\(\int (a+b x+c x^2)^{5/2} (A+C x^2) \, dx\) [26]

Optimal result

Integrand size = 22, antiderivative size = 267 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {5 \left (b^2-4 a c\right )^2 \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (32 A c^2+9 b^2 C-4 a c C\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{112 c^2}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}-\frac {5 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \] Output:

5/16384*(-4*a*c+b^2)^2*(32*A*c^2-4*C*a*c+9*C*b^2)*(2*c*x+b)*(c*x^2+b*x+a)^ 
(1/2)/c^5-5/6144*(-4*a*c+b^2)*(32*A*c^2-4*C*a*c+9*C*b^2)*(2*c*x+b)*(c*x^2+ 
b*x+a)^(3/2)/c^4+1/384*(32*A*c^2-4*C*a*c+9*C*b^2)*(2*c*x+b)*(c*x^2+b*x+a)^ 
(5/2)/c^3-9/112*b*C*(c*x^2+b*x+a)^(7/2)/c^2+1/8*C*x*(c*x^2+b*x+a)^(7/2)/c- 
5/32768*(-4*a*c+b^2)^3*(32*A*c^2-4*C*a*c+9*C*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 = 6.17 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.35 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (224 A c^2 (b+2 c x) \left (15 b^4-40 b^3 c x+32 b c^2 x \left (13 a+8 c x^2\right )+8 b^2 c \left (-20 a+11 c x^2\right )+16 c^2 \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right )+C \left (945 b^7-630 b^6 c x+8 b^4 c^2 x \left (791 a-54 c x^2\right )+84 b^5 c \left (-125 a+6 c x^2\right )+16 b^3 c^2 \left (2359 a^2-284 a c x^2+24 c^2 x^4\right )+96 b^2 c^3 x \left (-199 a^2+36 a c x^2+648 c^2 x^4\right )+896 c^4 x \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )+64 b c^3 \left (-663 a^3+174 a^2 c x^2+2456 a c^2 x^4+1584 c^3 x^6\right )\right )\right )-105 \left (b^2-4 a c\right )^3 \left (32 A c^2+9 b^2 C-4 a c C\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{344064 c^{11/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(224*A*c^2*(b + 2*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)) + C*(945*b^7 - 630*b^6*c*x + 8*b^4*c^2*x*(79 
1*a - 54*c*x^2) + 84*b^5*c*(-125*a + 6*c*x^2) + 16*b^3*c^2*(2359*a^2 - 284 
*a*c*x^2 + 24*c^2*x^4) + 96*b^2*c^3*x*(-199*a^2 + 36*a*c*x^2 + 648*c^2*x^4 
) + 896*c^4*x*(15*a^3 + 118*a^2*c*x^2 + 136*a*c^2*x^4 + 48*c^3*x^6) + 64*b 
*c^3*(-663*a^3 + 174*a^2*c*x^2 + 2456*a*c^2*x^4 + 1584*c^3*x^6))) - 105*(b 
^2 - 4*a*c)^3*(32*A*c^2 + 9*b^2*C - 4*a*c*C)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] 
 + Sqrt[a + x*(b + c*x)])])/(344064*c^(11/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\int \frac {1}{2} (16 A c-2 a C-9 b C x) \left (c x^2+b x+a\right )^{5/2}dx}{8 c}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (2 (8 A c-a C)-9 b C x) \left (c x^2+b x+a\right )^{5/2}dx}{16 c}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (-4 a c C+32 A c^2+9 b^2 C\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{2 c}-\frac {9 b C \left (a+b x+c x^2\right )^{7/2}}{7 c}}{16 c}+\frac {C x \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(C*x*(a + b*x + c*x^2)^(7/2))/(8*c) + ((-9*b*C*(a + b*x + c*x^2)^(7/2))/(7 
*c) + ((32*A*c^2 + 9*b^2*C - 4*a*c*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)))/(2*c))/(16*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 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]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(237)=474\).

