\(\int (d+e x)^3 (a+b x+c x^2)^{5/2} \, dx\) [593]

Optimal result

Integrand size = 22, antiderivative size = 400 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}} \] Output:

5/32768*(-4*a*c+b^2)^2*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8* 
b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6-5/12288*(-4*a*c+b^2)*(-b*e+2*c*d)* 
(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8*b*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/ 
c^5+1/768*(-b*e+2*c*d)*(32*c^2*d^2+11*b^2*e^2-4*c*e*(3*a*e+8*b*d))*(2*c*x+ 
b)*(c*x^2+b*x+a)^(5/2)/c^4+1/9*e*(e*x+d)^2*(c*x^2+b*x+a)^(7/2)/c+1/2016*e* 
(640*c^2*d^2+99*b^2*e^2-2*c*e*(32*a*e+243*b*d)+154*c*e*(-b*e+2*c*d)*x)*(c* 
x^2+b*x+a)^(7/2)/c^3-5/65536*(-4*a*c+b^2)^3*(-b*e+2*c*d)*(32*c^2*d^2+11*b^ 
2*e^2-4*c*e*(3*a*e+8*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/ 
2))/c^(13/2)
 

Mathematica [A] (verified)

Time = 10.57 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.70 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {e (d+e x)^2 (a+x (b+c x))^{7/2}+\frac {e (a+x (b+c x))^{7/2} \left (99 b^2 e^2+4 c^2 d (160 d+77 e x)-2 c e (243 b d+32 a e+77 b e x)\right )}{224 c^2}+\frac {3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^4-40 b^3 c x+32 b c^2 x \left (13 a+8 c x^2\right )+8 b^2 c \left (-20 a+11 c x^2\right )+16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right )-15 \left (b^2-4 a c\right )^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{65536 c^{11/2}}}{9 c} \] Input:

Integrate[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(e*(d + e*x)^2*(a + x*(b + c*x))^(7/2) + (e*(a + x*(b + c*x))^(7/2)*(99*b^ 
2*e^2 + 4*c^2*d*(160*d + 77*e*x) - 2*c*e*(243*b*d + 32*a*e + 77*b*e*x)))/( 
224*c^2) + (3*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8*b*d + 3*a* 
e))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^4 - 40*b^3*c*x + 32 
*b*c^2*x*(13*a + 8*c*x^2) + 8*b^2*c*(-20*a + 11*c*x^2) + 16*c^2*(33*a^2 + 
26*a*c*x^2 + 8*c^2*x^4)) - 15*(b^2 - 4*a*c)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[ 
c]*Sqrt[a + x*(b + c*x)])]))/(65536*c^(11/2)))/(9*c)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.78, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1166, 27, 1225, 1087, 1087, 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.

\(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {\int \frac {1}{2} (d+e x) \left (18 c d^2-e (7 b d+4 a e)+11 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{9 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) \left (18 c d^2-7 b e d-4 a e^2+11 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {9 (2 c d-b e) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right ) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32 c^2}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{112 c^2}}{18 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}\)

Input:

Int[(d + e*x)^3*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(e*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/(9*c) + ((e*(640*c^2*d^2 + 99*b^2* 
e^2 - 2*c*e*(243*b*d + 32*a*e) + 154*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2 
)^(7/2))/(112*c^2) + (9*(2*c*d - b*e)*(32*c^2*d^2 + 11*b^2*e^2 - 4*c*e*(8* 
b*d + 3*a*e))*(((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(12*c) - (5*(b^2 - 4* 
a*c)*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b 
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x) 
/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c)))/(24*c)))/(32*c 
^2))/(18*c)
 

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 1166

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[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1206\) vs. \(2(370)=740\).

Time = 1.06 (sec) , antiderivative size = 1207, normalized size of antiderivative = 3.02

method	result	size
default	\(\text {Expression too large to display}\)	\(1207\)
risch	\(\text {Expression too large to display}\)	\(1257\)

Input:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

d^3*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b 
)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1 
/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))) 
))+e^3*(1/9*x^2*(c*x^2+b*x+a)^(7/2)/c-11/18*b/c*(1/8*x*(c*x^2+b*x+a)^(7/2) 
/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+ 
a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4 
*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*l 
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c* 
x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c 
+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c 
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))-2/9*a/c*(1/7*(c*x^2 
+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c- 
b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x 
+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+( 
c*x^2+b*x+a)^(1/2)))))))+3*d*e^2*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/ 
7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24 
*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/ 
4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c 
^(1/2)+(c*x^2+b*x+a)^(1/2))))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2 
)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (370) = 740\).

