\(\int x (A+B x) (a+b x+c x^2)^{5/2} \, dx\) [132]

Optimal result

Integrand size = 21, antiderivative size = 252 \[ \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (9 b^2 B-16 A b c-4 a B c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {(9 b B-16 A c-14 B c x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \] Output:

5/16384*(-4*a*c+b^2)^2*(-16*A*b*c-4*B*a*c+9*B*b^2)*(2*c*x+b)*(c*x^2+b*x+a) 
^(1/2)/c^5-5/6144*(-4*a*c+b^2)*(-16*A*b*c-4*B*a*c+9*B*b^2)*(2*c*x+b)*(c*x^ 
2+b*x+a)^(3/2)/c^4+1/384*(-16*A*b*c-4*B*a*c+9*B*b^2)*(2*c*x+b)*(c*x^2+b*x+ 
a)^(5/2)/c^3-1/112*(-14*B*c*x-16*A*c+9*B*b)*(c*x^2+b*x+a)^(7/2)/c^2-5/3276 
8*(-4*a*c+b^2)^3*(-16*A*b*c-4*B*a*c+9*B*b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2) 
/(c*x^2+b*x+a)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 2.77 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.55 \[ \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^7 B-210 b^6 c (8 A+3 B x)+28 b^5 c (-375 a B+2 c x (20 A+9 B x))+16 b^3 c^2 \left (2359 a^2 B+24 c^2 x^3 (2 A+B x)-4 a c x (168 A+71 B x)\right )+8 b^4 c^2 \left (-2 c x^2 (56 A+27 B x)+7 a (320 A+113 B x)\right )+32 b^2 c^3 \left (12 a c x^2 (20 A+9 B x)-3 a^2 (616 A+199 B x)+8 c^2 x^4 (296 A+243 B x)\right )+64 b c^3 \left (-663 a^3 B+6 a^2 c x (76 A+29 B x)+16 c^3 x^5 (116 A+99 B x)+8 a c^2 x^3 (394 A+307 B x)\right )+128 c^4 \left (48 c^3 x^6 (8 A+7 B x)+3 a^3 (128 A+35 B x)+8 a c^2 x^4 (144 A+119 B x)+2 a^2 c x^2 (576 A+413 B x)\right )\right )+105 \left (b^2-4 a c\right )^3 \left (9 b^2 B-16 A b c-4 a B c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{688128 c^{11/2}} \] Input:

Integrate[x*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^7*B - 210*b^6*c*(8*A + 3*B*x) + 28 
*b^5*c*(-375*a*B + 2*c*x*(20*A + 9*B*x)) + 16*b^3*c^2*(2359*a^2*B + 24*c^2 
*x^3*(2*A + B*x) - 4*a*c*x*(168*A + 71*B*x)) + 8*b^4*c^2*(-2*c*x^2*(56*A + 
 27*B*x) + 7*a*(320*A + 113*B*x)) + 32*b^2*c^3*(12*a*c*x^2*(20*A + 9*B*x) 
- 3*a^2*(616*A + 199*B*x) + 8*c^2*x^4*(296*A + 243*B*x)) + 64*b*c^3*(-663* 
a^3*B + 6*a^2*c*x*(76*A + 29*B*x) + 16*c^3*x^5*(116*A + 99*B*x) + 8*a*c^2* 
x^3*(394*A + 307*B*x)) + 128*c^4*(48*c^3*x^6*(8*A + 7*B*x) + 3*a^3*(128*A 
+ 35*B*x) + 8*a*c^2*x^4*(144*A + 119*B*x) + 2*a^2*c*x^2*(576*A + 413*B*x)) 
) + 105*(b^2 - 4*a*c)^3*(9*b^2*B - 16*A*b*c - 4*a*B*c)*Log[b + 2*c*x - 2*S 
qrt[c]*Sqrt[a + x*(b + c*x)]])/(688128*c^(11/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (-4 a B c-16 A b c+9 b^2 B\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{32 c^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2}\)

Input:

Int[x*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

-1/112*((9*b*B - 16*A*c - 14*B*c*x)*(a + b*x + c*x^2)^(7/2))/c^2 + ((9*b^2 
*B - 16*A*b*c - 4*a*B*c)*(((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)
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(510\) vs. \(2(226)=452\).