Time = 1.55 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.79

method	result	size
risch	\(\frac {\left (43008 C \,c^{7} x^{7}+101376 b \,c^{6} C \,x^{6}+57344 A \,c^{7} x^{5}+121856 C a \,c^{6} x^{5}+62208 C \,b^{2} c^{5} x^{5}+143360 A b \,c^{6} x^{4}+157184 C a b \,c^{5} x^{4}+384 C \,b^{3} c^{4} x^{4}+186368 A a \,c^{6} x^{3}+96768 A \,b^{2} c^{5} x^{3}+105728 C \,a^{2} c^{5} x^{3}+3456 C a \,b^{2} c^{4} x^{3}-432 C \,b^{4} c^{3} x^{3}+279552 A a b \,c^{5} x^{2}+1792 A \,b^{3} c^{4} x^{2}+11136 C \,a^{2} b \,c^{4} x^{2}-4544 C a \,b^{3} c^{3} x^{2}+504 C \,b^{5} c^{2} x^{2}+236544 A \,a^{2} c^{5} x +21504 A a \,b^{2} c^{4} x -2240 A \,b^{4} c^{3} x +13440 C \,a^{3} c^{4} x -19104 C \,a^{2} b^{2} c^{3} x +6328 C a \,b^{4} c^{2} x -630 C \,b^{6} c x +118272 A \,a^{2} b \,c^{4}-35840 A a \,b^{3} c^{3}+3360 A \,b^{5} c^{2}-42432 C \,a^{3} b \,c^{3}+37744 C \,a^{2} b^{3} c^{2}-10500 C a \,b^{5} c +945 C \,b^{7}\right ) \sqrt {c \,x^{2}+b x +a}}{344064 c^{5}}+\frac {5 \left (2048 A \,a^{3} c^{5}-1536 A \,a^{2} b^{2} c^{4}+384 A a \,b^{4} c^{3}-32 A \,b^{6} c^{2}-256 C \,a^{4} c^{4}+768 C \,a^{3} b^{2} c^{3}-480 C \,a^{2} b^{4} c^{2}+112 C a \,b^{6} c -9 C \,b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32768 c^{\frac {11}{2}}}\)	\(479\)
default	\(A \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+C \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )\)	\(487\)

Input:

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

Output:

1/344064/c^5*(43008*C*c^7*x^7+101376*C*b*c^6*x^6+57344*A*c^7*x^5+121856*C* 
a*c^6*x^5+62208*C*b^2*c^5*x^5+143360*A*b*c^6*x^4+157184*C*a*b*c^5*x^4+384* 
C*b^3*c^4*x^4+186368*A*a*c^6*x^3+96768*A*b^2*c^5*x^3+105728*C*a^2*c^5*x^3+ 
3456*C*a*b^2*c^4*x^3-432*C*b^4*c^3*x^3+279552*A*a*b*c^5*x^2+1792*A*b^3*c^4 
*x^2+11136*C*a^2*b*c^4*x^2-4544*C*a*b^3*c^3*x^2+504*C*b^5*c^2*x^2+236544*A 
*a^2*c^5*x+21504*A*a*b^2*c^4*x-2240*A*b^4*c^3*x+13440*C*a^3*c^4*x-19104*C* 
a^2*b^2*c^3*x+6328*C*a*b^4*c^2*x-630*C*b^6*c*x+118272*A*a^2*b*c^4-35840*A* 
a*b^3*c^3+3360*A*b^5*c^2-42432*C*a^3*b*c^3+37744*C*a^2*b^3*c^2-10500*C*a*b 
^5*c+945*C*b^7)*(c*x^2+b*x+a)^(1/2)+5/32768*(2048*A*a^3*c^5-1536*A*a^2*b^2 
*c^4+384*A*a*b^4*c^3-32*A*b^6*c^2-256*C*a^4*c^4+768*C*a^3*b^2*c^3-480*C*a^ 
2*b^4*c^2+112*C*a*b^6*c-9*C*b^8)/c^(11/2)*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. 475 vs. \(2 (237) = 474\).

