\(\int \frac {A+C x^2}{(a+b x+c x^2)^{9/2}} \, dx\) [33]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \left (b (A c+a C)+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{7 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac {2 \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x)}{35 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac {32 \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x)}{105 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {256 c \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x)}{105 \left (b^2-4 a c\right )^4 \sqrt {a+b x+c x^2}} \] Output:

1/7*(-2*b*(A*c+C*a)-2*(2*A*c^2+(-2*a*c+b^2)*C)*x)/c/(-4*a*c+b^2)/(c*x^2+b* 
x+a)^(7/2)+2/35*(24*A*c^2+4*C*a*c+5*C*b^2)*(2*c*x+b)/c/(-4*a*c+b^2)^2/(c*x 
^2+b*x+a)^(5/2)-32/105*(24*A*c^2+4*C*a*c+5*C*b^2)*(2*c*x+b)/(-4*a*c+b^2)^3 
/(c*x^2+b*x+a)^(3/2)+256/105*c*(24*A*c^2+4*C*a*c+5*C*b^2)*(2*c*x+b)/(-4*a* 
c+b^2)^4/(c*x^2+b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 15.75 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.90 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\frac {2 \left (3 \left (b^2-4 a c\right )^2 \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x) (a+x (b+c x))-16 c \left (b^2-4 a c\right ) \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x) (a+x (b+c x))^2+128 c^2 \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x) (a+x (b+c x))^3-15 \left (b^2-4 a c\right )^3 \left (b^2 C x+a C (b-2 c x)+A c (b+2 c x)\right )\right )}{105 c \left (b^2-4 a c\right )^4 (a+x (b+c x))^{7/2}} \] Input:

Integrate[(A + C*x^2)/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(2*(3*(b^2 - 4*a*c)^2*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a + x*(b 
 + c*x)) - 16*c*(b^2 - 4*a*c)*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*( 
a + x*(b + c*x))^2 + 128*c^2*(24*A*c^2 + 5*b^2*C + 4*a*c*C)*(b + 2*c*x)*(a 
 + x*(b + c*x))^3 - 15*(b^2 - 4*a*c)^3*(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b 
 + 2*c*x))))/(105*c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))^(7/2))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2191, 27, 1089, 1089, 1088}

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.

\(\displaystyle \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 2191

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (4 a C+24 A c+\frac {5 b^2 C}{c}\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{7/2}}dx}{7 \left (b^2-4 a c\right )}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{7 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}\)

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1089

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

\(\Big \downarrow \) 1088

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

Input:

Int[(A + C*x^2)/(a + b*x + c*x^2)^(9/2),x]
 

Output:

(-2*(b*c*(A + (a*C)/c) + (2*A*c^2 + (b^2 - 2*a*c)*C)*x))/(7*c*(b^2 - 4*a*c 
)*(a + b*x + c*x^2)^(7/2)) - ((24*A*c + 4*a*C + (5*b^2*C)/c)*((-2*(b + 2*c 
*x))/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (16*c*((-2*(b + 2*c*x))/( 
3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*(b^2 - 4* 
a*c)^2*Sqrt[a + b*x + c*x^2])))/(5*(b^2 - 4*a*c))))/(7*(b^2 - 4*a*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 1088

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

rule 1089

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

rule 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(207)=414\).

