Integrand size = 22, antiderivative size = 220 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \left (b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{7 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}+\frac {2 \left (24 A c+4 a C+\frac {5 b^2 C}{c}\right ) (b+2 c x)}{35 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}}-\frac {32 \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x)}{105 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^{3/2}}+\frac {256 c \left (24 A c^2+5 b^2 C+4 a c C\right ) (b+2 c x)}{105 \left (b^2-4 a c\right )^4 \sqrt {a+b x+c x^2}} \]
-2/7*(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)^(7/2)+2/35*(24*A*c+4*C*a+5*b^2*C/c)*(2*c*x+b)/(-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)
Time = 5.87 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.86 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\frac {-6 A (b+2 c x) \left (5 b^6-24 b^5 c x+64 b^3 c^2 x \left (7 a-12 c x^2\right )+4 b^4 c \left (-21 a+26 c x^2\right )-128 b c^3 x \left (35 a^2+56 a c x^2+24 c^2 x^4\right )-16 b^2 c^2 \left (-35 a^2+196 a c x^2+184 c^2 x^4\right )-64 c^3 \left (35 a^3+70 a^2 c x^2+56 a c^2 x^4+16 c^3 x^6\right )\right )+2 C \left (1920 a^4 b c^2+320 a^3 c \left (b^3+21 b^2 c x+21 b c^2 x^2+14 c^3 x^3\right )+5 b^2 x^2 \left (-7 b^5+70 b^4 c x+560 b^3 c^2 x^2+1120 b^2 c^3 x^3+896 b c^4 x^4+256 c^5 x^5\right )+8 a^2 \left (-b^5+140 b^4 c x+1190 b^3 c^2 x^2+1540 b^2 c^3 x^3+1120 b c^4 x^4+448 c^5 x^5\right )+4 a x \left (-7 b^6+343 b^5 c x+2170 b^4 c^2 x^2+3360 b^3 c^3 x^3+2240 b^2 c^4 x^4+896 b c^5 x^5+256 c^6 x^6\right )\right )}{105 \left (b^2-4 a c\right )^4 (a+x (b+c x))^{7/2}} \]
(-6*A*(b + 2*c*x)*(5*b^6 - 24*b^5*c*x + 64*b^3*c^2*x*(7*a - 12*c*x^2) + 4* b^4*c*(-21*a + 26*c*x^2) - 128*b*c^3*x*(35*a^2 + 56*a*c*x^2 + 24*c^2*x^4) - 16*b^2*c^2*(-35*a^2 + 196*a*c*x^2 + 184*c^2*x^4) - 64*c^3*(35*a^3 + 70*a ^2*c*x^2 + 56*a*c^2*x^4 + 16*c^3*x^6)) + 2*C*(1920*a^4*b*c^2 + 320*a^3*c*( b^3 + 21*b^2*c*x + 21*b*c^2*x^2 + 14*c^3*x^3) + 5*b^2*x^2*(-7*b^5 + 70*b^4 *c*x + 560*b^3*c^2*x^2 + 1120*b^2*c^3*x^3 + 896*b*c^4*x^4 + 256*c^5*x^5) + 8*a^2*(-b^5 + 140*b^4*c*x + 1190*b^3*c^2*x^2 + 1540*b^2*c^3*x^3 + 1120*b* c^4*x^4 + 448*c^5*x^5) + 4*a*x*(-7*b^6 + 343*b^5*c*x + 2170*b^4*c^2*x^2 + 3360*b^3*c^3*x^3 + 2240*b^2*c^4*x^4 + 896*b*c^5*x^5 + 256*c^6*x^6)))/(105* (b^2 - 4*a*c)^4*(a + x*(b + c*x))^(7/2))
Time = 0.34 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.98, 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 |
| \(\displaystyle -\frac {2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )\right )}{7 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}}-\frac {\left (-\frac {2 (b+2 c x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {16 c \left (\frac {16 c (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (b+2 c x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}\right ) \left (4 a C+24 A c+\frac {5 b^2 C}{c}\right )}{7 \left (b^2-4 a c\right )}\) |
(-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))
3.2.85.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(204)=408\).