Time = 0.14 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.57 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/1376256*(105*(9*C*b^8 - 112*C*a*b^6*c - 2048*A*a^3*c^5 + 256*(C*a^4 + 
6*A*a^2*b^2)*c^4 - 384*(2*C*a^3*b^2 + A*a*b^4)*c^3 + 32*(15*C*a^2*b^4 + A* 
b^6)*c^2)*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*C*c^8*x^7 + 101376*C*b*c^7*x^6 + 
945*C*b^7*c - 10500*C*a*b^5*c^2 + 118272*A*a^2*b*c^5 + 256*(243*C*b^2*c^6 
+ 476*C*a*c^7 + 224*A*c^8)*x^5 - 64*(663*C*a^3*b + 560*A*a*b^3)*c^4 + 128* 
(3*C*b^3*c^5 + 1228*C*a*b*c^6 + 1120*A*b*c^7)*x^4 + 112*(337*C*a^2*b^3 + 3 
0*A*b^5)*c^3 - 16*(27*C*b^4*c^4 - 216*C*a*b^2*c^5 - 11648*A*a*c^7 - 112*(5 
9*C*a^2 + 54*A*b^2)*c^6)*x^3 + 8*(63*C*b^5*c^3 - 568*C*a*b^3*c^4 + 34944*A 
*a*b*c^6 + 16*(87*C*a^2*b + 14*A*b^3)*c^5)*x^2 - 2*(315*C*b^6*c^2 - 3164*C 
*a*b^4*c^3 - 118272*A*a^2*c^6 - 1344*(5*C*a^3 + 8*A*a*b^2)*c^5 + 16*(597*C 
*a^2*b^2 + 70*A*b^4)*c^4)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/688128*(105*(9* 
C*b^8 - 112*C*a*b^6*c - 2048*A*a^3*c^5 + 256*(C*a^4 + 6*A*a^2*b^2)*c^4 - 3 
84*(2*C*a^3*b^2 + A*a*b^4)*c^3 + 32*(15*C*a^2*b^4 + A*b^6)*c^2)*sqrt(-c)*a 
rctan(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*C*c^8*x^7 + 101376*C*b*c^7*x^6 + 945*C*b^7*c - 10500*C*a*b^ 
5*c^2 + 118272*A*a^2*b*c^5 + 256*(243*C*b^2*c^6 + 476*C*a*c^7 + 224*A*c^8) 
*x^5 - 64*(663*C*a^3*b + 560*A*a*b^3)*c^4 + 128*(3*C*b^3*c^5 + 1228*C*a*b* 
c^6 + 1120*A*b*c^7)*x^4 + 112*(337*C*a^2*b^3 + 30*A*b^5)*c^3 - 16*(27*C*b^ 
4*c^4 - 216*C*a*b^2*c^5 - 11648*A*a*c^7 - 112*(59*C*a^2 + 54*A*b^2)*c^6...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2691 vs. \(2 (270) = 540\).

Time = 0.58 (sec) , antiderivative size = 2691, normalized size of antiderivative = 10.08 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(33*C*b*c*x**6/112 + C*c**2*x**7/8 + x** 
5*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(6*c) + x**4*(3*A*b*c**2 + 2 
37*C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/ 
(12*c))/(5*c) + x**3*(3*A*a*c**2 + 3*A*b**2*c + 3*C*a**2*c + 3*C*a*b**2 - 
5*a*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(6*c) - 9*b*(3*A*b*c**2 + 
237*C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224) 
/(12*c))/(10*c))/(4*c) + x**2*(6*A*a*b*c + A*b**3 + 3*C*a**2*b - 4*a*(3*A* 
b*c**2 + 237*C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b* 
*2*c/224)/(12*c))/(5*c) - 7*b*(3*A*a*c**2 + 3*A*b**2*c + 3*C*a**2*c + 3*C* 
a*b**2 - 5*a*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(6*c) - 9*b*(3*A* 
b*c**2 + 237*C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b* 
*2*c/224)/(12*c))/(10*c))/(8*c))/(3*c) + x*(3*A*a**2*c + 3*A*a*b**2 + C*a* 
*3 - 3*a*(3*A*a*c**2 + 3*A*b**2*c + 3*C*a**2*c + 3*C*a*b**2 - 5*a*(A*c**3 
+ 17*C*a*c**2/8 + 243*C*b**2*c/224)/(6*c) - 9*b*(3*A*b*c**2 + 237*C*a*b*c/ 
56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(12*c))/(10 
*c))/(4*c) - 5*b*(6*A*a*b*c + A*b**3 + 3*C*a**2*b - 4*a*(3*A*b*c**2 + 237* 
C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(12 
*c))/(5*c) - 7*b*(3*A*a*c**2 + 3*A*b**2*c + 3*C*a**2*c + 3*C*a*b**2 - 5*a* 
(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/(6*c) - 9*b*(3*A*b*c**2 + 237* 
C*a*b*c/56 + C*b**3 - 11*b*(A*c**3 + 17*C*a*c**2/8 + 243*C*b**2*c/224)/...
 

