\(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^3} \, dx\) [39]

Optimal result

Integrand size = 40, antiderivative size = 411 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {\left (7 c d^2+a e^2\right ) (a e+c d x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^2 d^2 e^3}-\frac {\left (105 c^3 d^6-35 a c^2 d^4 e^2-5 a^2 c d^2 e^4-a^3 e^6\right ) \left (3 c d^2-5 a e^2-2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^2 d^2 e^5 \left (c d^2-a e^2\right )}-\frac {2 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e^3 \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 c d e^3 (d+e x)}+\frac {3 \left (c d^2-a e^2\right ) \left (105 c^3 d^6-35 a c^2 d^4 e^2-5 a^2 c d^2 e^4-a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{5/2} d^{5/2} e^{11/2}} \] Output:

-1/8*(a*e^2+7*c*d^2)*(c*d*x+a*e)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
/c^2/d^2/e^3-1/64*(-a^3*e^6-5*a^2*c*d^2*e^4-35*a*c^2*d^4*e^2+105*c^3*d^6)* 
(-2*c*d*e*x-5*a*e^2+3*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d 
^2/e^5/(-a*e^2+c*d^2)-2*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e^3/(- 
a*e^2+c*d^2)/(e*x+d)^3+1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e^3 
/(e*x+d)+3/64*(-a*e^2+c*d^2)*(-a^3*e^6-5*a^2*c*d^2*e^4-35*a*c^2*d^4*e^2+10 
5*c^3*d^6)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(1/2))/c^(5/2)/d^(5/2)/e^(11/2)
 

Mathematica [A] (verified)

Time = 10.88 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {c} \sqrt {d} \sqrt {e} \left (3 a^3 e^6 (d+e x)+a^2 c d e^4 \left (13 d^2+11 d e x-2 e^2 x^2\right )-a c^2 d^2 e^2 \left (315 d^3+119 d^2 e x-44 d e^2 x^2+24 e^3 x^3\right )+c^3 d^3 \left (315 d^4+105 d^3 e x-42 d^2 e^2 x^2+24 d e^3 x^3-16 e^4 x^4\right )\right )+\frac {3 \left (c d^2-a e^2\right )^{3/2} \left (105 c^3 d^6-35 a c^2 d^4 e^2-5 a^2 c d^2 e^4-a^3 e^6\right ) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {c d} \sqrt {a e+c d x}}\right )}{64 c^{5/2} d^{5/2} e^{11/2} (d+e x)} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(3*a^3*e^6*(d + 
e*x) + a^2*c*d*e^4*(13*d^2 + 11*d*e*x - 2*e^2*x^2) - a*c^2*d^2*e^2*(315*d^ 
3 + 119*d^2*e*x - 44*d*e^2*x^2 + 24*e^3*x^3) + c^3*d^3*(315*d^4 + 105*d^3* 
e*x - 42*d^2*e^2*x^2 + 24*d*e^3*x^3 - 16*e^4*x^4))) + (3*(c*d^2 - a*e^2)^( 
3/2)*(105*c^3*d^6 - 35*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 - a^3*e^6)*Sqrt[(c* 
d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + 
c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[a*e + c*d*x])))/ 
(64*c^(5/2)*d^(5/2)*e^(11/2)*(d + e*x))
 

Rubi [A] (verified)