Time = 1.54 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.37

method	result	size
trager	\(\frac {128 C \,a^{3} b \,c^{3} x^{2}+\frac {704}{3} C \,a^{2} b^{2} c^{3} x^{3}+\frac {392}{15} C a \,b^{5} c \,x^{2}+192 A \,a^{2} b^{2} c^{3} x -16 A a \,b^{4} c^{2} x +128 C \,a^{3} b^{2} c^{2} x +768 A \,a^{2} b \,c^{4} x^{2}+128 A a \,b^{3} c^{3} x^{2}+\frac {2048}{105} C a \,c^{6} x^{7}+\frac {512}{21} C \,b^{2} c^{5} x^{7}+\frac {2048}{5} A b \,c^{6} x^{6}+\frac {544}{3} C \,a^{2} b^{3} c^{2} x^{2}+\frac {64}{3} C \,a^{2} b^{4} c x +\frac {1024}{15} C \,a^{2} c^{5} x^{5}+\frac {496}{3} C a \,b^{4} c^{2} x^{3}+256 C a \,b^{3} c^{3} x^{4}+768 A a \,b^{2} c^{4} x^{3}+\frac {1024}{15} C a b \,c^{5} x^{6}+\frac {512}{3} C \,a^{2} b \,c^{4} x^{4}+\frac {512}{3} C a \,b^{2} c^{4} x^{5}+1024 A a b \,c^{5} x^{4}+\frac {4}{5} A \,b^{6} c x +256 A \,a^{3} c^{4} x -\frac {8}{15} C a \,b^{6} x +\frac {256}{7} C \,a^{4} b \,c^{2}+\frac {128}{21} C \,a^{3} b^{3} c +\frac {2048}{5} A a \,c^{6} x^{5}+\frac {320}{3} C \,b^{4} c^{3} x^{5}+256 A \,b^{3} c^{4} x^{4}+32 A \,b^{4} c^{3} x^{3}+\frac {256}{3} C \,a^{3} c^{4} x^{3}+\frac {256}{3} C \,b^{3} c^{4} x^{6}+\frac {20}{3} C \,b^{6} c \,x^{3}-\frac {16}{5} A \,b^{5} c^{2} x^{2}+128 A \,a^{3} b \,c^{3}-32 A \,a^{2} b^{3} c^{2}+\frac {24}{5} A a \,b^{5} c +\frac {160}{3} C \,b^{5} c^{2} x^{4}+512 A \,a^{2} c^{5} x^{3}+512 A \,b^{2} c^{5} x^{5}-\frac {2}{7} A \,b^{7}-\frac {16}{105} C \,a^{2} b^{5}+\frac {4096}{35} A \,c^{7} x^{7}-\frac {2}{3} C \,b^{7} x^{2}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}\)	\(524\)
default	\(A \left (\frac {\frac {4 c x}{7}+\frac {2 b}{7}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}+\frac {24 c \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{7 \left (4 a c -b^{2}\right )}\right )+C \left (-\frac {x}{6 c \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {5 b \left (-\frac {1}{7 c \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}-\frac {b \left (\frac {\frac {4 c x}{7}+\frac {2 b}{7}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}+\frac {24 c \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{7 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )}{12 c}+\frac {a \left (\frac {\frac {4 c x}{7}+\frac {2 b}{7}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}+\frac {24 c \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{7 \left (4 a c -b^{2}\right )}\right )}{6 c}\right )\)	\(547\)
