Integrand size = 22, antiderivative size = 167 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {2 \left (16 A c+4 a C+\frac {3 b^2 C}{c}\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {16 \left (16 A c^2+3 b^2 C+4 a c C\right ) (b+2 c x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \]
-2/5*(b*c*(A+a*C/c)+(2*A*c^2+(-2*a*c+b^2)*C)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+ a)^(5/2)+2/15*(16*A*c+4*C*a+3*b^2*C/c)*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^2+b*x +a)^(3/2)-16/15*(16*A*c^2+4*C*a*c+3*C*b^2)*(2*c*x+b)/(-4*a*c+b^2)^3/(c*x^2 +b*x+a)^(1/2)
Time = 2.52 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.40 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (A (b+2 c x) \left (3 b^4-16 b^3 c x+64 b c^2 x \left (5 a+4 c x^2\right )+8 b^2 c \left (-5 a+14 c x^2\right )+16 c^2 \left (15 a^2+20 a c x^2+8 c^2 x^4\right )\right )+C \left (96 a^3 b c+3 b^2 x^2 \left (5 b^3+30 b^2 c x+40 b c^2 x^2+16 c^3 x^3\right )+8 a^2 \left (b^3+30 b^2 c x+30 b c^2 x^2+20 c^3 x^3\right )+4 a x \left (5 b^4+50 b^3 c x+60 b^2 c^2 x^2+40 b c^3 x^3+16 c^4 x^4\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
(-2*(A*(b + 2*c*x)*(3*b^4 - 16*b^3*c*x + 64*b*c^2*x*(5*a + 4*c*x^2) + 8*b^ 2*c*(-5*a + 14*c*x^2) + 16*c^2*(15*a^2 + 20*a*c*x^2 + 8*c^2*x^4)) + C*(96* a^3*b*c + 3*b^2*x^2*(5*b^3 + 30*b^2*c*x + 40*b*c^2*x^2 + 16*c^3*x^3) + 8*a ^2*(b^3 + 30*b^2*c*x + 30*b*c^2*x^2 + 20*c^3*x^3) + 4*a*x*(5*b^4 + 50*b^3* c*x + 60*b^2*c^2*x^2 + 40*b*c^3*x^3 + 16*c^4*x^4))))/(15*(b^2 - 4*a*c)^3*( a + x*(b + c*x))^(5/2))
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2191, 27, 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.
| \(\displaystyle \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle -\frac {2 \int \frac {\frac {3 C b^2}{c}+16 A c+4 a C}{2 \left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (4 a C+16 A c+\frac {3 b^2 C}{c}\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1089 |
| \(\displaystyle -\frac {\left (4 a C+16 A c+\frac {3 b^2 C}{c}\right ) \left (-\frac {8 c \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle -\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {\left (\frac {16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) \left (4 a C+16 A c+\frac {3 b^2 C}{c}\right )}{5 \left (b^2-4 a c\right )}\) |
(-2*(b*c*(A + (a*C)/c) + (2*A*c^2 + (b^2 - 2*a*c)*C)*x))/(5*c*(b^2 - 4*a*c )*(a + b*x + c*x^2)^(5/2)) - ((16*A*c + 4*a*C + (3*b^2*C)/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))
3.2.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
Time = 0.67 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.76
| method | result | size |
| trager | \(\frac {\frac {512}{15} A \,c^{5} x^{5}+128 A a b \,c^{3} x^{2}+32 C \,a^{2} b \,c^{2} x^{2}+32 A \,a^{2} b \,c^{2}-\frac {16}{3} A a \,b^{3} c +\frac {2}{5} A \,b^{5}+\frac {64}{5} C \,a^{3} b c +12 C \,b^{4} c \,x^{3}+\frac {32}{3} A \,b^{3} c^{2} x^{2}+64 A \,a^{2} c^{3} x -\frac {4}{3} A \,b^{4} c x +\frac {8}{3} C a \,b^{4} x +\frac {256}{3} A a \,c^{4} x^{3}+64 A \,b^{2} c^{3} x^{3}+\frac {64}{3} C \,a^{2} c^{3} x^{3}+\frac {256}{3} A b \,c^{4} x^{4}+16 C \,b^{3} c^{2} x^{4}+\frac {32}{5} C \,b^{2} c^{3} x^{5}+\frac {128}{15} C a \,c^{4} x^{5}+\frac {80}{3} C a \,b^{3} c \,x^{2}+\frac {64}{3} C a b \,c^{3} x^{4}+32 C a \,b^{2} c^{2} x^{3}+2 C \,b^{5} x^{2}+\frac {16}{15} C \,a^{2} b^{3}+32 A a \,b^{2} c^{2} x +32 C \,a^{2} b^{2} c x}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}\) | \(294\) |
