\(\int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [313]

Optimal result

Integrand size = 39, antiderivative size = 140 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {c d \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}} \] Output:

(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)^(3/2)-c*d*a 
rctan(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)^(3/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {e} \sqrt {c d^2-a e^2} (a e+c d x)+c d \sqrt {a e+c d x} (d+e x) \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2} \sqrt {d+e x} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {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.

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

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

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 = 1.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c d e x +\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c \,d^{2}-\sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(162\)

Input:

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETU 
RNVERBOSE)
 

Output:

((e*x+d)*(c*d*x+a*e))^(1/2)*(arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e) 
^(1/2))*c*d*e*x+arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d^2 
-(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2))/(e*x+d)^(3/2)/(c*d*x+a*e)^(1/2 
)/(a*e^2-c*d^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. 268 vs. \(2 (124) = 248\).

Time = 0.10 (sec) , antiderivative size = 557, normalized size of antiderivative = 3.98 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{2 \, {\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}}, -\frac {{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x}\right ) - \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} e - a e^{3}\right )} \sqrt {e x + d}}{c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x}\right ] \] Input:

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algor 
ithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**(3/2)/(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)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x+d)^(3/2)/(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)^(3/2)), 
 x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {c d e^{2} {\left (\frac {\arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c d^{2} e - a e^{3}\right )}^{\frac {3}{2}}} + \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{{\left (c d^{2} e - a e^{3}\right )} {\left (e x + d\right )} c d e}\right )}}{{\left | e \right |}} \] Input:

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algor 
ithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)
 

Output:

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c \,d^{2}+\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c d e x -\sqrt {c d x +a e}\, a \,e^{3}+\sqrt {c d x +a e}\, c \,d^{2} e}{e \left (a^{2} e^{5} x -2 a c \,d^{2} e^{3} x +c^{2} d^{4} e x +a^{2} d \,e^{4}-2 a c \,d^{3} e^{2}+c^{2} d^{5}\right )} \] Input:

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

(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*d**2 + sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqr 
t(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c*d*e*x - sqrt(a*e + 
 c*d*x)*a*e**3 + sqrt(a*e + c*d*x)*c*d**2*e)/(e*(a**2*d*e**4 + a**2*e**5*x 
 - 2*a*c*d**3*e**2 - 2*a*c*d**2*e**3*x + c**2*d**5 + c**2*d**4*e*x))