Integrand size = 26, antiderivative size = 118 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 \left (b^2-4 a c\right ) d^{12} (b+2 c x)^{11}}+\frac {8 \left (a+b x+c x^2\right )^{7/2}}{99 \left (b^2-4 a c\right )^2 d^{12} (b+2 c x)^9}+\frac {16 \left (a+b x+c x^2\right )^{7/2}}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^7} \]
2/11*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^12/(2*c*x+b)^11+8/99*(c*x^2+b*x+a) ^(7/2)/(-4*a*c+b^2)^2/d^12/(2*c*x+b)^9+16/693*(c*x^2+b*x+a)^(7/2)/(-4*a*c+ b^2)^3/d^12/(2*c*x+b)^7
Time = 10.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {2 (a+x (b+c x))^{7/2} \left (99 b^4+176 b^3 c x+64 b c^2 x \left (-7 a+4 c x^2\right )+8 b^2 c \left (-77 a+38 c x^2\right )+16 c^2 \left (63 a^2-28 a c x^2+8 c^2 x^4\right )\right )}{693 \left (b^2-4 a c\right )^3 d^{12} (b+2 c x)^{11}} \]
(2*(a + x*(b + c*x))^(7/2)*(99*b^4 + 176*b^3*c*x + 64*b*c^2*x*(-7*a + 4*c* x^2) + 8*b^2*c*(-77*a + 38*c*x^2) + 16*c^2*(63*a^2 - 28*a*c*x^2 + 8*c^2*x^ 4)))/(693*(b^2 - 4*a*c)^3*d^12*(b + 2*c*x)^11)
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1117, 27, 1117, 1106}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{d^{10} (b+2 c x)^{10}}dx}{11 d^2 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{(b+2 c x)^{10}}dx}{11 d^{12} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {4 \left (\frac {2 \int \frac {\left (c x^2+b x+a\right )^{5/2}}{(b+2 c x)^8}dx}{9 \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) (b+2 c x)^9}\right )}{11 d^{12} \left (b^2-4 a c\right )}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}\) |
| \(\Big \downarrow \) 1106 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{7/2}}{11 d^{12} \left (b^2-4 a c\right ) (b+2 c x)^{11}}+\frac {4 \left (\frac {4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac {2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) (b+2 c x)^9}\right )}{11 d^{12} \left (b^2-4 a c\right )}\) |
(2*(a + b*x + c*x^2)^(7/2))/(11*(b^2 - 4*a*c)*d^12*(b + 2*c*x)^11) + (4*(( 2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*(b + 2*c*x)^9) + (4*(a + b*x + c*x^2)^(7/2))/(63*(b^2 - 4*a*c)^2*(b + 2*c*x)^7)))/(11*(b^2 - 4*a*c)*d^12 )
3.13.33.3.1 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[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[-2*b*d*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(d^2*(m + 1)*(b^2 - 4*a*c))), x] + Simp[b^2*((m + 2*p + 3)/(d^2*(m + 1)*(b^2 - 4*a* c))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & & (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) /2])
Time = 67.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13
| method | result | size |
| gosper | \(-\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}-448 x^{2} c^{3} a +304 b^{2} c^{2} x^{2}-448 a b \,c^{2} x +176 b^{3} c x +1008 a^{2} c^{2}-616 a \,b^{2} c +99 b^{4}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{693 \left (2 c x +b \right )^{11} d^{12} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}\) | \(133\) |
| default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{11 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{11}}-\frac {16 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{9 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{9}}+\frac {32 c^{3} \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{63 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{7}}\right )}{11 \left (4 a c -b^{2}\right )}}{4096 d^{12} c^{12}}\) | \(195\) |
| trager | \(-\frac {2 \left (128 c^{7} x^{10}+640 b \,c^{6} x^{9}-64 a \,c^{6} x^{8}+1456 b^{2} c^{5} x^{8}-256 a b \,c^{5} x^{7}+1984 c^{4} b^{3} x^{7}+48 a^{2} c^{5} x^{6}-472 a \,b^{2} c^{4} x^{6}+1795 b^{4} c^{3} x^{6}+144 a^{2} b \,c^{4} x^{5}-520 a \,b^{3} c^{3} x^{5}+1129 b^{5} c^{2} x^{5}+1808 a^{3} c^{4} x^{4}-1176 a^{2} b^{2} c^{3} x^{4}-31 c^{2} x^{4} a \,b^{4}+473 b^{6} c \,x^{4}+3616 a^{3} c^{3} b \,x^{3}-2592 a^{2} c^{2} b^{3} x^{3}+506 a \,b^{5} c \,x^{3}+99 b^{7} x^{3}+2576 a^{4} c^{3} x^{2}+136 a^{3} b^{2} c^{2} x^{2}-1023 a^{2} b^{4} c \,x^{2}+297 a \,b^{6} x^{2}+2576 a^{4} b \,c^{2} x -1672 a^{3} b^{3} c x +297 a^{2} x \,b^{5}+1008 a^{5} c^{2}-616 a^{4} b^{2} c +99 a^{3} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{693 d^{12} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (2 c x +b \right )^{11}}\) | \(378\) |
-2/693*(128*c^4*x^4+256*b*c^3*x^3-448*a*c^3*x^2+304*b^2*c^2*x^2-448*a*b*c^ 2*x+176*b^3*c*x+1008*a^2*c^2-616*a*b^2*c+99*b^4)*(c*x^2+b*x+a)^(7/2)/(2*c* x+b)^11/d^12/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)
Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (106) = 212\).