| gosper | \(\frac {\frac {512}{15} A \,c^{5} x^{5}+128 A a b \,c^{3} x^{2}+32 C \,a^{2} b \,c^{2} x^{2}+32 A \,a^{2} b \,c^{2}-\frac {16}{3} A a \,b^{3} c +\frac {2}{5} A \,b^{5}+\frac {64}{5} C \,a^{3} b c +12 C \,b^{4} c \,x^{3}+\frac {32}{3} A \,b^{3} c^{2} x^{2}+64 A \,a^{2} c^{3} x -\frac {4}{3} A \,b^{4} c x +\frac {8}{3} C a \,b^{4} x +\frac {256}{3} A a \,c^{4} x^{3}+64 A \,b^{2} c^{3} x^{3}+\frac {64}{3} C \,a^{2} c^{3} x^{3}+\frac {256}{3} A b \,c^{4} x^{4}+16 C \,b^{3} c^{2} x^{4}+\frac {32}{5} C \,b^{2} c^{3} x^{5}+\frac {128}{15} C a \,c^{4} x^{5}+\frac {80}{3} C a \,b^{3} c \,x^{2}+\frac {64}{3} C a b \,c^{3} x^{4}+32 C a \,b^{2} c^{2} x^{3}+2 C \,b^{5} x^{2}+\frac {16}{15} C \,a^{2} b^{3}+32 A a \,b^{2} c^{2} x +32 C \,a^{2} b^{2} c x}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) | \(316\) |
| default | \(A \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )+C \left (-\frac {x}{4 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {1}{5 c \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}-\frac {b \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{2 c}\right )}{8 c}+\frac {a \left (\frac {\frac {4 c x}{5}+\frac {2 b}{5}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {16 c \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{5 \left (4 a c -b^{2}\right )}\right )}{4 c}\right )\) | \(403\) |
2/15*(256*A*c^5*x^5+64*C*a*c^4*x^5+48*C*b^2*c^3*x^5+640*A*b*c^4*x^4+160*C* a*b*c^3*x^4+120*C*b^3*c^2*x^4+640*A*a*c^4*x^3+480*A*b^2*c^3*x^3+160*C*a^2* c^3*x^3+240*C*a*b^2*c^2*x^3+90*C*b^4*c*x^3+960*A*a*b*c^3*x^2+80*A*b^3*c^2* x^2+240*C*a^2*b*c^2*x^2+200*C*a*b^3*c*x^2+15*C*b^5*x^2+480*A*a^2*c^3*x+240 *A*a*b^2*c^2*x-10*A*b^4*c*x+240*C*a^2*b^2*c*x+20*C*a*b^4*x+240*A*a^2*b*c^2 -40*A*a*b^3*c+3*A*b^5+96*C*a^3*b*c+8*C*a^2*b^3)/(4*a*c-b^2)^3/(c*x^2+b*x+a )^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (155) = 310\).
Time = 5.16 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.37 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (8 \, C a^{2} b^{3} + 3 \, A b^{5} + 240 \, A a^{2} b c^{2} + 16 \, {\left (3 \, C b^{2} c^{3} + 4 \, C a c^{4} + 16 \, A c^{5}\right )} x^{5} + 40 \, {\left (3 \, C b^{3} c^{2} + 4 \, C a b c^{3} + 16 \, A b c^{4}\right )} x^{4} + 10 \, {\left (9 \, C b^{4} c + 24 \, C a b^{2} c^{2} + 64 \, A a c^{4} + 16 \, {\left (C a^{2} + 3 \, A b^{2}\right )} c^{3}\right )} x^{3} + 5 \, {\left (3 \, C b^{5} + 40 \, C a b^{3} c + 192 \, A a b c^{3} + 16 \, {\left (3 \, C a^{2} b + A b^{3}\right )} c^{2}\right )} x^{2} + 8 \, {\left (12 \, C a^{3} b - 5 \, A a b^{3}\right )} c + 10 \, {\left (2 \, C a b^{4} + 24 \, A a b^{2} c^{2} + 48 \, A a^{2} c^{3} + {\left (24 \, C a^{2} b^{2} - A b^{4}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15 \, {\left (a^{3} b^{6} - 12 \, a^{4} b^{4} c + 48 \, a^{5} b^{2} c^{2} - 64 \, a^{6} c^{3} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{6} + 3 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{5} + 3 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} x^{4} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} x^{3} + 3 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} x\right )}} \]
-2/15*(8*C*a^2*b^3 + 3*A*b^5 + 240*A*a^2*b*c^2 + 16*(3*C*b^2*c^3 + 4*C*a*c ^4 + 16*A*c^5)*x^5 + 40*(3*C*b^3*c^2 + 4*C*a*b*c^3 + 16*A*b*c^4)*x^4 + 10* (9*C*b^4*c + 24*C*a*b^2*c^2 + 64*A*a*c^4 + 16*(C*a^2 + 3*A*b^2)*c^3)*x^3 + 5*(3*C*b^5 + 40*C*a*b^3*c + 192*A*a*b*c^3 + 16*(3*C*a^2*b + A*b^3)*c^2)*x ^2 + 8*(12*C*a^3*b - 5*A*a*b^3)*c + 10*(2*C*a*b^4 + 24*A*a*b^2*c^2 + 48*A* a^2*c^3 + (24*C*a^2*b^2 - A*b^4)*c)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 - 12 *a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^ 2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2 *c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c ^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^ 4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)
Timed out. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (155) = 310\).
