\(\int \frac {(d+e x)^3}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [248]

Optimal result

Integrand size = 37, antiderivative size = 240 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {5 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3}+\frac {5 \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2}+\frac {(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d}+\frac {5 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} \sqrt {e}} \] Output:

5/8*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3+5/12* 
(-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2+1/3 
*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d+5/8*(-a*e^2+c*d^2)^ 
3*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^(7/2)/d^(7/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c} \sqrt {d} (a e+c d x) (d+e x) \left (15 a^2 e^4-10 a c d e^2 (4 d+e x)+c^2 d^2 \left (33 d^2+26 d e x+8 e^2 x^2\right )\right )+\frac {15 \left (c d^2-a e^2\right )^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {e}}}{24 c^{7/2} d^{7/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*(d + e*x)*(15*a^2*e^4 - 10*a*c*d*e^2*(4*d + 
 e*x) + c^2*d^2*(33*d^2 + 26*d*e*x + 8*e^2*x^2)) + (15*(c*d^2 - a*e^2)^3*S 
qrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sq 
rt[e]*Sqrt[a*e + c*d*x])])/Sqrt[e])/(24*c^(7/2)*d^(7/2)*Sqrt[(a*e + c*d*x) 
*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1134, 1134, 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.

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {3 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} d^{3/2} \sqrt {e}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{4 d}+\frac {(d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d}\right )}{6 d}+\frac {(d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\)

Input:

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

Output:

((d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d) + (5*(d^ 
2 - (a*e^2)/c)*(((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2 
*c*d) + (3*(d^2 - (a*e^2)/c)*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/ 
(c*d) + ((d^2 - (a*e^2)/c)*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*Sqrt[c]* 
d^(3/2)*Sqrt[e])))/(4*d)))/(6*d)
 

Defintions of rubi rules used

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 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(887\) vs. \(2(212)=424\).

Time = 1.31 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.70

method	result	size
default	\(\frac {d^{3} \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{\sqrt {d e c}}+e^{3} \left (\frac {x^{2} \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{3 d e c}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )}{6 d e c}-\frac {2 a \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{3 c}\right )+3 d \,e^{2} \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )+3 d^{2} e \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )\)	\(888\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.22 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 33 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (13 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{4} d^{4} e}, -\frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 33 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5} + 2 \, {\left (13 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{4} d^{4} e}\right ] \] Input:

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

Output:

[-1/96*(15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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*(8*c^3*d^3*e^3*x^2 + 33*c^3*d^5*e - 
40*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 + 2*(13*c^3*d^4*e^2 - 5*a*c^2*d^2*e^4)*x 
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e), -1/48*(15*(c^3 
*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*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*(8*c^3*d^3*e^3*x^2 + 33*c^3*d^5*e - 40*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5 
+ 2*(13*c^3*d^4*e^2 - 5*a*c^2*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 
+ a*e^2)*x))/(c^4*d^4*e)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 782 vs. \(2 (230) = 460\).

Time = 0.89 (sec) , antiderivative size = 782, normalized size of antiderivative = 3.26 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c) + 
 d**3 - (a*e**2 + c*d**2)*(-2*a*e**3/(3*c) + 3*d**2*e - (3*a*e**2/2 + 3*c* 
d**2/2)*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c*d*e))/(2* 
c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a* 
d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + 
 c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (- 
a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))* 
*2), True)) + sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(e**2*x**2/(3 
*c*d) + x*(3*d*e**2 - e**2*(5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c*d*e) + 
(-2*a*e**3/(3*c) + 3*d**2*e - (3*a*e**2/2 + 3*c*d**2/2)*(3*d*e**2 - e**2*( 
5*a*e**2/2 + 5*c*d**2/2)/(3*c*d))/(2*c*d*e))/(c*d*e)), Ne(c*d*e, 0)), (2*( 
c**3*d**9*sqrt(a*d*e + x*(a*e**2 + c*d**2))/(a**3*e**6 + 3*a**2*c*d**2*e** 
4 + 3*a*c**2*d**4*e**2 + c**3*d**6) + c**2*d**6*e*(a*d*e + x*(a*e**2 + c*d 
**2))**(3/2)/(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d 
**6) + 3*c*d**3*e**2*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/(5*(a**3*e**6 + 
3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)) + e**3*(a*d*e + x*(a 
*e**2 + c*d**2))**(7/2)/(7*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4 
*e**2 + c**3*d**6)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), (Piecewis 
e((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True))/sqrt(a*d*e), True))
 

Maxima [F(-2)]

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

integrate((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (\frac {4 \, e^{2} x}{c d} + \frac {13 \, c^{2} d^{3} e^{3} - 5 \, a c d e^{5}}{c^{3} d^{3} e^{2}}\right )} x + \frac {33 \, c^{2} d^{4} e^{2} - 40 \, a c d^{2} e^{4} + 15 \, a^{2} e^{6}}{c^{3} d^{3} e^{2}}\right )} - \frac {5 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c^{3} d^{3}} \] Input:

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

Output:

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*e^2*x/(c*d) + (13*c 
^2*d^3*e^3 - 5*a*c*d*e^5)/(c^3*d^3*e^2))*x + (33*c^2*d^4*e^2 - 40*a*c*d^2* 
e^4 + 15*a^2*e^6)/(c^3*d^3*e^2)) - 5/16*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2 
*c*d^2*e^4 - a^3*e^6)*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^3*d^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{5}-40 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{2} d^{3} e^{3}-10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{2} d^{2} e^{4} x +33 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{5} e +26 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{4} e^{2} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{3} d^{3} e^{3} x^{2}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}+45 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}-45 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}+15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}}{24 c^{4} d^{4} e} \] Input:

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

Output:

(15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d*e**5 - 40*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a*c**2*d**3*e**3 - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2* 
d**2*e**4*x + 33*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e + 26*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**2*x + 8*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*c**3*d**3*e**3*x**2 - 15*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))*a**3*e**6 
 + 45*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqr 
t(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**2*e**4 - 45*sqrt(e)*s 
qrt(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))*a*c**2*d**4*e**2 + 15*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**3*d**6)/(24*c**4*d**4*e)