\(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [28]

Optimal result

Integrand size = 40, antiderivative size = 434 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 c^{9/2} d^{9/2} e^{11/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+5 a^4 c d e^8 (11 d+14 e x)+2 a^3 c^2 d^2 e^6 \left (27 d^2-16 d e x-28 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (13 d^3-6 d^2 e x+4 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (-525 d^4+336 d^3 e x-264 d^2 e^2 x^2+224 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (315 d^5-210 d^4 e x+168 d^3 e^2 x^2-144 d^2 e^3 x^3+128 d e^4 x^4+1280 e^5 x^5\right )\right )-\frac {15 \left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{9/2} d^{9/2} e^{11/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 + 5 
*a^4*c*d*e^8*(11*d + 14*e*x) + 2*a^3*c^2*d^2*e^6*(27*d^2 - 16*d*e*x - 28*e 
^2*x^2) + 6*a^2*c^3*d^3*e^4*(13*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 + 8*e^3*x^3) 
 + a*c^4*d^4*e^2*(-525*d^4 + 336*d^3*e*x - 264*d^2*e^2*x^2 + 224*d*e^3*x^3 
 + 1664*e^4*x^4) + c^5*d^5*(315*d^5 - 210*d^4*e*x + 168*d^3*e^2*x^2 - 144* 
d^2*e^3*x^3 + 128*d*e^4*x^4 + 1280*e^5*x^5)) - (15*(c*d^2 - a*e^2)^3*(21*c 
^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*ArcTanh[(Sqrt[c] 
*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*S 
qrt[d + e*x])))/(7680*c^(9/2)*d^(9/2)*e^(11/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1215, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

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. \(1425\) vs. \(2(402)=804\).

Time = 2.58 (sec) , antiderivative size = 1426, normalized size of antiderivative = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1426\)

Input:

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

Output:

d^2/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)))+1/e*(1/6*x*(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-7/12*(a*e^2+c*d^2)/d/e/c*(1/5*(a*d*e+(a* 
e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^2+c*d^2)/d/e/c*(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*l 
n((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))))-1/6*a/c*(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))))-d/e^2*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/d/e/c-1/2*(a*e^ 
2+c*d^2)/d/e/c*(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*...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.41 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/30720*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^ 
3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*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 + 315*c^6*d^11 
*e - 525*a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4* 
c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 
- 16*(9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c^ 
6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*x^2 
- 2*(105*c^6*d^10*e^2 - 168*a*c^5*d^8*e^4 + 18*a^2*c^4*d^6*e^6 + 16*a^3*c^ 
3*d^4*e^8 - 35*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^5*d^5*e^6), 1/15360*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^ 
2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 
 7*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 + 315*c^6*d^1 
1*e - 525*a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4 
*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 
 - 16*(9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c 
^6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*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. 984 vs. \(2 (442) = 884\).

Time = 82.71 (sec) , antiderivative size = 2502, normalized size of antiderivative = 5.76 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

a*e*Piecewise(((-a*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c 
*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c 
*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(-2*a*(a*d*e/5 - (a*e**2/10 + 
c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c* 
d**2/2)*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a 
*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(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*(a*e**2 + c 
*d**2))*(x**4/5 + x**3*(a*e**2/10 + c*d**2/10)/(4*c*d*e) + x**2*(a*d*e/5 - 
 (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e) + 
x*(-3*a*(a*e**2/10 + c*d**2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 
 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e)) 
/(2*c*d*e) + (-2*a*(a*d*e/5 - (a*e**2/10 + c*d**2/10)*(7*a*e**2/2 + 7*c*d* 
*2/2)/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/2)*(-3*a*(a*e**2/10 + c*d* 
*2/10)/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(a*d*e/5 - (a*e**2/10 + c*d**2/10 
)*(7*a*e**2/2 + 7*c*d**2/2)/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(c*d*e)), Ne( 
c*d*e, 0)), (2*(-a**3*d**3*e**3*(a*d*e + x*(a*e**2 + c*d**2))**(3/2)/3 ...
 

Maxima [F(-2)]

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

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),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 = 0.15 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.18 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d x + \frac {c^{6} d^{7} e^{4} + 13 \, a c^{5} d^{5} e^{6}}{c^{5} d^{5} e^{5}}\right )} x - \frac {9 \, c^{6} d^{8} e^{3} - 14 \, a c^{5} d^{6} e^{5} - 3 \, a^{2} c^{4} d^{4} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {21 \, c^{6} d^{9} e^{2} - 33 \, a c^{5} d^{7} e^{4} + 3 \, a^{2} c^{4} d^{5} e^{6} - 7 \, a^{3} c^{3} d^{3} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{6} d^{10} e - 168 \, a c^{5} d^{8} e^{3} + 18 \, a^{2} c^{4} d^{6} e^{5} + 16 \, a^{3} c^{3} d^{4} e^{7} - 35 \, a^{4} c^{2} d^{2} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {315 \, c^{6} d^{11} - 525 \, a c^{5} d^{9} e^{2} + 78 \, a^{2} c^{4} d^{7} e^{4} + 54 \, a^{3} c^{3} d^{5} e^{6} + 55 \, a^{4} c^{2} d^{3} e^{8} - 105 \, a^{5} c d e^{10}}{c^{5} d^{5} e^{5}}\right )} + \frac {{\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{4} d^{4} e^{5}} \] Input:

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),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*c*d*x + 
 (c^6*d^7*e^4 + 13*a*c^5*d^5*e^6)/(c^5*d^5*e^5))*x - (9*c^6*d^8*e^3 - 14*a 
*c^5*d^6*e^5 - 3*a^2*c^4*d^4*e^7)/(c^5*d^5*e^5))*x + (21*c^6*d^9*e^2 - 33* 
a*c^5*d^7*e^4 + 3*a^2*c^4*d^5*e^6 - 7*a^3*c^3*d^3*e^8)/(c^5*d^5*e^5))*x - 
(105*c^6*d^10*e - 168*a*c^5*d^8*e^3 + 18*a^2*c^4*d^6*e^5 + 16*a^3*c^3*d^4* 
e^7 - 35*a^4*c^2*d^2*e^9)/(c^5*d^5*e^5))*x + (315*c^6*d^11 - 525*a*c^5*d^9 
*e^2 + 78*a^2*c^4*d^7*e^4 + 54*a^3*c^3*d^5*e^6 + 55*a^4*c^2*d^3*e^8 - 105* 
a^5*c*d*e^10)/(c^5*d^5*e^5)) + 1/1024*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 1 
5*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^ 
10 - 7*a^6*e^12)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - s 
qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^4*d^4*e^5)
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.38 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d*e**11 + 55*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**4*c**2*d**3*e**9 + 70*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**4*c**2*d**2*e**10*x + 54*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d** 
5*e**7 - 32*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**4*e**8*x - 56*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**9*x**2 + 78*sqrt(d + e*x)*s 
qrt(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 + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**5* 
e**7*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**8*x**3 - 
525*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**9*e**3 + 336*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a*c**5*d**8*e**4*x - 264*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 + 1664*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5*e**7*x**4 + 315 
*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**11*e - 210*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*c**6*d**10*e**2*x + 168*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d** 
9*e**3*x**2 - 144*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**8*e**4*x**3 + 12 
8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**5*x**4 + 1280*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*c**6*d**6*e**6*x**5 + 105*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**6*e**12 - 90*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**...