Time = 0.30 (sec) , antiderivative size = 2119, normalized size of antiderivative = 5.30 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/8257536*(315*(64*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6 
)*d^3 - 96*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*e 
+ 6*(9*b^8*c - 112*a*b^6*c^2 + 480*a^2*b^4*c^3 - 768*a^3*b^2*c^4 + 256*a^4 
*c^5)*d*e^2 - (11*b^9 - 144*a*b^7*c + 672*a^2*b^5*c^2 - 1280*a^3*b^3*c^3 + 
 768*a^4*b*c^4)*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 
 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(229376*c^9*e^3*x^8 + 14336*( 
54*c^9*d*e^2 + 37*b*c^8*e^3)*x^7 + 1024*(864*c^9*d^2*e + 1782*b*c^8*d*e^2 
+ (309*b^2*c^7 + 608*a*c^8)*e^3)*x^6 + 256*(1344*c^9*d^3 + 8352*b*c^8*d^2* 
e + 18*(243*b^2*c^7 + 476*a*c^8)*d*e^2 + (5*b^3*c^6 + 3012*a*b*c^7)*e^3)*x 
^5 + 128*(6720*b*c^8*d^3 + 288*(37*b^2*c^7 + 72*a*c^8)*d^2*e + 18*(3*b^3*c 
^6 + 1228*a*b*c^7)*d*e^2 - (11*b^4*c^5 - 84*a*b^2*c^6 - 3840*a^2*c^7)*e^3) 
*x^4 + 1344*(15*b^5*c^4 - 160*a*b^3*c^5 + 528*a^2*b*c^6)*d^3 - 288*(105*b^ 
6*c^3 - 1120*a*b^4*c^4 + 3696*a^2*b^2*c^5 - 3072*a^3*c^6)*d^2*e + 18*(945* 
b^7*c^2 - 10500*a*b^5*c^3 + 37744*a^2*b^3*c^4 - 42432*a^3*b*c^5)*d*e^2 - ( 
3465*b^8*c - 40740*a*b^6*c^2 + 162288*a^2*b^4*c^3 - 234432*a^3*b^2*c^4 + 6 
5536*a^4*c^5)*e^3 + 16*(1344*(27*b^2*c^7 + 52*a*c^8)*d^3 + 288*(3*b^3*c^6 
+ 788*a*b*c^7)*d^2*e - 18*(27*b^4*c^5 - 216*a*b^2*c^6 - 6608*a^2*c^7)*d*e^ 
2 + (99*b^5*c^4 - 856*a*b^3*c^5 + 1968*a^2*b*c^6)*e^3)*x^3 + 8*(1344*(b^3* 
c^6 + 156*a*b*c^7)*d^3 - 288*(7*b^4*c^5 - 60*a*b^2*c^6 - 1152*a^2*c^7)*d^2 
*e + 18*(63*b^5*c^4 - 568*a*b^3*c^5 + 1392*a^2*b*c^6)*d*e^2 - (231*b^6*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11123 vs. \(2 (401) = 802\).

Time = 1.29 (sec) , antiderivative size = 11123, normalized size of antiderivative = 27.81 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**3*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e**3*x**8/9 + x**7*(37*b*c**2*e**3 
/18 + 3*c**3*d*e**2)/(8*c) + x**6*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b* 
c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d** 
2*e)/(7*c) + x**5*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 
 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 
13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2 
*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c 
) + x**4*(3*a**2*c*e**3 + 3*a*b**2*e**3 + 18*a*b*c*d*e**2 + 9*a*c**2*d**2* 
e - 6*a*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c 
**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/(7*c) + 3*b**3*d*e**2 
 + 9*b**2*c*d**2*e + 3*b*c**2*d**3 - 11*b*(6*a*b*c*e**3 + 9*a*c**2*d*e**2 
- 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e**3 + 9*b**2*c*d*e 
**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c*e**3 + 9*b*c**2* 
d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 3*c**3*d**2*e)/ 
(14*c) + c**3*d**3)/(12*c))/(5*c) + x**3*(3*a**2*b*e**3 + 9*a**2*c*d*e**2 
+ 9*a*b**2*d*e**2 + 18*a*b*c*d**2*e + 3*a*c**2*d**3 - 5*a*(6*a*b*c*e**3 + 
9*a*c**2*d*e**2 - 7*a*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(8*c) + b**3*e** 
3 + 9*b**2*c*d*e**2 + 9*b*c**2*d**2*e - 13*b*(19*a*c**2*e**3/9 + 3*b**2*c* 
e**3 + 9*b*c**2*d*e**2 - 15*b*(37*b*c**2*e**3/18 + 3*c**3*d*e**2)/(16*c) + 
 3*c**3*d**2*e)/(14*c) + c**3*d**3)/(6*c) + 3*b**3*d**2*e + 3*b**2*c*d*...
 