Time = 0.76 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.38
| method | result | size |
| trager | \(\frac {-\frac {2}{7} A \,b^{7}+\frac {128}{21} C \,a^{3} b^{3} c +\frac {256}{7} C \,a^{4} b \,c^{2}+\frac {4}{5} A \,b^{6} c x -\frac {8}{15} C a \,b^{6} x +256 A \,a^{3} c^{4} x +\frac {160}{3} C \,b^{5} c^{2} x^{4}+512 A \,a^{2} c^{5} x^{3}+32 A \,b^{4} c^{3} x^{3}+\frac {256}{3} C \,a^{3} c^{4} x^{3}+\frac {20}{3} C \,b^{6} c \,x^{3}-\frac {16}{5} A \,b^{5} c^{2} x^{2}+256 A \,b^{3} c^{4} x^{4}+\frac {1024}{15} C \,a^{2} c^{5} x^{5}+\frac {320}{3} C \,b^{4} c^{3} x^{5}+\frac {2048}{5} A a \,c^{6} x^{5}+512 A \,b^{2} c^{5} x^{5}+\frac {256}{3} C \,b^{3} c^{4} x^{6}+\frac {2048}{105} C a \,c^{6} x^{7}+\frac {512}{21} C \,b^{2} c^{5} x^{7}+\frac {2048}{5} A b \,c^{6} x^{6}+128 A \,a^{3} b \,c^{3}-32 A \,a^{2} b^{3} c^{2}+\frac {24}{5} A a \,b^{5} c -\frac {2}{3} C \,b^{7} x^{2}-\frac {16}{105} C \,a^{2} b^{5}+\frac {512}{3} C a \,b^{2} c^{4} x^{5}+\frac {4096}{35} A \,c^{7} x^{7}+128 C \,a^{3} b \,c^{3} x^{2}+\frac {1024}{15} C a b \,c^{5} x^{6}+\frac {512}{3} C \,a^{2} b \,c^{4} x^{4}+\frac {64}{3} C \,a^{2} b^{4} c x +\frac {496}{3} C a \,b^{4} c^{2} x^{3}+\frac {392}{15} C a \,b^{5} c \,x^{2}+192 A \,a^{2} b^{2} c^{3} x +\frac {544}{3} C \,a^{2} b^{3} c^{2} x^{2}+128 C \,a^{3} b^{2} c^{2} x +768 A a \,b^{2} c^{4} x^{3}+\frac {704}{3} C \,a^{2} b^{2} c^{3} x^{3}+768 A \,a^{2} b \,c^{4} x^{2}+128 A a \,b^{3} c^{3} x^{2}-16 A a \,b^{4} c^{2} x +1024 A a b \,c^{5} x^{4}+256 C a \,b^{3} c^{3} x^{4}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}\) | \(524\) |
| 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 | \(\frac {-\frac {2}{7} A \,b^{7}+\frac {128}{21} C \,a^{3} b^{3} c +\frac {256}{7} C \,a^{4} b \,c^{2}+\frac {4}{5} A \,b^{6} c x -\frac {8}{15} C a \,b^{6} x +256 A \,a^{3} c^{4} x +\frac {160}{3} C \,b^{5} c^{2} x^{4}+512 A \,a^{2} c^{5} x^{3}+32 A \,b^{4} c^{3} x^{3}+\frac {256}{3} C \,a^{3} c^{4} x^{3}+\frac {20}{3} C \,b^{6} c \,x^{3}-\frac {16}{5} A \,b^{5} c^{2} x^{2}+256 A \,b^{3} c^{4} x^{4}+\frac {1024}{15} C \,a^{2} c^{5} x^{5}+\frac {320}{3} C \,b^{4} c^{3} x^{5}+\frac {2048}{5} A a \,c^{6} x^{5}+512 A \,b^{2} c^{5} x^{5}+\frac {256}{3} C \,b^{3} c^{4} x^{6}+\frac {2048}{105} C a \,c^{6} x^{7}+\frac {512}{21} C \,b^{2} c^{5} x^{7}+\frac {2048}{5} A b \,c^{6} x^{6}+128 A \,a^{3} b \,c^{3}-32 A \,a^{2} b^{3} c^{2}+\frac {24}{5} A a \,b^{5} c -\frac {2}{3} C \,b^{7} x^{2}-\frac {16}{105} C \,a^{2} b^{5}+\frac {512}{3} C a \,b^{2} c^{4} x^{5}+\frac {4096}{35} A \,c^{7} x^{7}+128 C \,a^{3} b \,c^{3} x^{2}+\frac {1024}{15} C a b \,c^{5} x^{6}+\frac {512}{3} C \,a^{2} b \,c^{4} x^{4}+\frac {64}{3} C \,a^{2} b^{4} c x +\frac {496}{3} C a \,b^{4} c^{2} x^{3}+\frac {392}{15} C a \,b^{5} c \,x^{2}+192 A \,a^{2} b^{2} c^{3} x +\frac {544}{3} C \,a^{2} b^{3} c^{2} x^{2}+128 C \,a^{3} b^{2} c^{2} x +768 A a \,b^{2} c^{4} x^{3}+\frac {704}{3} C \,a^{2} b^{2} c^{3} x^{3}+768 A \,a^{2} b \,c^{4} x^{2}+128 A a \,b^{3} c^{3} x^{2}-16 A a \,b^{4} c^{2} x +1024 A a b \,c^{5} x^{4}+256 C a \,b^{3} c^{3} x^{4}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}\) | \(555\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (204) = 408\).