Time = 2.41 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {1213, 25, 2192, 27, 2192, 27, 2192, 27, 1160, 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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {-3 c^2 d^2 \left (5 c d^2-3 a e^2\right ) x^3 e^6+2 c d \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x^2 e^5-8 c d^2 \left (c d^2-a e^2\right )^2 x e^4+8 c d^3 \left (c d^2-a e^2\right )^2 e^3}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-3 c^2 d^2 \left (5 c d^2-3 a e^2\right ) x^3 e^6+2 c d \left (4 c^2 d^4-11 a c e^2 d^2+4 a^2 e^4\right ) x^2 e^5-8 c d^2 \left (c d^2-a e^2\right )^2 x e^4+8 c d^3 \left (c d^2-a e^2\right )^2 e^3}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {3 \left (c^2 d^2 \left (41 c^2 d^4-34 a c e^2 d^2+a^2 e^4\right ) x^2 e^6-4 c^2 d^3 \left (4 c^2 d^4-13 a c e^2 d^2+7 a^2 e^4\right ) x e^5+16 c^2 d^4 \left (c d^2-a e^2\right )^2 e^4\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {c^2 d^2 \left (41 c^2 d^4-34 a c e^2 d^2+a^2 e^4\right ) x^2 e^6-4 c^2 d^3 \left (4 c^2 d^4-13 a c e^2 d^2+7 a^2 e^4\right ) x e^5+16 c^2 d^4 \left (c d^2-a e^2\right )^2 e^4}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {c^2 d^2 e^5 \left (2 d \left (32 c^3 d^6-105 a c^2 e^2 d^4+66 a^2 c e^4 d^2-a^3 e^6\right )-e \left (187 c^3 d^6-187 a c^2 e^2 d^4+13 a^2 c e^4 d^2+3 a^3 e^6\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c d e^5 x \left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c d e^4 \int \frac {2 d \left (32 c^3 d^6-105 a c^2 e^2 d^4+66 a^2 c e^4 d^2-a^3 e^6\right )-e \left (187 c^3 d^6-187 a c^2 e^2 d^4+13 a^2 c e^4 d^2+3 a^3 e^6\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c d e^5 x \left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c d e^4 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c d e^5 x \left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {\frac {1}{4} c d e^4 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )+\frac {1}{2} c d e^5 x \left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {1}{2} c d e^5 x \left (a^2 e^4-34 a c d^2 e^2+41 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {1}{4} c d e^4 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-a^3 e^6-5 a^2 c d^2 e^4-35 a c^2 d^4 e^2+105 c^3 d^6\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{3/2} d^{3/2} \sqrt {e}}-\frac {\left (3 a^3 e^6+13 a^2 c d^2 e^4-187 a c^2 d^4 e^2+187 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{2 c d e}-c d e^5 x^2 \left (5 c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 c d e}+\frac {1}{4} c d e^5 x^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^7}-\frac {2 d^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^5 (d+e x)}\)

Input:

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

Output:

(-2*d^3*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^5* 
(d + e*x)) + ((c*d*e^5*x^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/4 
+ (-(c*d*e^5*(5*c*d^2 - 3*a*e^2)*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2]) + ((c*d*e^5*(41*c^2*d^4 - 34*a*c*d^2*e^2 + a^2*e^4)*x*Sqrt[a*d*e + 
 (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (c*d*e^4*(-(((187*c^3*d^6 - 187*a*c^2 
*d^4*e^2 + 13*a^2*c*d^2*e^4 + 3*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2])/(c*d)) + (3*(c*d^2 - a*e^2)*(105*c^3*d^6 - 35*a*c^2*d^4*e^2 - 
5*a^2*c*d^2*e^4 - a^3*e^6)*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])])/(2*c^(3/2)* 
d^(3/2)*Sqrt[e])))/4)/(2*c*d*e))/(8*c*d*e))/e^7
 

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 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1213

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

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1191\) vs. \(2(381)=762\).

Time = 2.99 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1192\)

Input:

int(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^3,x,method=_RETURN 
VERBOSE)
 

Output:

1/e^3*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2) 
/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c* 
d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e 
^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+( 
a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))-3*d/e^4*(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)))+3/e^5*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))))-d^3/e^6*(-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...
 