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (370) = 740\).

Time = 0.46 (sec) , antiderivative size = 1193, normalized size of antiderivative = 2.98 \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/2064384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*e^3*x + (54* 
c^10*d*e^2 + 37*b*c^9*e^3)/c^8)*x + (864*c^10*d^2*e + 1782*b*c^9*d*e^2 + 3 
09*b^2*c^8*e^3 + 608*a*c^9*e^3)/c^8)*x + (1344*c^10*d^3 + 8352*b*c^9*d^2*e 
 + 4374*b^2*c^8*d*e^2 + 8568*a*c^9*d*e^2 + 5*b^3*c^7*e^3 + 3012*a*b*c^8*e^ 
3)/c^8)*x + (6720*b*c^9*d^3 + 10656*b^2*c^8*d^2*e + 20736*a*c^9*d^2*e + 54 
*b^3*c^7*d*e^2 + 22104*a*b*c^8*d*e^2 - 11*b^4*c^6*e^3 + 84*a*b^2*c^7*e^3 + 
 3840*a^2*c^8*e^3)/c^8)*x + (36288*b^2*c^8*d^3 + 69888*a*c^9*d^3 + 864*b^3 
*c^7*d^2*e + 226944*a*b*c^8*d^2*e - 486*b^4*c^6*d*e^2 + 3888*a*b^2*c^7*d*e 
^2 + 118944*a^2*c^8*d*e^2 + 99*b^5*c^5*e^3 - 856*a*b^3*c^6*e^3 + 1968*a^2* 
b*c^7*e^3)/c^8)*x + (1344*b^3*c^7*d^3 + 209664*a*b*c^8*d^3 - 2016*b^4*c^6* 
d^2*e + 17280*a*b^2*c^7*d^2*e + 331776*a^2*c^8*d^2*e + 1134*b^5*c^5*d*e^2 
- 10224*a*b^3*c^6*d*e^2 + 25056*a^2*b*c^7*d*e^2 - 231*b^6*c^4*e^3 + 2232*a 
*b^4*c^5*e^3 - 6384*a^2*b^2*c^6*e^3 + 4096*a^3*c^7*e^3)/c^8)*x - (6720*b^4 
*c^6*d^3 - 64512*a*b^2*c^7*d^3 - 709632*a^2*c^8*d^3 - 10080*b^5*c^5*d^2*e 
+ 96768*a*b^3*c^6*d^2*e - 262656*a^2*b*c^7*d^2*e + 5670*b^6*c^4*d*e^2 - 56 
952*a*b^4*c^5*d*e^2 + 171936*a^2*b^2*c^6*d*e^2 - 120960*a^3*c^7*d*e^2 - 11 
55*b^7*c^3*e^3 + 12348*a*b^5*c^4*e^3 - 42192*a^2*b^3*c^5*e^3 + 44096*a^3*b 
*c^6*e^3)/c^8)*x + (20160*b^5*c^5*d^3 - 215040*a*b^3*c^6*d^3 + 709632*a^2* 
b*c^7*d^3 - 30240*b^6*c^4*d^2*e + 322560*a*b^4*c^5*d^2*e - 1064448*a^2*b^2 
*c^6*d^2*e + 884736*a^3*c^7*d^2*e + 17010*b^7*c^3*d*e^2 - 189000*a*b^5*...
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int((d + e*x)^3*(a + b*x + c*x^2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int((d + e*x)^3*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (e x +d \right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x \] Input:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x)
 

Output:

int((e*x+d)^3*(c*x^2+b*x+a)^(5/2),x)