\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^4} \, dx\) [216]

Optimal result

Integrand size = 37, antiderivative size = 157 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 (d+e x)}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^3}+\frac {2 c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {e} \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}+\frac {3 c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{3 e^{5/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1130, 1125, 25, 27, 1092, 219}

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.

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,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) + (c* 
d*((-2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*(d + e*x)) + (Sqrt[ 
c]*Sqrt[d]*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]* 
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/e^(3/2)))/e
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1125

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 
2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[(1/Sqrt[a + b*x + c*x^2])*Expan 
dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x 
), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && ILtQ[m, 0] && EqQ[m + p, -3/2]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(137)=274\).

Time = 2.43 (sec) , antiderivative size = 473, normalized size of antiderivative = 3.01

method	result	size
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {2 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{4}}\)	\(473\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^4,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^4*(-2/3/(a*e^2-c*d^2)/(x+d/e)^4*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e) 
)^(5/2)+2/3*d*e*c/(a*e^2-c*d^2)*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e) 
^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/(x+ 
d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c*d^2) 
*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4 
*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e 
))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/ 
(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2) 
)))))
 

Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\left [\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {\frac {c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, {\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {c d}{e}} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-\frac {c d}{e}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {c d}{e}}}{2 \, {\left (c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm=" 
fricas")
 

Output:

[1/6*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(c*d/e)*log(8*c^2*d^2*e^2* 
x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3 
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + 
 a*c*d*e^3)*x) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x 
+ 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/3*(3*(c*d*e^2*x^2 
+ 2*c*d^2*e*x + c*d^3)*sqrt(-c*d/e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c 
*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + 
 a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 
 + a*e^2)*x)*(4*c*d*e*x + 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2 
)]
 

Sympy [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{4}}\, dx \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**4,x)
 

Output:

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)/(d + e*x)**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm=" 
maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x, algorithm=" 
giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[0,0,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,4 
]%%%}+%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^4,x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^4, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,e^{3}}{3}-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c \,d^{2} e -\frac {8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c d \,e^{2} x}{3}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{3}+4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2} e x +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c d \,e^{2} x^{2}}{e^{3} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^4,x)
 

Output:

(2*( - sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*e**3 - 3*sqrt(d + e*x)*sqrt(a*e + 
 c*d*x)*c*d**2*e - 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c*d*e**2*x + 3*sqrt(e 
)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d 
+ e*x))/sqrt(a*e**2 - c*d**2))*c*d**3 + 6*sqrt(e)*sqrt(d)*sqrt(c)*log((sqr 
t(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d* 
*2))*c*d**2*e*x + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) 
 + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c*d*e**2*x**2))/( 
3*e**3*(d**2 + 2*d*e*x + e**2*x**2))