Fricas [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 928, normalized size of antiderivative = 2.26 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/256*(3*(105*c^4*d^9 - 140*a*c^3*d^7*e^2 + 30*a^2*c^2*d^5*e^4 + 4*a^3*c* 
d^3*e^6 + a^4*d*e^8 + (105*c^4*d^8*e - 140*a*c^3*d^6*e^3 + 30*a^2*c^2*d^4* 
e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9)*x)*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*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) + 8*(c^2*d^3*e + a*c*d*e^3)* 
x) + 4*(16*c^4*d^4*e^5*x^4 - 315*c^4*d^8*e + 315*a*c^3*d^6*e^3 - 13*a^2*c^ 
2*d^4*e^5 - 3*a^3*c*d^2*e^7 - 24*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 + 2*(21 
*c^4*d^6*e^3 - 22*a*c^3*d^4*e^5 + a^2*c^2*d^2*e^7)*x^2 - (105*c^4*d^7*e^2 
- 119*a*c^3*d^5*e^4 + 11*a^2*c^2*d^3*e^6 + 3*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^7*x + c^3*d^4*e^6), -1/128*(3*( 
105*c^4*d^9 - 140*a*c^3*d^7*e^2 + 30*a^2*c^2*d^5*e^4 + 4*a^3*c*d^3*e^6 + a 
^4*d*e^8 + (105*c^4*d^8*e - 140*a*c^3*d^6*e^3 + 30*a^2*c^2*d^4*e^5 + 4*a^3 
*c*d^2*e^7 + a^4*e^9)*x)*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^2*x 
^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(16*c^4*d^4*e^5*x^4 - 3 
15*c^4*d^8*e + 315*a*c^3*d^6*e^3 - 13*a^2*c^2*d^4*e^5 - 3*a^3*c*d^2*e^7 - 
24*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 + 2*(21*c^4*d^6*e^3 - 22*a*c^3*d^4*e^ 
5 + a^2*c^2*d^2*e^7)*x^2 - (105*c^4*d^7*e^2 - 119*a*c^3*d^5*e^4 + 11*a^2*c 
^2*d^3*e^6 + 3*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) 
)/(c^3*d^3*e^7*x + c^3*d^4*e^6)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^3,x, algorit 
hm="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>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {1}{64} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (\frac {2 \, c d x}{e^{2}} - \frac {5 \, c^{4} d^{5} e^{16} - 3 \, a c^{3} d^{3} e^{18}}{c^{3} d^{3} e^{19}}\right )} x + \frac {41 \, c^{4} d^{6} e^{15} - 34 \, a c^{3} d^{4} e^{17} + a^{2} c^{2} d^{2} e^{19}}{c^{3} d^{3} e^{19}}\right )} x - \frac {187 \, c^{4} d^{7} e^{14} - 187 \, a c^{3} d^{5} e^{16} + 13 \, a^{2} c^{2} d^{3} e^{18} + 3 \, a^{3} c d e^{20}}{c^{3} d^{3} e^{19}}\right )} - \frac {2 \, {\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4}\right )}}{{\left ({\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} e + \sqrt {c d e} d\right )} e^{5}} - \frac {3 \, {\left (105 \, c^{4} d^{8} - 140 \, a c^{3} d^{6} e^{2} + 30 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left ({\left | c d^{2} + a e^{2} + 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{2} d^{2} e^{5}} \] Input:

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

Output:

1/64*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*c*d*x/e^2 - (5*c 
^4*d^5*e^16 - 3*a*c^3*d^3*e^18)/(c^3*d^3*e^19))*x + (41*c^4*d^6*e^15 - 34* 
a*c^3*d^4*e^17 + a^2*c^2*d^2*e^19)/(c^3*d^3*e^19))*x - (187*c^4*d^7*e^14 - 
 187*a*c^3*d^5*e^16 + 13*a^2*c^2*d^3*e^18 + 3*a^3*c*d*e^20)/(c^3*d^3*e^19) 
) - 2*(c^2*d^7 - 2*a*c*d^5*e^2 + a^2*d^3*e^4)/(((sqrt(c*d*e)*x - sqrt(c*d* 
e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*e + sqrt(c*d*e)*d)*e^5) - 3/128*(105*c 
^4*d^8 - 140*a*c^3*d^6*e^2 + 30*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^ 
8)*log(abs(c*d^2 + a*e^2 + 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + 
 c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^2*d^2*e^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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