\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [292]

Optimal result

Integrand size = 39, antiderivative size = 181 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {e} \sqrt {a e+c d x} \left (4 a e^2+c d (-3 d+e x)\right )+3 \left (c d^2-a e^2\right )^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 e^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1131, 1131, 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.

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) - 
((c*d^2 - a*e^2)*((2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*Sqrt[ 
d + e*x]) - (2*Sqrt[c*d^2 - a*e^2]*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])])/e^(3/2)))/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 1131

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 + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*p + 1, 0] && IntegerQ[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 = 265, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a^{2} e^{4}-6 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) a c \,d^{2} e^{2}+3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{4}-c d e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-4 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,e^{2}+3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c \,d^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, e^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(265\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(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))*a^2*e^4-6*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/ 
2))*a*c*d^2*e^2+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2 
*d^4-c*d*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-4*((a*e^2-c*d^2)*e) 
^(1/2)*(c*d*x+a*e)^(1/2)*a*e^2+3*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2) 
*c*d^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/e^2/((a*e^2-c*d^2)*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{e}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} e \sqrt {-\frac {c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x - 3 \, c d^{2} + 4 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (e^{3} x + d e^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{e}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} e \sqrt {\frac {c d^{2} - a e^{2}}{e}}}{c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x}\right ) - \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x - 3 \, c d^{2} + 4 \, a e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, {\left (e^{3} x + d e^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (c^{2} d^{4} {\left | e \right |} - 2 \, a c d^{2} e^{2} {\left | e \right |} + a^{2} e^{4} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e^{5} {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{7} {\left | e \right |} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{4} {\left | e \right |}}{e^{6}}\right )}}{3 \, e^{3}} \] Input:

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

Output:

2/3*(3*(c^2*d^4*abs(e) - 2*a*c*d^2*e^2*abs(e) + a^2*e^4*abs(e))*arctan(sqr 
t((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))/sqrt(c*d^2*e - 
 a*e^3) - (3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d^2*e^5*abs(e) - 3* 
sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*e^7*abs(e) - ((e*x + d)*c*d*e - 
c*d^2*e + a*e^3)^(3/2)*e^4*abs(e))/e^6)/e^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {-2 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) a \,e^{2}+2 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c \,d^{2}+\frac {8 \sqrt {c d x +a e}\, a \,e^{3}}{3}-2 \sqrt {c d x +a e}\, c \,d^{2} e +\frac {2 \sqrt {c d x +a e}\, c d \,e^{2} x}{3}}{e^{3}} \] Input:

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

Output:

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