Time = 1.23 (sec) , antiderivative size = 511, normalized size of antiderivative = 2.03

method	result	size
default	\(A \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )+B \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )\)	\(511\)
risch	\(\frac {\left (43008 B \,c^{7} x^{7}+49152 A \,c^{7} x^{6}+101376 B b \,c^{6} x^{6}+118784 A b \,c^{6} x^{5}+121856 B a \,c^{6} x^{5}+62208 B \,b^{2} c^{5} x^{5}+147456 A a \,c^{6} x^{4}+75776 A \,b^{2} c^{5} x^{4}+157184 B a b \,c^{5} x^{4}+384 B \,b^{3} c^{4} x^{4}+201728 A a b \,c^{5} x^{3}+768 A \,b^{3} c^{4} x^{3}+105728 B \,a^{2} c^{5} x^{3}+3456 B a \,b^{2} c^{4} x^{3}-432 B \,b^{4} c^{3} x^{3}+147456 A \,a^{2} c^{5} x^{2}+7680 A a \,b^{2} c^{4} x^{2}-896 A \,b^{4} c^{3} x^{2}+11136 B \,a^{2} b \,c^{4} x^{2}-4544 B a \,b^{3} c^{3} x^{2}+504 B \,b^{5} c^{2} x^{2}+29184 A \,a^{2} b \,c^{4} x -10752 A a \,b^{3} c^{3} x +1120 A \,b^{5} c^{2} x +13440 B \,a^{3} c^{4} x -19104 B \,a^{2} b^{2} c^{3} x +6328 B a \,b^{4} c^{2} x -630 B \,b^{6} c x +49152 A \,a^{3} c^{4}-59136 A \,a^{2} b^{2} c^{3}+17920 A a \,b^{4} c^{2}-1680 A \,b^{6} c -42432 B \,a^{3} b \,c^{3}+37744 B \,a^{2} b^{3} c^{2}-10500 B a \,b^{5} c +945 B \,b^{7}\right ) \sqrt {c \,x^{2}+b x +a}}{344064 c^{5}}-\frac {5 \left (1024 A \,a^{3} b \,c^{4}-768 A \,a^{2} b^{3} c^{3}+192 A a \,b^{5} c^{2}-16 A \,b^{7} c +256 B \,a^{4} c^{4}-768 B \,a^{3} b^{2} c^{3}+480 B \,a^{2} b^{4} c^{2}-112 B a \,b^{6} c +9 B \,b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32768 c^{\frac {11}{2}}}\)	\(525\)

Input:

int(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

A*(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))))))+B*(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-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))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (226) = 452\).

Time = 0.17 (sec) , antiderivative size = 1039, normalized size of antiderivative = 4.12 \[ \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/1376256*(105*(9*B*b^8 + 256*(B*a^4 + 4*A*a^3*b)*c^4 - 768*(B*a^3*b^2 + 
A*a^2*b^3)*c^3 + 96*(5*B*a^2*b^4 + 2*A*a*b^5)*c^2 - 16*(7*B*a*b^6 + A*b^7) 
*c)*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*(43008*B*c^8*x^7 + 945*B*b^7*c + 49152*A*a^3*c 
^5 + 3072*(33*B*b*c^7 + 16*A*c^8)*x^6 + 256*(243*B*b^2*c^6 + 4*(119*B*a + 
116*A*b)*c^7)*x^5 - 192*(221*B*a^3*b + 308*A*a^2*b^2)*c^4 + 128*(3*B*b^3*c 
^5 + 1152*A*a*c^7 + 4*(307*B*a*b + 148*A*b^2)*c^6)*x^4 + 112*(337*B*a^2*b^ 
3 + 160*A*a*b^4)*c^3 - 16*(27*B*b^4*c^4 - 16*(413*B*a^2 + 788*A*a*b)*c^6 - 
 24*(9*B*a*b^2 + 2*A*b^3)*c^5)*x^3 - 420*(25*B*a*b^5 + 4*A*b^6)*c^2 + 8*(6 
3*B*b^5*c^3 + 18432*A*a^2*c^6 + 48*(29*B*a^2*b + 20*A*a*b^2)*c^5 - 8*(71*B 
*a*b^3 + 14*A*b^4)*c^4)*x^2 - 2*(315*B*b^6*c^2 - 192*(35*B*a^3 + 76*A*a^2* 
b)*c^5 + 48*(199*B*a^2*b^2 + 112*A*a*b^3)*c^4 - 28*(113*B*a*b^4 + 20*A*b^5 
)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/688128*(105*(9*B*b^8 + 256*(B*a^4 
+ 4*A*a^3*b)*c^4 - 768*(B*a^3*b^2 + A*a^2*b^3)*c^3 + 96*(5*B*a^2*b^4 + 2*A 
*a*b^5)*c^2 - 16*(7*B*a*b^6 + A*b^7)*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b 
*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(43008*B*c^8*x^7 
 + 945*B*b^7*c + 49152*A*a^3*c^5 + 3072*(33*B*b*c^7 + 16*A*c^8)*x^6 + 256* 
(243*B*b^2*c^6 + 4*(119*B*a + 116*A*b)*c^7)*x^5 - 192*(221*B*a^3*b + 308*A 
*a^2*b^2)*c^4 + 128*(3*B*b^3*c^5 + 1152*A*a*c^7 + 4*(307*B*a*b + 148*A*b^2 
)*c^6)*x^4 + 112*(337*B*a^2*b^3 + 160*A*a*b^4)*c^3 - 16*(27*B*b^4*c^4 -...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4009 vs. \(2 (262) = 524\).