Time = 14.68 (sec) , antiderivative size = 978, normalized size of antiderivative = 4.45 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (8 \, C a^{2} b^{5} + 15 \, A b^{7} - 6720 \, A a^{3} b c^{3} - 256 \, {\left (5 \, C b^{2} c^{5} + 4 \, C a c^{6} + 24 \, A c^{7}\right )} x^{7} - 896 \, {\left (5 \, C b^{3} c^{4} + 4 \, C a b c^{5} + 24 \, A b c^{6}\right )} x^{6} - 224 \, {\left (25 \, C b^{4} c^{3} + 40 \, C a b^{2} c^{4} + 96 \, A a c^{6} + 8 \, {\left (2 \, C a^{2} + 15 \, A b^{2}\right )} c^{5}\right )} x^{5} - 560 \, {\left (5 \, C b^{5} c^{2} + 24 \, C a b^{3} c^{3} + 96 \, A a b c^{5} + 8 \, {\left (2 \, C a^{2} b + 3 \, A b^{3}\right )} c^{4}\right )} x^{4} - 70 \, {\left (5 \, C b^{6} c + 124 \, C a b^{4} c^{2} + 384 \, A a^{2} c^{5} + 64 \, {\left (C a^{3} + 9 \, A a b^{2}\right )} c^{4} + 8 \, {\left (22 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} c^{3}\right )} x^{3} - 240 \, {\left (8 \, C a^{4} b - 7 \, A a^{2} b^{3}\right )} c^{2} + 7 \, {\left (5 \, C b^{7} - 196 \, C a b^{5} c - 5760 \, A a^{2} b c^{4} - 960 \, {\left (C a^{3} b + A a b^{3}\right )} c^{3} - 8 \, {\left (170 \, C a^{2} b^{3} - 3 \, A b^{5}\right )} c^{2}\right )} x^{2} - 4 \, {\left (80 \, C a^{3} b^{3} + 63 \, A a b^{5}\right )} c + 14 \, {\left (2 \, C a b^{6} - 720 \, A a^{2} b^{2} c^{3} - 960 \, A a^{3} c^{4} - 60 \, {\left (8 \, C a^{3} b^{2} - A a b^{4}\right )} c^{2} - {\left (80 \, C a^{2} b^{4} + 3 \, A b^{6}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{105 \, {\left (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} + {\left (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}\right )} x^{8} + 4 \, {\left (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}\right )} x^{7} + 2 \, {\left (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}\right )} x^{6} + 4 \, {\left (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}\right )} x^{5} + {\left (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}\right )} x^{4} + 4 \, {\left (a b^{11} - 13 \, 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}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{10} - 46 \, a^{3} b^{8} c + 256 \, a^{4} b^{6} c^{2} - 576 \, a^{5} b^{4} c^{3} + 256 \, a^{6} b^{2} c^{4} + 512 \, a^{7} c^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{9} - 16 \, a^{4} b^{7} c + 96 \, a^{5} b^{5} c^{2} - 256 \, a^{6} b^{3} c^{3} + 256 \, a^{7} b c^{4}\right )} x\right )}} \]
-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...
