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

Optimal result

Integrand size = 38, antiderivative size = 443 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {\left (7 c^4 d^8+8 a c^3 d^6 e^2+6 a^2 c^2 d^4 e^4-21 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e^4}+\frac {\left (c d^2-3 a e^2\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2 e}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (35 c^3 d^6+33 a c^2 d^4 e^2+21 a^2 c d^2 e^4-105 a^3 e^6-6 c d e \left (7 c^2 d^4+6 a c d^2 e^2-21 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^4 d^4 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^{11/2} d^{11/2} e^{9/2}} \] Output:

-1/512*(-21*a^4*e^8+6*a^2*c^2*d^4*e^4+8*a*c^3*d^6*e^2+7*c^4*d^8)*(2*c*d*e* 
x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/e^4+1/20*(- 
3*a*e^2+c*d^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e+1/6*x 
^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d+1/960*(35*c^3*d^6+33*a*c^2* 
d^4*e^2+21*a^2*c*d^2*e^4-105*a^3*e^6-6*c*d*e*(-21*a^2*e^4+6*a*c*d^2*e^2+7* 
c^2*d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/e^3+1/512*(-a* 
e^2+c*d^2)^3*(21*a^3*e^6+21*a^2*c*d^2*e^4+15*a*c^2*d^4*e^2+7*c^3*d^6)*arct 
anh(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^(11/2)/d^(11/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.87 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (315 a^5 e^{10}-105 a^4 c d e^8 (5 d+2 e x)+6 a^3 c^2 d^2 e^6 \left (13 d^2+56 d e x+28 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (9 d^3-6 d^2 e x-44 d e^2 x^2-24 e^3 x^3\right )+a c^4 d^4 e^2 \left (55 d^4-32 d^3 e x+24 d^2 e^2 x^2+224 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )+\frac {15 \left (c d^2-a e^2\right )^3 \left (7 c^3 d^6+15 a c^2 d^4 e^2+21 a^2 c d^2 e^4+21 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{11/2} d^{11/2} e^{9/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(315*a^5*e^10 - 10 
5*a^4*c*d*e^8*(5*d + 2*e*x) + 6*a^3*c^2*d^2*e^6*(13*d^2 + 56*d*e*x + 28*e^ 
2*x^2) + 6*a^2*c^3*d^3*e^4*(9*d^3 - 6*d^2*e*x - 44*d*e^2*x^2 - 24*e^3*x^3) 
 + a*c^4*d^4*e^2*(55*d^4 - 32*d^3*e*x + 24*d^2*e^2*x^2 + 224*d*e^3*x^3 + 1 
28*e^4*x^4) + c^5*d^5*(-105*d^5 + 70*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2*e^3 
*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5)) + (15*(c*d^2 - a*e^2)^3*(7*c^3*d^6 
+ 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*ArcTanh[(Sqrt[e]*Sqrt[ 
a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d 
+ e*x])))/(7680*c^(11/2)*d^(11/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1236, 27, 1236, 27, 1225, 1087, 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.

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\int \frac {1}{2} x \left (4 a d e \left (c d^2-3 a e^2\right )+\left (7 c^2 d^4+6 a c e^2 d^2-21 a^2 e^4\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{5 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\int x \left (4 a d e \left (c d^2-3 a e^2\right )+\left (7 c^2 d^4+6 a c e^2 d^2-21 a^2 e^4\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-6 c d e x \left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right )+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-6 c d e x \left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right )+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-6 c d e x \left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right )+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d}-\frac {\frac {\frac {5 \left (-21 a^4 e^8+6 a^2 c^2 d^4 e^4+8 a c^3 d^6 e^2+7 c^4 d^8\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c^2 d^2 e^2}-\frac {\left (-105 a^3 e^6-6 c d e x \left (-21 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right )+21 a^2 c d^2 e^4+33 a c^2 d^4 e^2+35 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2 e^2}}{10 c d e}-\frac {1}{5} x^2 \left (\frac {d}{e}-\frac {3 a e}{c d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d}\)

Input:

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

Output:

(x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*c*d) - (-1/5*((d/e 
- (3*a*e)/(c*d))*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (-1/ 
24*((35*c^3*d^6 + 33*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 - 105*a^3*e^6 - 6*c* 
d*e*(7*c^2*d^4 + 6*a*c*d^2*e^2 - 21*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2)^(3/2))/(c^2*d^2*e^2) + (5*(7*c^4*d^8 + 8*a*c^3*d^6*e^2 + 6*a 
^2*c^2*d^4*e^4 - 21*a^4*e^8)*(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c*d*e) - ((c*d^2 - a*e^2)^2*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])])/(8*c^(3/2)*d^(3/2)*e^(3/2))))/(16*c^2*d^2*e^2))/(10 
*c*d*e))/(4*c*d)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1948\) vs. \(2(411)=822\).