gosper	\(\frac {128 C \,a^{3} b \,c^{3} x^{2}+\frac {704}{3} C \,a^{2} b^{2} c^{3} x^{3}+\frac {392}{15} C a \,b^{5} c \,x^{2}+192 A \,a^{2} b^{2} c^{3} x -16 A a \,b^{4} c^{2} x +128 C \,a^{3} b^{2} c^{2} x +768 A \,a^{2} b \,c^{4} x^{2}+128 A a \,b^{3} c^{3} x^{2}+\frac {2048}{105} C a \,c^{6} x^{7}+\frac {512}{21} C \,b^{2} c^{5} x^{7}+\frac {2048}{5} A b \,c^{6} x^{6}+\frac {544}{3} C \,a^{2} b^{3} c^{2} x^{2}+\frac {64}{3} C \,a^{2} b^{4} c x +\frac {1024}{15} C \,a^{2} c^{5} x^{5}+\frac {496}{3} C a \,b^{4} c^{2} x^{3}+256 C a \,b^{3} c^{3} x^{4}+768 A a \,b^{2} c^{4} x^{3}+\frac {1024}{15} C a b \,c^{5} x^{6}+\frac {512}{3} C \,a^{2} b \,c^{4} x^{4}+\frac {512}{3} C a \,b^{2} c^{4} x^{5}+1024 A a b \,c^{5} x^{4}+\frac {4}{5} A \,b^{6} c x +256 A \,a^{3} c^{4} x -\frac {8}{15} C a \,b^{6} x +\frac {256}{7} C \,a^{4} b \,c^{2}+\frac {128}{21} C \,a^{3} b^{3} c +\frac {2048}{5} A a \,c^{6} x^{5}+\frac {320}{3} C \,b^{4} c^{3} x^{5}+256 A \,b^{3} c^{4} x^{4}+32 A \,b^{4} c^{3} x^{3}+\frac {256}{3} C \,a^{3} c^{4} x^{3}+\frac {256}{3} C \,b^{3} c^{4} x^{6}+\frac {20}{3} C \,b^{6} c \,x^{3}-\frac {16}{5} A \,b^{5} c^{2} x^{2}+128 A \,a^{3} b \,c^{3}-32 A \,a^{2} b^{3} c^{2}+\frac {24}{5} A a \,b^{5} c +\frac {160}{3} C \,b^{5} c^{2} x^{4}+512 A \,a^{2} c^{5} x^{3}+512 A \,b^{2} c^{5} x^{5}-\frac {2}{7} A \,b^{7}-\frac {16}{105} C \,a^{2} b^{5}+\frac {4096}{35} A \,c^{7} x^{7}-\frac {2}{3} C \,b^{7} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(555\)
orering	\(\frac {128 C \,a^{3} b \,c^{3} x^{2}+\frac {704}{3} C \,a^{2} b^{2} c^{3} x^{3}+\frac {392}{15} C a \,b^{5} c \,x^{2}+192 A \,a^{2} b^{2} c^{3} x -16 A a \,b^{4} c^{2} x +128 C \,a^{3} b^{2} c^{2} x +768 A \,a^{2} b \,c^{4} x^{2}+128 A a \,b^{3} c^{3} x^{2}+\frac {2048}{105} C a \,c^{6} x^{7}+\frac {512}{21} C \,b^{2} c^{5} x^{7}+\frac {2048}{5} A b \,c^{6} x^{6}+\frac {544}{3} C \,a^{2} b^{3} c^{2} x^{2}+\frac {64}{3} C \,a^{2} b^{4} c x +\frac {1024}{15} C \,a^{2} c^{5} x^{5}+\frac {496}{3} C a \,b^{4} c^{2} x^{3}+256 C a \,b^{3} c^{3} x^{4}+768 A a \,b^{2} c^{4} x^{3}+\frac {1024}{15} C a b \,c^{5} x^{6}+\frac {512}{3} C \,a^{2} b \,c^{4} x^{4}+\frac {512}{3} C a \,b^{2} c^{4} x^{5}+1024 A a b \,c^{5} x^{4}+\frac {4}{5} A \,b^{6} c x +256 A \,a^{3} c^{4} x -\frac {8}{15} C a \,b^{6} x +\frac {256}{7} C \,a^{4} b \,c^{2}+\frac {128}{21} C \,a^{3} b^{3} c +\frac {2048}{5} A a \,c^{6} x^{5}+\frac {320}{3} C \,b^{4} c^{3} x^{5}+256 A \,b^{3} c^{4} x^{4}+32 A \,b^{4} c^{3} x^{3}+\frac {256}{3} C \,a^{3} c^{4} x^{3}+\frac {256}{3} C \,b^{3} c^{4} x^{6}+\frac {20}{3} C \,b^{6} c \,x^{3}-\frac {16}{5} A \,b^{5} c^{2} x^{2}+128 A \,a^{3} b \,c^{3}-32 A \,a^{2} b^{3} c^{2}+\frac {24}{5} A a \,b^{5} c +\frac {160}{3} C \,b^{5} c^{2} x^{4}+512 A \,a^{2} c^{5} x^{3}+512 A \,b^{2} c^{5} x^{5}-\frac {2}{7} A \,b^{7}-\frac {16}{105} C \,a^{2} b^{5}+\frac {4096}{35} A \,c^{7} x^{7}-\frac {2}{3} C \,b^{7} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\)	\(555\)