Time = 0.96 (sec) , antiderivative size = 4009, normalized size of antiderivative = 15.91 \[ \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate(x*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise(((-a*(3*A*a**2*b + B*a**3 - 3*a*(6*A*a*b*c + A*b**3 + 3*B*a**2*c 
 + 3*B*a*b**2 - 5*a*(3*A*b*c**2 + 17*B*a*c**2/8 + 3*B*b**2*c - 13*b*(A*c** 
3 + 33*B*b*c**2/16)/(14*c))/(6*c) - 9*b*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b 
*c + B*b**3 - 6*a*(A*c**3 + 33*B*b*c**2/16)/(7*c) - 11*b*(3*A*b*c**2 + 17* 
B*a*c**2/8 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16)/(14*c))/(12*c))/( 
10*c))/(4*c) - 5*b*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b - 4*a*(3*A*a*c**2 
 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 6*a*(A*c**3 + 33*B*b*c**2/16)/(7*c) - 
 11*b*(3*A*b*c**2 + 17*B*a*c**2/8 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c** 
2/16)/(14*c))/(12*c))/(5*c) - 7*b*(6*A*a*b*c + A*b**3 + 3*B*a**2*c + 3*B*a 
*b**2 - 5*a*(3*A*b*c**2 + 17*B*a*c**2/8 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B 
*b*c**2/16)/(14*c))/(6*c) - 9*b*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b 
**3 - 6*a*(A*c**3 + 33*B*b*c**2/16)/(7*c) - 11*b*(3*A*b*c**2 + 17*B*a*c**2 
/8 + 3*B*b**2*c - 13*b*(A*c**3 + 33*B*b*c**2/16)/(14*c))/(12*c))/(10*c))/( 
8*c))/(6*c))/(2*c) - b*(A*a**3 - 2*a*(3*A*a**2*c + 3*A*a*b**2 + 3*B*a**2*b 
 - 4*a*(3*A*a*c**2 + 3*A*b**2*c + 6*B*a*b*c + B*b**3 - 6*a*(A*c**3 + 33*B* 
b*c**2/16)/(7*c) - 11*b*(3*A*b*c**2 + 17*B*a*c**2/8 + 3*B*b**2*c - 13*b*(A 
*c**3 + 33*B*b*c**2/16)/(14*c))/(12*c))/(5*c) - 7*b*(6*A*a*b*c + A*b**3 + 
3*B*a**2*c + 3*B*a*b**2 - 5*a*(3*A*b*c**2 + 17*B*a*c**2/8 + 3*B*b**2*c - 1 
3*b*(A*c**3 + 33*B*b*c**2/16)/(14*c))/(6*c) - 9*b*(3*A*a*c**2 + 3*A*b**2*c 
 + 6*B*a*b*c + B*b**3 - 6*a*(A*c**3 + 33*B*b*c**2/16)/(7*c) - 11*b*(3*A...
 

