*e + a*e^3)*a^2*e^6 - 8*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d^2*e 
 + 8*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e 
- c*d^2*e + a*e^3)^(5/2))/((c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*(e*x + 
d)^3*c^3*d^3*e^3))*abs(e)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{9/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(9/2),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx=\frac {-3 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{6}-9 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{5} e x -9 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{4} e^{2} x^{2}-3 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{3} e^{3} x^{3}-8 \sqrt {c d x +a e}\, a^{3} e^{7}+22 \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{5}-2 \sqrt {c d x +a e}\, a^{2} c d \,e^{6} x -17 \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{3}+10 \sqrt {c d x +a e}\, a \,c^{2} d^{3} e^{4} x +3 \sqrt {c d x +a e}\, a \,c^{2} d^{2} e^{5} x^{2}+3 \sqrt {c d x +a e}\, c^{3} d^{6} e -8 \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x -3 \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}}{24 e^{2} \left (a^{3} e^{9} x^{3}-3 a^{2} c \,d^{2} e^{7} x^{3}+3 a \,c^{2} d^{4} e^{5} x^{3}-c^{3} d^{6} e^{3} x^{3}+3 a^{3} d \,e^{8} x^{2}-9 a^{2} c \,d^{3} e^{6} x^{2}+9 a \,c^{2} d^{5} e^{4} x^{2}-3 c^{3} d^{7} e^{2} x^{2}+3 a^{3} d^{2} e^{7} x -9 a^{2} c \,d^{4} e^{5} x +9 a \,c^{2} d^{6} e^{3} x -3 c^{3} d^{8} e x +a^{3} d^{3} e^{6}-3 a^{2} c \,d^{5} e^{4}+3 a \,c^{2} d^{7} e^{2}-c^{3} d^{9}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(9/2),x)
 

Output:

( - 3*sqrt(e)*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**6 - 9*sqrt(e)*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 - 9*sqrt(e)*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 - 3*sqrt(e)*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 - 8*sqrt(a*e + c*d*x)*a**3*e**7 + 22*sqrt(a*e + c*d* 
x)*a**2*c*d**2*e**5 - 2*sqrt(a*e + c*d*x)*a**2*c*d*e**6*x - 17*sqrt(a*e + 
c*d*x)*a*c**2*d**4*e**3 + 10*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x + 3*sqrt 
(a*e + c*d*x)*a*c**2*d**2*e**5*x**2 + 3*sqrt(a*e + c*d*x)*c**3*d**6*e - 8* 
sqrt(a*e + c*d*x)*c**3*d**5*e**2*x - 3*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x* 
*2)/(24*e**2*(a**3*d**3*e**6 + 3*a**3*d**2*e**7*x + 3*a**3*d*e**8*x**2 + a 
**3*e**9*x**3 - 3*a**2*c*d**5*e**4 - 9*a**2*c*d**4*e**5*x - 9*a**2*c*d**3* 
e**6*x**2 - 3*a**2*c*d**2*e**7*x**3 + 3*a*c**2*d**7*e**2 + 9*a*c**2*d**6*e 
**3*x + 9*a*c**2*d**5*e**4*x**2 + 3*a*c**2*d**4*e**5*x**3 - c**3*d**9 - 3* 
c**3*d**8*e*x - 3*c**3*d**7*e**2*x**2 - c**3*d**6*e**3*x**3))