Input:

int((C*x^2+A)/(c*x^2+b*x+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

2/105*(6144*A*c^7*x^7+1024*C*a*c^6*x^7+1280*C*b^2*c^5*x^7+21504*A*b*c^6*x^ 
6+3584*C*a*b*c^5*x^6+4480*C*b^3*c^4*x^6+21504*A*a*c^6*x^5+26880*A*b^2*c^5* 
x^5+3584*C*a^2*c^5*x^5+8960*C*a*b^2*c^4*x^5+5600*C*b^4*c^3*x^5+53760*A*a*b 
*c^5*x^4+13440*A*b^3*c^4*x^4+8960*C*a^2*b*c^4*x^4+13440*C*a*b^3*c^3*x^4+28 
00*C*b^5*c^2*x^4+26880*A*a^2*c^5*x^3+40320*A*a*b^2*c^4*x^3+1680*A*b^4*c^3* 
x^3+4480*C*a^3*c^4*x^3+12320*C*a^2*b^2*c^3*x^3+8680*C*a*b^4*c^2*x^3+350*C* 
b^6*c*x^3+40320*A*a^2*b*c^4*x^2+6720*A*a*b^3*c^3*x^2-168*A*b^5*c^2*x^2+672 
0*C*a^3*b*c^3*x^2+9520*C*a^2*b^3*c^2*x^2+1372*C*a*b^5*c*x^2-35*C*b^7*x^2+1 
3440*A*a^3*c^4*x+10080*A*a^2*b^2*c^3*x-840*A*a*b^4*c^2*x+42*A*b^6*c*x+6720 
*C*a^3*b^2*c^2*x+1120*C*a^2*b^4*c*x-28*C*a*b^6*x+6720*A*a^3*b*c^3-1680*A*a 
^2*b^3*c^2+252*A*a*b^5*c-15*A*b^7+1920*C*a^4*b*c^2+320*C*a^3*b^3*c-8*C*a^2 
*b^5)/(4*a*c-b^2)^4/(c*x^2+b*x+a)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (205) = 410\).

Time = 13.63 (sec) , antiderivative size = 978, normalized size of antiderivative = 4.43 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((C*x^2+A)/(c*x^2+b*x+a)^(9/2),x, algorithm="fricas")
 

Output:

-2/105*(8*C*a^2*b^5 + 15*A*b^7 - 6720*A*a^3*b*c^3 - 256*(5*C*b^2*c^5 + 4*C 
*a*c^6 + 24*A*c^7)*x^7 - 896*(5*C*b^3*c^4 + 4*C*a*b*c^5 + 24*A*b*c^6)*x^6 
- 224*(25*C*b^4*c^3 + 40*C*a*b^2*c^4 + 96*A*a*c^6 + 8*(2*C*a^2 + 15*A*b^2) 
*c^5)*x^5 - 560*(5*C*b^5*c^2 + 24*C*a*b^3*c^3 + 96*A*a*b*c^5 + 8*(2*C*a^2* 
b + 3*A*b^3)*c^4)*x^4 - 70*(5*C*b^6*c + 124*C*a*b^4*c^2 + 384*A*a^2*c^5 + 
64*(C*a^3 + 9*A*a*b^2)*c^4 + 8*(22*C*a^2*b^2 + 3*A*b^4)*c^3)*x^3 - 240*(8* 
C*a^4*b - 7*A*a^2*b^3)*c^2 + 7*(5*C*b^7 - 196*C*a*b^5*c - 5760*A*a^2*b*c^4 
 - 960*(C*a^3*b + A*a*b^3)*c^3 - 8*(170*C*a^2*b^3 - 3*A*b^5)*c^2)*x^2 - 4* 
(80*C*a^3*b^3 + 63*A*a*b^5)*c + 14*(2*C*a*b^6 - 720*A*a^2*b^2*c^3 - 960*A* 
a^3*c^4 - 60*(8*C*a^3*b^2 - A*a*b^4)*c^2 - (80*C*a^2*b^4 + 3*A*b^6)*c)*x)* 
sqrt(c*x^2 + b*x + a)/(a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b 
^2*c^3 + 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3* 
b^2*c^7 + 256*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 
256*a^3*b^3*c^6 + 256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256* 
a^2*b^6*c^4 - 576*a^3*b^4*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^ 
11*c - 13*a*b^9*c^2 + 48*a^2*b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 
768*a^5*b*c^6)*x^5 + (b^12 - 4*a*b^10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 
 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1536*a^6*c^6)*x^4 + 4*(a*b^11 - 1 
3*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512*a^5*b^3*c^4 + 768*a^6* 
b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6*c^2 - 576*a^5*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x**2+A)/(c*x**2+b*x+a)**(9/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((C*x^2+A)/(c*x^2+b*x+a)^(9/2),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. 805 vs. \(2 (205) = 410\).