Time = 50.42 (sec) , antiderivative size = 851, normalized size of antiderivative = 7.21 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {2 \, {\left (128 \, c^{7} x^{10} + 640 \, b c^{6} x^{9} + 16 \, {\left (91 \, b^{2} c^{5} - 4 \, a c^{6}\right )} x^{8} + 64 \, {\left (31 \, b^{3} c^{4} - 4 \, a b c^{5}\right )} x^{7} + 99 \, a^{3} b^{4} - 616 \, a^{4} b^{2} c + 1008 \, a^{5} c^{2} + {\left (1795 \, b^{4} c^{3} - 472 \, a b^{2} c^{4} + 48 \, a^{2} c^{5}\right )} x^{6} + {\left (1129 \, b^{5} c^{2} - 520 \, a b^{3} c^{3} + 144 \, a^{2} b c^{4}\right )} x^{5} + {\left (473 \, b^{6} c - 31 \, a b^{4} c^{2} - 1176 \, a^{2} b^{2} c^{3} + 1808 \, a^{3} c^{4}\right )} x^{4} + {\left (99 \, b^{7} + 506 \, a b^{5} c - 2592 \, a^{2} b^{3} c^{2} + 3616 \, a^{3} b c^{3}\right )} x^{3} + {\left (297 \, a b^{6} - 1023 \, a^{2} b^{4} c + 136 \, a^{3} b^{2} c^{2} + 2576 \, a^{4} c^{3}\right )} x^{2} + {\left (297 \, a^{2} b^{5} - 1672 \, a^{3} b^{3} c + 2576 \, a^{4} b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{693 \, {\left (2048 \, {\left (b^{6} c^{11} - 12 \, a b^{4} c^{12} + 48 \, a^{2} b^{2} c^{13} - 64 \, a^{3} c^{14}\right )} d^{12} x^{11} + 11264 \, {\left (b^{7} c^{10} - 12 \, a b^{5} c^{11} + 48 \, a^{2} b^{3} c^{12} - 64 \, a^{3} b c^{13}\right )} d^{12} x^{10} + 28160 \, {\left (b^{8} c^{9} - 12 \, a b^{6} c^{10} + 48 \, a^{2} b^{4} c^{11} - 64 \, a^{3} b^{2} c^{12}\right )} d^{12} x^{9} + 42240 \, {\left (b^{9} c^{8} - 12 \, a b^{7} c^{9} + 48 \, a^{2} b^{5} c^{10} - 64 \, a^{3} b^{3} c^{11}\right )} d^{12} x^{8} + 42240 \, {\left (b^{10} c^{7} - 12 \, a b^{8} c^{8} + 48 \, a^{2} b^{6} c^{9} - 64 \, a^{3} b^{4} c^{10}\right )} d^{12} x^{7} + 29568 \, {\left (b^{11} c^{6} - 12 \, a b^{9} c^{7} + 48 \, a^{2} b^{7} c^{8} - 64 \, a^{3} b^{5} c^{9}\right )} d^{12} x^{6} + 14784 \, {\left (b^{12} c^{5} - 12 \, a b^{10} c^{6} + 48 \, a^{2} b^{8} c^{7} - 64 \, a^{3} b^{6} c^{8}\right )} d^{12} x^{5} + 5280 \, {\left (b^{13} c^{4} - 12 \, a b^{11} c^{5} + 48 \, a^{2} b^{9} c^{6} - 64 \, a^{3} b^{7} c^{7}\right )} d^{12} x^{4} + 1320 \, {\left (b^{14} c^{3} - 12 \, a b^{12} c^{4} + 48 \, a^{2} b^{10} c^{5} - 64 \, a^{3} b^{8} c^{6}\right )} d^{12} x^{3} + 220 \, {\left (b^{15} c^{2} - 12 \, a b^{13} c^{3} + 48 \, a^{2} b^{11} c^{4} - 64 \, a^{3} b^{9} c^{5}\right )} d^{12} x^{2} + 22 \, {\left (b^{16} c - 12 \, a b^{14} c^{2} + 48 \, a^{2} b^{12} c^{3} - 64 \, a^{3} b^{10} c^{4}\right )} d^{12} x + {\left (b^{17} - 12 \, a b^{15} c + 48 \, a^{2} b^{13} c^{2} - 64 \, a^{3} b^{11} c^{3}\right )} d^{12}\right )}} \]