Timed out. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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. 805 vs. \(2 (204) = 408\).
Time = 0.29 (sec) , antiderivative size = 805, normalized size of antiderivative = 3.66 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\frac {2 \, {\left ({\left ({\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, {\left (5 \, C b^{2} c^{5} + 4 \, C a c^{6} + 24 \, A c^{7}\right )} 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}} + \frac {7 \, {\left (5 \, C b^{3} c^{4} + 4 \, C a b c^{5} + 24 \, A b c^{6}\right )}}{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}}\right )} x + \frac {7 \, {\left (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}\right )}}{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}}\right )} x + \frac {35 \, {\left (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}\right )}}{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}}\right )} x + \frac {35 \, {\left (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}\right )}}{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}}\right )} x - \frac {7 \, {\left (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}\right )}}{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}}\right )} x - \frac {14 \, {\left (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}\right )}}{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}}\right )} x - \frac {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}}\right )}}{105 \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}}} \]
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)
Time = 14.48 (sec) , antiderivative size = 1018, normalized size of antiderivative = 4.63 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^{9/2}} \, dx=\frac {x\,\left (\frac {2\,c^2\,\left (160\,C\,b^2+768\,A\,c^2+96\,C\,a\,c\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}-\frac {64\,C\,a\,c^3}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {32\,C\,b^2\,c^2}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}\right )+\frac {b\,c\,\left (160\,C\,b^2+768\,A\,c^2+96\,C\,a\,c\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {32\,C\,a\,b\,c^2}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}-\frac {\frac {8\,C\,b}{105\,{\left (4\,a\,c-b^2\right )}^2}-\frac {16\,C\,c\,x}{105\,{\left (4\,a\,c-b^2\right )}^2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}}+\frac {\frac {8\,C\,b\,c}{105\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,C\,c^2\,x}{105\,\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}}-\frac {\frac {4\,C\,x}{35\,\left (4\,a\,c-b^2\right )}-\frac {2\,C\,b}{35\,c\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}+\frac {\frac {b\,c\,\left (1312\,C\,b^2\,c+6144\,A\,c^3+896\,C\,a\,c^2\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}+\frac {2\,c^2\,x\,\left (1312\,C\,b^2\,c+6144\,A\,c^3+896\,C\,a\,c^2\right )}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^3}}{\sqrt {c\,x^2+b\,x+a}}+\frac {x\,\left (\frac {4\,A\,c^2}{7\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,C\,b^2}{7\,\left (4\,a\,c^2-b^2\,c\right )}-\frac {4\,C\,a\,c}{7\,\left (4\,a\,c^2-b^2\,c\right )}\right )+\frac {2\,A\,b\,c}{7\,\left (4\,a\,c^2-b^2\,c\right )}+\frac {2\,C\,a\,b}{7\,\left (4\,a\,c^2-b^2\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{7/2}}+\frac {x\,\left (\frac {2\,c\,\left (12\,C\,b^2+48\,A\,c^2+8\,C\,a\,c\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}+\frac {16\,C\,a\,c^2}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,C\,b^2\,c}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}\right )+\frac {b\,\left (12\,C\,b^2+48\,A\,c^2+8\,C\,a\,c\right )}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}-\frac {8\,C\,a\,b\,c}{35\,\left (4\,a\,c^2-b^2\,c\right )\,\left (4\,a\,c-b^2\right )}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}}-\frac {\frac {32\,C\,b\,c^2}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {64\,C\,c^3\,x}{105\,\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 {64\,C\,b\,c^2}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}+\frac {128\,C\,c^3\,x}{105\,\left (4\,a\,c^2-b^2\,c\right )\,{\left (4\,a\,c-b^2\right )}^2}}{\sqrt {c\,x^2+b\,x+a}} \]
(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) + ...