Time = 0.35 (sec) , antiderivative size = 805, normalized size of antiderivative = 3.64 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

integrate((C*x^2+A)/(c*x^2+b*x+a)^(9/2),x, algorithm="giac")
 

Output:

2/105*(((2*(8*(2*(4*(2*(5*C*b^2*c^5 + 4*C*a*c^6 + 24*A*c^7)*x/(b^8 - 16*a* 
b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4) + 7*(5*C*b^3*c^4 + 
 4*C*a*b*c^5 + 24*A*b*c^6)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^ 
2*c^3 + 256*a^4*c^4))*x + 7*(25*C*b^4*c^3 + 40*C*a*b^2*c^4 + 16*C*a^2*c^5 
+ 120*A*b^2*c^5 + 96*A*a*c^6)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3 
*b^2*c^3 + 256*a^4*c^4))*x + 35*(5*C*b^5*c^2 + 24*C*a*b^3*c^3 + 16*C*a^2*b 
*c^4 + 24*A*b^3*c^4 + 96*A*a*b*c^5)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 2 
56*a^3*b^2*c^3 + 256*a^4*c^4))*x + 35*(5*C*b^6*c + 124*C*a*b^4*c^2 + 176*C 
*a^2*b^2*c^3 + 24*A*b^4*c^3 + 64*C*a^3*c^4 + 576*A*a*b^2*c^4 + 384*A*a^2*c 
^5)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x 
 - 7*(5*C*b^7 - 196*C*a*b^5*c - 1360*C*a^2*b^3*c^2 + 24*A*b^5*c^2 - 960*C* 
a^3*b*c^3 - 960*A*a*b^3*c^3 - 5760*A*a^2*b*c^4)/(b^8 - 16*a*b^6*c + 96*a^2 
*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))*x - 14*(2*C*a*b^6 - 80*C*a^2*b^ 
4*c - 3*A*b^6*c - 480*C*a^3*b^2*c^2 + 60*A*a*b^4*c^2 - 720*A*a^2*b^2*c^3 - 
 960*A*a^3*c^4)/(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256 
*a^4*c^4))*x - (8*C*a^2*b^5 + 15*A*b^7 - 320*C*a^3*b^3*c - 252*A*a*b^5*c - 
 1920*C*a^4*b*c^2 + 1680*A*a^2*b^3*c^2 - 6720*A*a^3*b*c^3)/(b^8 - 16*a*b^6 
*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4))/(c*x^2 + b*x + a)^(7 
/2)
 

Mupad [B] (verification not implemented)

Time = 17.43 (sec) , antiderivative size = 1018, normalized size of antiderivative = 4.61 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

int((A + C*x^2)/(a + b*x + c*x^2)^(9/2),x)
 

Output:

(x*((2*c^2*(768*A*c^2 + 160*C*b^2 + 96*C*a*c))/(105*(4*a*c^2 - b^2*c)*(4*a 
*c - b^2)^2) - (64*C*a*c^3)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) + (32* 
C*b^2*c^2)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2)) + (b*c*(768*A*c^2 + 16 
0*C*b^2 + 96*C*a*c))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) + (32*C*a*b*c 
^2)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2))/(a + b*x + c*x^2)^(3/2) - ((8 
*C*b)/(105*(4*a*c - b^2)^2) - (16*C*c*x)/(105*(4*a*c - b^2)^2))/(a + b*x + 
 c*x^2)^(3/2) + ((8*C*b*c)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*C*c 
^2*x)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) - ((4 
*C*x)/(35*(4*a*c - b^2)) - (2*C*b)/(35*c*(4*a*c - b^2)))/(a + b*x + c*x^2) 
^(5/2) + ((b*c*(6144*A*c^3 + 896*C*a*c^2 + 1312*C*b^2*c))/(105*(4*a*c^2 - 
b^2*c)*(4*a*c - b^2)^3) + (2*c^2*x*(6144*A*c^3 + 896*C*a*c^2 + 1312*C*b^2* 
c))/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^3))/(a + b*x + c*x^2)^(1/2) + (x* 
((4*A*c^2)/(7*(4*a*c^2 - b^2*c)) + (2*C*b^2)/(7*(4*a*c^2 - b^2*c)) - (4*C* 
a*c)/(7*(4*a*c^2 - b^2*c))) + (2*A*b*c)/(7*(4*a*c^2 - b^2*c)) + (2*C*a*b)/ 
(7*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^2)^(7/2) + (x*((2*c*(48*A*c^2 + 12*C 
*b^2 + 8*C*a*c))/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*C*a*c^2)/(35*( 
4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*C*b^2*c)/(35*(4*a*c^2 - b^2*c)*(4*a*c 
 - b^2))) + (b*(48*A*c^2 + 12*C*b^2 + 8*C*a*c))/(35*(4*a*c^2 - b^2*c)*(4*a 
*c - b^2)) - (8*C*a*b*c)/(35*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + 
c*x^2)^(5/2) - ((32*C*b*c^2)/(105*(4*a*c^2 - b^2*c)*(4*a*c - b^2)^2) + ...
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 1509, normalized size of antiderivative = 6.83 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx =\text {Too large to display} \] Input:

int((C*x^2+A)/(c*x^2+b*x+a)^(9/2),x)
 

Output:

(2*(8640*sqrt(a + b*x + c*x**2)*a**4*b*c**3 + 13440*sqrt(a + b*x + c*x**2) 
*a**4*c**4*x - 1360*sqrt(a + b*x + c*x**2)*a**3*b**3*c**2 + 16800*sqrt(a + 
 b*x + c*x**2)*a**3*b**2*c**3*x + 47040*sqrt(a + b*x + c*x**2)*a**3*b*c**4 
*x**2 + 31360*sqrt(a + b*x + c*x**2)*a**3*c**5*x**3 + 244*sqrt(a + b*x + c 
*x**2)*a**2*b**5*c + 280*sqrt(a + b*x + c*x**2)*a**2*b**4*c**2*x + 16240*s 
qrt(a + b*x + c*x**2)*a**2*b**3*c**3*x**2 + 52640*sqrt(a + b*x + c*x**2)*a 
**2*b**2*c**4*x**3 + 62720*sqrt(a + b*x + c*x**2)*a**2*b*c**5*x**4 + 25088 
*sqrt(a + b*x + c*x**2)*a**2*c**6*x**5 - 15*sqrt(a + b*x + c*x**2)*a*b**7 
+ 14*sqrt(a + b*x + c*x**2)*a*b**6*c*x + 1204*sqrt(a + b*x + c*x**2)*a*b** 
5*c**2*x**2 + 10360*sqrt(a + b*x + c*x**2)*a*b**4*c**3*x**3 + 26880*sqrt(a 
 + b*x + c*x**2)*a*b**3*c**4*x**4 + 35840*sqrt(a + b*x + c*x**2)*a*b**2*c* 
*5*x**5 + 25088*sqrt(a + b*x + c*x**2)*a*b*c**6*x**6 + 7168*sqrt(a + b*x + 
 c*x**2)*a*c**7*x**7 - 35*sqrt(a + b*x + c*x**2)*b**7*c*x**2 + 350*sqrt(a 
+ b*x + c*x**2)*b**6*c**2*x**3 + 2800*sqrt(a + b*x + c*x**2)*b**5*c**3*x** 
4 + 5600*sqrt(a + b*x + c*x**2)*b**4*c**4*x**5 + 4480*sqrt(a + b*x + c*x** 
2)*b**3*c**5*x**6 + 1280*sqrt(a + b*x + c*x**2)*b**2*c**6*x**7 - 7168*sqrt 
(c)*a**5*c**3 - 1280*sqrt(c)*a**4*b**2*c**2 - 28672*sqrt(c)*a**4*b*c**3*x 
- 28672*sqrt(c)*a**4*c**4*x**2 - 5120*sqrt(c)*a**3*b**3*c**2*x - 48128*sqr 
t(c)*a**3*b**2*c**3*x**2 - 86016*sqrt(c)*a**3*b*c**4*x**3 - 43008*sqrt(c)* 
a**3*c**5*x**4 - 7680*sqrt(c)*a**2*b**4*c**2*x**2 - 44032*sqrt(c)*a**2*...