Time = 0.29 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.71 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (3 \, C b^{2} c^{3} + 4 \, C a c^{4} + 16 \, A c^{5}\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (3 \, C b^{3} c^{2} + 4 \, C a b c^{3} + 16 \, A b c^{4}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (9 \, C b^{4} c + 24 \, C a b^{2} c^{2} + 16 \, C a^{2} c^{3} + 48 \, A b^{2} c^{3} + 64 \, A a c^{4}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (3 \, C b^{5} + 40 \, C a b^{3} c + 48 \, C a^{2} b c^{2} + 16 \, A b^{3} c^{2} + 192 \, A a b c^{3}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {10 \, {\left (2 \, C a b^{4} + 24 \, C a^{2} b^{2} c - A b^{4} c + 24 \, A a b^{2} c^{2} + 48 \, A a^{2} c^{3}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {8 \, C a^{2} b^{3} + 3 \, A b^{5} + 96 \, C a^{3} b c - 40 \, A a b^{3} c + 240 \, A a^{2} b c^{2}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \]
-2/15*(((2*(4*(2*(3*C*b^2*c^3 + 4*C*a*c^4 + 16*A*c^5)*x/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) + 5*(3*C*b^3*c^2 + 4*C*a*b*c^3 + 16*A*b*c^4 )/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(9*C*b^4*c + 24* C*a*b^2*c^2 + 16*C*a^2*c^3 + 48*A*b^2*c^3 + 64*A*a*c^4)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(3*C*b^5 + 40*C*a*b^3*c + 48*C*a^2*b *c^2 + 16*A*b^3*c^2 + 192*A*a*b*c^3)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 10*(2*C*a*b^4 + 24*C*a^2*b^2*c - A*b^4*c + 24*A*a*b^2*c^2 + 48*A*a^2*c^3)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + (8* C*a^2*b^3 + 3*A*b^5 + 96*C*a^3*b*c - 40*A*a*b^3*c + 240*A*a^2*b*c^2)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))/(c*x^2 + b*x + a)^(5/2)
Time = 13.68 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.46 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {\frac {b\,c\,\left (56\,C\,b^2+256\,A\,c^2+32\,C\,a\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {2\,c^2\,x\,\left (56\,C\,b^2+256\,A\,c^2+32\,C\,a\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}}+\frac {\frac {8\,C\,b\,c}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,C\,c^2\,x}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{\sqrt {c\,x^2+b\,x+a}}-\frac {\frac {4\,C\,x}{15\,\left (4\,a\,c-b^2\right )}-\frac {2\,C\,b}{15\,c\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {x\,\left (\frac {4\,A\,c^2}{5\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,C\,b^2}{5\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {4\,C\,a\,c}{5\,\left (4\,a\,c^2-b^2\,c\right )}\right )+\frac {2\,A\,b\,c}{5\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,C\,a\,b}{5\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {x\,\left (\frac {2\,c\,\left (8\,C\,b^2+32\,A\,c^2+8\,C\,a\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,C\,a\,c^2}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,C\,b^2\,c}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,\left (8\,C\,b^2+32\,A\,c^2+8\,C\,a\,c\right )}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,C\,a\,b\,c}{15\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
((b*c*(256*A*c^2 + 56*C*b^2 + 32*C*a*c))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^ 2)^2) + (2*c^2*x*(256*A*c^2 + 56*C*b^2 + 32*C*a*c))/(15*(4*a*c^2 - b^2*c)* (4*a*c - b^2)^2))/(a + b*x + c*x^2)^(1/2) + ((8*C*b*c)/(15*(4*a*c^2 - b^2* c)*(4*a*c - b^2)) + (16*C*c^2*x)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) - ((4*C*x)/(15*(4*a*c - b^2)) - (2*C*b)/(15*c*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((4*A*c^2)/(5*(4*a*c^2 - b^2*c)) + ( 2*C*b^2)/(5*(4*a*c^2 - b^2*c)) - (4*C*a*c)/(5*(4*a*c^2 - b^2*c))) + (2*A*b *c)/(5*(4*a*c^2 - b^2*c)) + (2*C*a*b)/(5*(4*a*c^2 - b^2*c)))/(a + b*x + c* x^2)^(5/2) + (x*((2*c*(32*A*c^2 + 8*C*b^2 + 8*C*a*c))/(15*(4*a*c^2 - b^2*c )*(4*a*c - b^2)) + (16*C*a*c^2)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8* C*b^2*c)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (b*(32*A*c^2 + 8*C*b^2 + 8*C*a*c))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*C*a*b*c)/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2)