2/693*(128*c^7*x^10 + 640*b*c^6*x^9 + 16*(91*b^2*c^5 - 4*a*c^6)*x^8 + 64*( 31*b^3*c^4 - 4*a*b*c^5)*x^7 + 99*a^3*b^4 - 616*a^4*b^2*c + 1008*a^5*c^2 + (1795*b^4*c^3 - 472*a*b^2*c^4 + 48*a^2*c^5)*x^6 + (1129*b^5*c^2 - 520*a*b^ 3*c^3 + 144*a^2*b*c^4)*x^5 + (473*b^6*c - 31*a*b^4*c^2 - 1176*a^2*b^2*c^3 + 1808*a^3*c^4)*x^4 + (99*b^7 + 506*a*b^5*c - 2592*a^2*b^3*c^2 + 3616*a^3* b*c^3)*x^3 + (297*a*b^6 - 1023*a^2*b^4*c + 136*a^3*b^2*c^2 + 2576*a^4*c^3) *x^2 + (297*a^2*b^5 - 1672*a^3*b^3*c + 2576*a^4*b*c^2)*x)*sqrt(c*x^2 + b*x + a)/(2048*(b^6*c^11 - 12*a*b^4*c^12 + 48*a^2*b^2*c^13 - 64*a^3*c^14)*d^1 2*x^11 + 11264*(b^7*c^10 - 12*a*b^5*c^11 + 48*a^2*b^3*c^12 - 64*a^3*b*c^13 )*d^12*x^10 + 28160*(b^8*c^9 - 12*a*b^6*c^10 + 48*a^2*b^4*c^11 - 64*a^3*b^ 2*c^12)*d^12*x^9 + 42240*(b^9*c^8 - 12*a*b^7*c^9 + 48*a^2*b^5*c^10 - 64*a^ 3*b^3*c^11)*d^12*x^8 + 42240*(b^10*c^7 - 12*a*b^8*c^8 + 48*a^2*b^6*c^9 - 6 4*a^3*b^4*c^10)*d^12*x^7 + 29568*(b^11*c^6 - 12*a*b^9*c^7 + 48*a^2*b^7*c^8 - 64*a^3*b^5*c^9)*d^12*x^6 + 14784*(b^12*c^5 - 12*a*b^10*c^6 + 48*a^2*b^8 *c^7 - 64*a^3*b^6*c^8)*d^12*x^5 + 5280*(b^13*c^4 - 12*a*b^11*c^5 + 48*a^2* b^9*c^6 - 64*a^3*b^7*c^7)*d^12*x^4 + 1320*(b^14*c^3 - 12*a*b^12*c^4 + 48*a ^2*b^10*c^5 - 64*a^3*b^8*c^6)*d^12*x^3 + 220*(b^15*c^2 - 12*a*b^13*c^3 + 4 8*a^2*b^11*c^4 - 64*a^3*b^9*c^5)*d^12*x^2 + 22*(b^16*c - 12*a*b^14*c^2 + 4 8*a^2*b^12*c^3 - 64*a^3*b^10*c^4)*d^12*x + (b^17 - 12*a*b^15*c + 48*a^2*b^ 13*c^2 - 64*a^3*b^11*c^3)*d^12)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{12} + 24 b^{11} c x + 264 b^{10} c^{2} x^{2} + 1760 b^{9} c^{3} x^{3} + 7920 b^{8} c^{4} x^{4} + 25344 b^{7} c^{5} x^{5} + 59136 b^{6} c^{6} x^{6} + 101376 b^{5} c^{7} x^{7} + 126720 b^{4} c^{8} x^{8} + 112640 b^{3} c^{9} x^{9} + 67584 b^{2} c^{10} x^{10} + 24576 b c^{11} x^{11} + 4096 c^{12} x^{12}}\, dx}{d^{12}} \]
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c* *2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 409 6*c**12*x**12), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24 *b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x* *4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3 *x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 112640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096*c**12*x**12), x) + Int egral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b**12 + 24*b**11*c*x + 264*b**10*c** 2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x**4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**7 + 126720*b**4*c**8*x**8 + 1 12640*b**3*c**9*x**9 + 67584*b**2*c**10*x**10 + 24576*b*c**11*x**11 + 4096 *c**12*x**12), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**12 + 24 *b**11*c*x + 264*b**10*c**2*x**2 + 1760*b**9*c**3*x**3 + 7920*b**8*c**4*x* *4 + 25344*b**7*c**5*x**5 + 59136*b**6*c**6*x**6 + 101376*b**5*c**7*x**...