Maxima [F(-2)]

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

integrate(x*(B*x+A)*(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. 526 vs. \(2 (226) = 452\).

Time = 0.28 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.09 \[ \int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, B c^{2} x + \frac {33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac {243 \, B b^{2} c^{7} + 476 \, B a c^{8} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac {3 \, B b^{3} c^{6} + 1228 \, B a b c^{7} + 592 \, A b^{2} c^{7} + 1152 \, A a c^{8}}{c^{7}}\right )} x - \frac {27 \, B b^{4} c^{5} - 216 \, B a b^{2} c^{6} - 48 \, A b^{3} c^{6} - 6608 \, B a^{2} c^{7} - 12608 \, A a b c^{7}}{c^{7}}\right )} x + \frac {63 \, B b^{5} c^{4} - 568 \, B a b^{3} c^{5} - 112 \, A b^{4} c^{5} + 1392 \, B a^{2} b c^{6} + 960 \, A a b^{2} c^{6} + 18432 \, A a^{2} c^{7}}{c^{7}}\right )} x - \frac {315 \, B b^{6} c^{3} - 3164 \, B a b^{4} c^{4} - 560 \, A b^{5} c^{4} + 9552 \, B a^{2} b^{2} c^{5} + 5376 \, A a b^{3} c^{5} - 6720 \, B a^{3} c^{6} - 14592 \, A a^{2} b c^{6}}{c^{7}}\right )} x + \frac {945 \, B b^{7} c^{2} - 10500 \, B a b^{5} c^{3} - 1680 \, A b^{6} c^{3} + 37744 \, B a^{2} b^{3} c^{4} + 17920 \, A a b^{4} c^{4} - 42432 \, B a^{3} b c^{5} - 59136 \, A a^{2} b^{2} c^{5} + 49152 \, A a^{3} c^{6}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, B b^{8} - 112 \, B a b^{6} c - 16 \, A b^{7} c + 480 \, B a^{2} b^{4} c^{2} + 192 \, A a b^{5} c^{2} - 768 \, B a^{3} b^{2} c^{3} - 768 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \] Input:

integrate(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x + (33*B*b*c^ 
8 + 16*A*c^9)/c^7)*x + (243*B*b^2*c^7 + 476*B*a*c^8 + 464*A*b*c^8)/c^7)*x 
+ (3*B*b^3*c^6 + 1228*B*a*b*c^7 + 592*A*b^2*c^7 + 1152*A*a*c^8)/c^7)*x - ( 
27*B*b^4*c^5 - 216*B*a*b^2*c^6 - 48*A*b^3*c^6 - 6608*B*a^2*c^7 - 12608*A*a 
*b*c^7)/c^7)*x + (63*B*b^5*c^4 - 568*B*a*b^3*c^5 - 112*A*b^4*c^5 + 1392*B* 
a^2*b*c^6 + 960*A*a*b^2*c^6 + 18432*A*a^2*c^7)/c^7)*x - (315*B*b^6*c^3 - 3 
164*B*a*b^4*c^4 - 560*A*b^5*c^4 + 9552*B*a^2*b^2*c^5 + 5376*A*a*b^3*c^5 - 
6720*B*a^3*c^6 - 14592*A*a^2*b*c^6)/c^7)*x + (945*B*b^7*c^2 - 10500*B*a*b^ 
5*c^3 - 1680*A*b^6*c^3 + 37744*B*a^2*b^3*c^4 + 17920*A*a*b^4*c^4 - 42432*B 
*a^3*b*c^5 - 59136*A*a^2*b^2*c^5 + 49152*A*a^3*c^6)/c^7) + 5/32768*(9*B*b^ 
8 - 112*B*a*b^6*c - 16*A*b^7*c + 480*B*a^2*b^4*c^2 + 192*A*a*b^5*c^2 - 768 
*B*a^3*b^2*c^3 - 768*A*a^2*b^3*c^3 + 256*B*a^4*c^4 + 1024*A*a^3*b*c^4)*log 
(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/2)
 

Mupad [F(-1)]

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

int(x*(A + B*x)*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int(x*(A + B*x)*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

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

int(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

int(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)