Maxima [F(-2)]

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

integrate((c*x^2+b*x+a)^(5/2)*(C*x^2+A),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. 480 vs. \(2 (237) = 474\).

Time = 0.37 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.80 \[ \int \left (a+b x+c x^2\right )^{5/2} \left (A+C x^2\right ) \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, C c^{2} x + 33 \, C b c\right )} x + \frac {243 \, C b^{2} c^{7} + 476 \, C a c^{8} + 224 \, A c^{9}}{c^{7}}\right )} x + \frac {3 \, C b^{3} c^{6} + 1228 \, C a b c^{7} + 1120 \, A b c^{8}}{c^{7}}\right )} x - \frac {27 \, C b^{4} c^{5} - 216 \, C a b^{2} c^{6} - 6608 \, C a^{2} c^{7} - 6048 \, A b^{2} c^{7} - 11648 \, A a c^{8}}{c^{7}}\right )} x + \frac {63 \, C b^{5} c^{4} - 568 \, C a b^{3} c^{5} + 1392 \, C a^{2} b c^{6} + 224 \, A b^{3} c^{6} + 34944 \, A a b c^{7}}{c^{7}}\right )} x - \frac {315 \, C b^{6} c^{3} - 3164 \, C a b^{4} c^{4} + 9552 \, C a^{2} b^{2} c^{5} + 1120 \, A b^{4} c^{5} - 6720 \, C a^{3} c^{6} - 10752 \, A a b^{2} c^{6} - 118272 \, A a^{2} c^{7}}{c^{7}}\right )} x + \frac {945 \, C b^{7} c^{2} - 10500 \, C a b^{5} c^{3} + 37744 \, C a^{2} b^{3} c^{4} + 3360 \, A b^{5} c^{4} - 42432 \, C a^{3} b c^{5} - 35840 \, A a b^{3} c^{5} + 118272 \, A a^{2} b c^{6}}{c^{7}}\right )} + \frac {5 \, {\left (9 \, C b^{8} - 112 \, C a b^{6} c + 480 \, C a^{2} b^{4} c^{2} + 32 \, A b^{6} c^{2} - 768 \, C a^{3} b^{2} c^{3} - 384 \, A a b^{4} c^{3} + 256 \, C a^{4} c^{4} + 1536 \, A a^{2} b^{2} c^{4} - 2048 \, A a^{3} c^{5}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \] Input:

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

Output:

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*C*c^2*x + 33*C*b*c)* 
x + (243*C*b^2*c^7 + 476*C*a*c^8 + 224*A*c^9)/c^7)*x + (3*C*b^3*c^6 + 1228 
*C*a*b*c^7 + 1120*A*b*c^8)/c^7)*x - (27*C*b^4*c^5 - 216*C*a*b^2*c^6 - 6608 
*C*a^2*c^7 - 6048*A*b^2*c^7 - 11648*A*a*c^8)/c^7)*x + (63*C*b^5*c^4 - 568* 
C*a*b^3*c^5 + 1392*C*a^2*b*c^6 + 224*A*b^3*c^6 + 34944*A*a*b*c^7)/c^7)*x - 
 (315*C*b^6*c^3 - 3164*C*a*b^4*c^4 + 9552*C*a^2*b^2*c^5 + 1120*A*b^4*c^5 - 
 6720*C*a^3*c^6 - 10752*A*a*b^2*c^6 - 118272*A*a^2*c^7)/c^7)*x + (945*C*b^ 
7*c^2 - 10500*C*a*b^5*c^3 + 37744*C*a^2*b^3*c^4 + 3360*A*b^5*c^4 - 42432*C 
*a^3*b*c^5 - 35840*A*a*b^3*c^5 + 118272*A*a^2*b*c^6)/c^7) + 5/32768*(9*C*b 
^8 - 112*C*a*b^6*c + 480*C*a^2*b^4*c^2 + 32*A*b^6*c^2 - 768*C*a^3*b^2*c^3 
- 384*A*a*b^4*c^3 + 256*C*a^4*c^4 + 1536*A*a^2*b^2*c^4 - 2048*A*a^3*c^5)*l 
og(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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