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Exception raised: ValueError} \]
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. 2365 vs. \(2 (106) = 212\).
Time = 1.37 (sec) , antiderivative size = 2365, normalized size of antiderivative = 20.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Too large to display} \]
1/5544*(29568*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*c^8 + 236544*(sqrt(c) *x - sqrt(c*x^2 + b*x + a))^15*b*c^(15/2) + 868560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b^2*c^7 + 73920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*a*c ^8 + 1940400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^3*c^(13/2) + 517440* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*a*b*c^(15/2) + 2953104*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^4*c^6 + 1600368*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*b^2*c^7 + 162624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*c^ 8 + 3256176*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^5*c^(11/2) + 2875488* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*b^3*c^(13/2) + 975744*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*b*c^(15/2) + 2709168*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^10*b^6*c^5 + 3307920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10 *a*b^4*c^6 + 2583504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*b^2*c^7 + 133056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^3*c^8 + 1755600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^7*c^(9/2) + 2513280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^5*c^(11/2) + 3973200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) ^9*a^2*b^3*c^(13/2) + 665280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b*c ^(15/2) + 910800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^8*c^4 + 1227600*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^6*c^5 + 3944160*(sqrt(c)*x - sqrt (c*x^2 + b*x + a))^8*a^2*b^4*c^6 + 1401840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*b^2*c^7 + 95040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^...
Time = 25.57 (sec) , antiderivative size = 9995, normalized size of antiderivative = 84.70 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{12}} \, dx=\text {Too large to display} \]
(((b*((b*((b*((b*((b*((4*c^4*(40*a*c - 3*b^2))/(231*d^12*(4*a*c - b^2)^3*( 96*a*c^3 - 24*b^2*c^2)) - (8*b^2*c^4)/(231*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c^3*(120*a*c - 23*b^2))/(693*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (5168*a^2*c^4 + 58*b^4*c^2 - 138 4*a*b^2*c^3)/(693*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (b*c*(115*b^4 + 2584*a^2*c^2 - 1092*a*b^2*c))/(231*d^12*(4*a*c - b^2)^3*(9 6*a*c^3 - 24*b^2*c^2))))/(2*c) + (133*b^6 + 4704*a^3*c^3 + 348*a^2*b^2*c^2 - 906*a*b^4*c)/(693*d^12*(4*a*c - b^2)^3*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (133*a*b^5 - 1118*a^2*b^3*c + 2352*a^3*b*c^2)/(693*d^12*(4*a*c - b^2)^3 *(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*( (b*(b^2/(6160*c*d^12*(4*a*c - b^2)^5) - (32*a*c^3 - 5*b^2*c^2)/(9240*c^3*d ^12*(4*a*c - b^2)^5)))/(2*c) - (7*b^3*c - 32*a*b*c^2)/(9240*c^3*d^12*(4*a* c - b^2)^5)))/(2*c) - (6*b^4 + 126*a^2*c^2 - 55*a*b^2*c)/(9240*c^3*d^12*(4 *a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*(b^2/(5280 *c*d^12*(4*a*c - b^2)^5) - (76*a*c^3 - 13*b^2*c^2)/(15840*c^3*d^12*(4*a*c - b^2)^5)))/(2*c) - (17*b^3*c - 76*a*b*c^2)/(15840*c^3*d^12*(4*a*c - b^2)^ 5)))/(2*c) + (23*b^4 + 296*a^2*c^2 - 167*a*b^2*c)/(15840*c^3*d^12*(4*a*c - b^2)^5))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((5*a*c - b^2)/( 1155*c*d^12*(4*a*c - b^2)^5) - b^2/(9240*c*d^12*(4*a*c - b^2)^5)))/(2*c) - (b*(30*a*c - 7*b^2))/(6930*c^2*d^12*(4*a*c - b^2)^5)))/(2*c) + (147*b^...