\(\int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [314]

Optimal result

Integrand size = 39, antiderivative size = 207 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {3 c^2 d^2 \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2}} \] Output:

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)/(e*x+d)^(5/2)+3 
/4*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^(3 
/2)-3/4*c^2*d^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))/e^(1/2)/(-a*e^2+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {e} \sqrt {c d^2-a e^2} \left (-2 a^2 e^3+a c d e (5 d+e x)+c^2 d^2 x (5 d+3 e x)\right )+3 c^2 d^2 \sqrt {a e+c d x} (d+e x)^2 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{4 \sqrt {e} \left (c d^2-a e^2\right )^{5/2} (d+e x)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {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.

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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))
 

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 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 = 1.09 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{2} x^{2}+6 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{4}-3 c d e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+2 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,e^{2}-5 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c \,d^{2}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(282\)

Input:

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

Output:

-1/4/(e*x+d)^(5/2)*((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^2*d^2*e^2*x^2+6*arctanh(e*(c*d*x+a*e)^(1/2) 
/((a*e^2-c*d^2)*e)^(1/2))*c^2*d^3*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^ 
2-c*d^2)*e)^(1/2))*c^2*d^4-3*c*d*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^( 
1/2)+2*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*e^2-5*((a*e^2-c*d^2)*e) 
^(1/2)*(c*d*x+a*e)^(1/2)*c*d^2)/(c*d*x+a*e)^(1/2)/(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. 424 vs. \(2 (181) = 362\).

Time = 0.11 (sec) , antiderivative size = 869, normalized size of antiderivative = 4.20 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*s 
qrt(-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*(5*c^2*d^4*e - 7*a*c*d^2*e^3 + 2* 
a^2*e^5 + 3*(c^2*d^3*e^2 - a*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
 a*e^2)*x)*sqrt(e*x + d))/(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - 
 a^3*d^3*e^7 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10 
)*x^3 + 3*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^ 
2 + 3*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*x), 
-1/4*(3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*sq 
rt(c*d^2*e - a*e^3)*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq 
rt(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)) 
 - (5*c^2*d^4*e - 7*a*c*d^2*e^3 + 2*a^2*e^5 + 3*(c^2*d^3*e^2 - a*c*d*e^4)* 
x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^9*e - 
 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7 + (c^3*d^6*e^4 - 3*a*c^2* 
d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10)*x^3 + 3*(c^3*d^7*e^3 - 3*a*c^2*d^5*e 
^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^2 + 3*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 
 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\frac {3 \, c^{3} d^{3} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} + \frac {5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} - 5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}}{4 \, c d {\left | e \right |}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-3 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{4}-6 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{3} e x -3 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{2} e^{2} x^{2}-2 \sqrt {c d x +a e}\, a^{2} e^{5}+7 \sqrt {c d x +a e}\, a c \,d^{2} e^{3}+3 \sqrt {c d x +a e}\, a c d \,e^{4} x -5 \sqrt {c d x +a e}\, c^{2} d^{4} e -3 \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x}{4 e \left (a^{3} e^{8} x^{2}-3 a^{2} c \,d^{2} e^{6} x^{2}+3 a \,c^{2} d^{4} e^{4} x^{2}-c^{3} d^{6} e^{2} x^{2}+2 a^{3} d \,e^{7} x -6 a^{2} c \,d^{3} e^{5} x +6 a \,c^{2} d^{5} e^{3} x -2 c^{3} d^{7} e x +a^{3} d^{2} e^{6}-3 a^{2} c \,d^{4} e^{4}+3 a \,c^{2} d^{6} e^{2}-c^{3} d^{8}\right )} \] Input:

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