Time = 2.40 (sec) , antiderivative size = 1949, normalized size of antiderivative = 4.40

method	result	size
default	\(\text {Expression too large to display}\)	\(1949\)

Input:

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

Output:

d*(1/5*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e/c-7/10*(a*e^2+c*d^2 
)/d/e/c*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e/c-5/8*(a*e^2+c* 
d^2)/d/e/c*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e/c-1/2*(a*e^2+c 
*d^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)))-1/4*a/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)))-2/5*a/c*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e 
/c-1/2*(a*e^2+c*d^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))))+e*(1/6*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3 
/2)/d/e/c-3/4*(a*e^2+c*d^2)/d/e/c*(1/5*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(3/2)/d/e/c-7/10*(a*e^2+c*d^2)/d/e/c*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d* 
x^2*e)^(3/2)/d/e/c-5/8*(a*e^2+c*d^2)/d/e/c*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d 
*x^2*e)^(3/2)/d/e/c-1/2*(a*e^2+c*d^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...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.36 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d^8*e^4 - 4*a^3*c 
^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 21*a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11 
*e + 55*a*c^5*d^9*e^3 + 54*a^2*c^4*d^7*e^5 + 78*a^3*c^3*d^5*e^7 - 525*a^4* 
c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 
+ 16*(3*c^6*d^8*e^4 + 14*a*c^5*d^6*e^6 - 9*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c^6 
*d^9*e^3 - 3*a*c^5*d^7*e^5 + 33*a^2*c^4*d^5*e^7 - 21*a^3*c^3*d^3*e^9)*x^2 
+ 2*(35*c^6*d^10*e^2 - 16*a*c^5*d^8*e^4 - 18*a^2*c^4*d^6*e^6 + 168*a^3*c^3 
*d^4*e^8 - 105*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^ 
2)*x))/(c^6*d^6*e^5), -1/15360*(15*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2* 
c^4*d^8*e^4 - 4*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 
 21*a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^ 
11*e + 55*a*c^5*d^9*e^3 + 54*a^2*c^4*d^7*e^5 + 78*a^3*c^3*d^5*e^7 - 525*a^ 
4*c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^ 
4 + 16*(3*c^6*d^8*e^4 + 14*a*c^5*d^6*e^6 - 9*a^2*c^4*d^4*e^8)*x^3 - 8*(7*c 
^6*d^9*e^3 - 3*a*c^5*d^7*e^5 + 33*a^2*c^4*d^5*e^7 - 21*a^3*c^3*d^3*e^9)...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2166 vs. \(2 (452) = 904\).

Time = 1.55 (sec) , antiderivative size = 2166, normalized size of antiderivative = 4.89 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*(-3*a*(7*a*d*e**2/6 + c*d**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a 
*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c) - ( 
5*a*e**2/2 + 5*c*d**2/2)*(a*d**2*e - 4*a*(a*e**3 + 2*c*d**2*e - e*(11*a*e* 
*2/2 + 11*c*d**2/2)/6)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(7*a*d*e**2/6 + c 
*d**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 
11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2) 
*(-2*a*(a*d**2*e - 4*a*(a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 11*c*d**2/2 
)/6)/(5*c) - (7*a*e**2/2 + 7*c*d**2/2)*(7*a*d*e**2/6 + c*d**3 - (9*a*e**2/ 
2 + 9*c*d**2/2)*(a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5 
*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(7*a*d*e**2/6 
+ c*d**3 - (9*a*e**2/2 + 9*c*d**2/2)*(a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 
 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d**2*e 
- 4*a*(a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c) - (7*a 
*e**2/2 + 7*c*d**2/2)*(7*a*d*e**2/6 + c*d**3 - (9*a*e**2/2 + 9*c*d**2/2)*( 
a*e**3 + 2*c*d**2*e - e*(11*a*e**2/2 + 11*c*d**2/2)/6)/(5*c*d*e))/(4*c*d*e 
))/(3*c*d*e))/(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*(...
 

Maxima [F(-2)]

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

integrate(x^3*(e*x+d)*(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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.14 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, e x + \frac {13 \, c^{5} d^{6} e^{5} + a c^{4} d^{4} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {3 \, c^{5} d^{7} e^{4} + 14 \, a c^{4} d^{5} e^{6} - 9 \, a^{2} c^{3} d^{3} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {7 \, c^{5} d^{8} e^{3} - 3 \, a c^{4} d^{6} e^{5} + 33 \, a^{2} c^{3} d^{4} e^{7} - 21 \, a^{3} c^{2} d^{2} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {35 \, c^{5} d^{9} e^{2} - 16 \, a c^{4} d^{7} e^{4} - 18 \, a^{2} c^{3} d^{5} e^{6} + 168 \, a^{3} c^{2} d^{3} e^{8} - 105 \, a^{4} c d e^{10}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{5} d^{10} e - 55 \, a c^{4} d^{8} e^{3} - 54 \, a^{2} c^{3} d^{6} e^{5} - 78 \, a^{3} c^{2} d^{4} e^{7} + 525 \, a^{4} c d^{2} e^{9} - 315 \, a^{5} e^{11}}{c^{5} d^{5} e^{5}}\right )} - \frac {{\left (7 \, c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} - 3 \, a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} - 15 \, a^{4} c^{2} d^{4} e^{8} + 42 \, a^{5} c d^{2} e^{10} - 21 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{5} d^{5} e^{4}} \] Input:

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

Output:

1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*e*x + ( 
13*c^5*d^6*e^5 + a*c^4*d^4*e^7)/(c^5*d^5*e^5))*x + (3*c^5*d^7*e^4 + 14*a*c 
^4*d^5*e^6 - 9*a^2*c^3*d^3*e^8)/(c^5*d^5*e^5))*x - (7*c^5*d^8*e^3 - 3*a*c^ 
4*d^6*e^5 + 33*a^2*c^3*d^4*e^7 - 21*a^3*c^2*d^2*e^9)/(c^5*d^5*e^5))*x + (3 
5*c^5*d^9*e^2 - 16*a*c^4*d^7*e^4 - 18*a^2*c^3*d^5*e^6 + 168*a^3*c^2*d^3*e^ 
8 - 105*a^4*c*d*e^10)/(c^5*d^5*e^5))*x - (105*c^5*d^10*e - 55*a*c^4*d^8*e^ 
3 - 54*a^2*c^3*d^6*e^5 - 78*a^3*c^2*d^4*e^7 + 525*a^4*c*d^2*e^9 - 315*a^5* 
e^11)/(c^5*d^5*e^5)) - 1/1024*(7*c^6*d^12 - 6*a*c^5*d^10*e^2 - 3*a^2*c^4*d 
^8*e^4 - 4*a^3*c^3*d^6*e^6 - 15*a^4*c^2*d^4*e^8 + 42*a^5*c*d^2*e^10 - 21*a 
^6*e^12)*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^5*d^5*e^4)
 

Mupad [B] (verification not implemented)

Time = 7.94 (sec) , antiderivative size = 1678, normalized size of antiderivative = 3.79 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((7*a*e^2 + 7*c*d^2)*((a*((x/2 + (a*e^2 + c*d^2)/(4*c*d*e))*(x*(a*e^2 + c* 
d^2) + a*d*e + c*d*e*x^2)^(1/2) - (log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*( 
c*d*e)^(1/2) + a*e^2 + c*d^2 + 2*c*d*e*x)*((a*e^2 + c*d^2)^2/4 - a*c*d^2*e 
^2))/(2*(c*d*e)^(3/2))))/(4*c) - (x*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2 
)^(3/2))/(4*c*d*e) + (((log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2 
) + a*e^2 + c*d^2 + 2*c*d*e*x)*((a*e^2 + c*d^2)^3 - 4*a*c*d^2*e^2*(a*e^2 + 
 c*d^2)))/(16*(c*d*e)^(5/2)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1 
/2)*(8*c*d*e*(a*d*e + c*d*e*x^2) - 3*(a*e^2 + c*d^2)^2 + 2*c*d*e*x*(a*e^2 
+ c*d^2)))/(24*c^2*d^2*e^2))*(5*a*e^2 + 5*c*d^2))/(8*c*d*e)))/(10*c*e) - ( 
2*a*d*((log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d^ 
2 + 2*c*d*e*x)*((a*e^2 + c*d^2)^3 - 4*a*c*d^2*e^2*(a*e^2 + c*d^2)))/(16*(c 
*d*e)^(5/2)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(8*c*d*e*(a* 
d*e + c*d*e*x^2) - 3*(a*e^2 + c*d^2)^2 + 2*c*d*e*x*(a*e^2 + c*d^2)))/(24*c 
^2*d^2*e^2)))/(5*c) + (x^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/ 
(6*c*d) + (x^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(5*c*e) + (a 
*e*((a*((x/2 + (a*e^2 + c*d^2)/(4*c*d*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d 
*e*x^2)^(1/2) - (log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e 
^2 + c*d^2 + 2*c*d*e*x)*((a*e^2 + c*d^2)^2/4 - a*c*d^2*e^2))/(2*(c*d*e)^(3 
/2))))/(4*c) - (x*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(4*c*d*e) 
 + (((log(2*((a*e + c*d*x)*(d + e*x))^(1/2)*(c*d*e)^(1/2) + a*e^2 + c*d...
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.33 \[ \int x^3 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(315*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d*e**11 - 525*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**4*c**2*d**3*e**9 - 210*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**4*c**2*d**2*e**10*x + 78*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**5 
*e**7 + 336*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**4*e**8*x + 168*sq 
rt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**9*x**2 + 54*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**2*c**4*d**7*e**5 - 36*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**2*c**4*d**6*e**6*x - 264*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d** 
5*e**7*x**2 - 144*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**8*x**3 
 + 55*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**9*e**3 - 32*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a*c**5*d**8*e**4*x + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a*c**5*d**7*e**5*x**2 + 224*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**6*e* 
*6*x**3 + 128*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5*e**7*x**4 - 105* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**11*e + 70*sqrt(d + e*x)*sqrt(a*e + 
 c*d*x)*c**6*d**10*e**2*x - 56*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**9*e 
**3*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**8*e**4*x**3 + 1664*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**5*x**4 + 1280*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*c**6*d**6*e**6*x**5 - 315*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))*a**6*e**12 + 630*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**5*c*d**2*...