Integrand size = 26, antiderivative size = 239 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{1024 c^3 d^{11} (b+2 c x)^6}+\frac {\sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^{11} (b+2 c x)^4}+\frac {3 \sqrt {a+b x+c x^2}}{8192 c^3 \left (b^2-4 a c\right )^2 d^{11} (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{128 c^2 d^{11} (b+2 c x)^8}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}+\frac {3 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16384 c^{7/2} \left (b^2-4 a c\right )^{5/2} d^{11}} \]
-1/128*(c*x^2+b*x+a)^(3/2)/c^2/d^11/(2*c*x+b)^8-1/20*(c*x^2+b*x+a)^(5/2)/c /d^11/(2*c*x+b)^10+3/16384*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^ 2)^(1/2))/c^(7/2)/(-4*a*c+b^2)^(5/2)/d^11-1/1024*(c*x^2+b*x+a)^(1/2)/c^3/d ^11/(2*c*x+b)^6+1/4096*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)/d^11/(2*c*x+b) ^4+3/8192*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)^2/d^11/(2*c*x+b)^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.26 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\frac {2 (a+x (b+c x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},6,\frac {9}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{7 \left (b^2-4 a c\right )^6 d^{11}} \]
(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^6*d^11)
Time = 0.44 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1108, 27, 1108, 1108, 1117, 1117, 1112, 218}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^9 (b+2 c x)^9}dx}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^9}dx}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {\frac {3 \int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^7}dx}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {1}{(b+2 c x)^5 \sqrt {c x^2+b x+a}}dx}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {3 \int \frac {1}{(b+2 c x)^3 \sqrt {c x^2+b x+a}}dx}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {3 \left (\frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 1112 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {3 \left (\frac {2 c \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{b^2-4 a c}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {3 \left (\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}\right )}{4 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) (b+2 c x)^4}}{24 c}-\frac {\sqrt {a+b x+c x^2}}{12 c (b+2 c x)^6}\right )}{32 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{16 c (b+2 c x)^8}}{8 c d^{11}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{20 c d^{11} (b+2 c x)^{10}}\) |
-1/20*(a + b*x + c*x^2)^(5/2)/(c*d^11*(b + 2*c*x)^10) + (-1/16*(a + b*x + c*x^2)^(3/2)/(c*(b + 2*c*x)^8) + (3*(-1/12*Sqrt[a + b*x + c*x^2]/(c*(b + 2 *c*x)^6) + (Sqrt[a + b*x + c*x^2]/(2*(b^2 - 4*a*c)*(b + 2*c*x)^4) + (3*(Sq rt[a + b*x + c*x^2]/((b^2 - 4*a*c)*(b + 2*c*x)^2) + ArcTan[(2*Sqrt[c]*Sqrt [a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[c]*(b^2 - 4*a*c)^(3/2))))/(4 *(b^2 - 4*a*c)))/(24*c)))/(32*c))/(8*c*d^11)
3.13.32.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[b*(p/(d*e*(m + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0] ) && IntegerQ[2*p]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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 = 38.27 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.36
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {15 \left (2 c x +b \right )^{10} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{65536}+\sqrt {4 c^{2} a -b^{2} c}\, \left (-\frac {15 c^{8} x^{8}}{128}+\frac {5 x^{6} \left (-6 b x +a \right ) c^{7}}{64}+\left (\frac {15}{64} a b \,x^{5}-\frac {215}{256} b^{2} x^{6}+\frac {31}{16} a^{2} x^{4}\right ) c^{6}+\frac {21 x^{2} \left (-\frac {75}{224} b^{3} x^{3}-\frac {173}{672} a \,b^{2} x^{2}+\frac {31}{21} a^{2} b x +a^{3}\right ) c^{5}}{8}+\left (-\frac {223}{128} a \,b^{3} x^{3}+\frac {15}{16} a^{2} b^{2} x^{2}+\frac {21}{8} a^{3} b x -\frac {119}{256} b^{4} x^{4}+a^{4}\right ) c^{4}+\left (-a^{2} b^{3} x -\frac {909}{1024} b^{4} x^{2} a -\frac {3}{256} x^{3} b^{5}-\frac {11}{32} a^{3} b^{2}\right ) c^{3}+\frac {b^{4} \left (\frac {291}{16} b^{2} x^{2}+\frac {23}{4} a b x +a^{2}\right ) c^{2}}{256}+\frac {5 b^{6} \left (7 b x +a \right ) c}{4096}+\frac {15 b^{8}}{32768}\right ) \sqrt {c \,x^{2}+b x +a}}{20 \sqrt {4 c^{2} a -b^{2} c}\, d^{11} \left (2 c x +b \right )^{10} c^{3} \left (-\frac {b^{2}}{4}+a c \right )^{2}}\) | \(324\) |
| default | \(\frac {-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{5 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{10}}-\frac {6 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}+\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {3 c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {10 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}\right )}{5 \left (4 a c -b^{2}\right )}}{2048 d^{11} c^{11}}\) | \(610\) |
-1/20/(4*a*c^2-b^2*c)^(1/2)*(15/65536*(2*c*x+b)^10*arctanh(2*c*(c*x^2+b*x+ a)^(1/2)/(4*a*c^2-b^2*c)^(1/2))+(4*a*c^2-b^2*c)^(1/2)*(-15/128*c^8*x^8+5/6 4*x^6*(-6*b*x+a)*c^7+(15/64*a*b*x^5-215/256*b^2*x^6+31/16*a^2*x^4)*c^6+21/ 8*x^2*(-75/224*b^3*x^3-173/672*a*b^2*x^2+31/21*a^2*b*x+a^3)*c^5+(-223/128* a*b^3*x^3+15/16*a^2*b^2*x^2+21/8*a^3*b*x-119/256*b^4*x^4+a^4)*c^4+(-a^2*b^ 3*x-909/1024*b^4*x^2*a-3/256*x^3*b^5-11/32*a^3*b^2)*c^3+1/256*b^4*(291/16* b^2*x^2+23/4*a*b*x+a^2)*c^2+5/4096*b^6*(7*b*x+a)*c+15/32768*b^8)*(c*x^2+b* x+a)^(1/2))/d^11/(2*c*x+b)^10/c^3/(-1/4*b^2+a*c)^2
Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (207) = 414\).
Time = 26.83 (sec) , antiderivative size = 2114, normalized size of antiderivative = 8.85 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\text {Too large to display} \]
[-1/163840*(15*(1024*c^10*x^10 + 5120*b*c^9*x^9 + 11520*b^2*c^8*x^8 + 1536 0*b^3*c^7*x^7 + 13440*b^4*c^6*x^6 + 8064*b^5*c^5*x^5 + 3360*b^6*c^4*x^4 + 960*b^7*c^3*x^3 + 180*b^8*c^2*x^2 + 20*b^9*c*x + b^10)*sqrt(-b^2*c + 4*a*c ^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(-b^2*c + 4*a*c^2)*sqr t(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(15*b^10*c - 20*a*b^8 *c^2 - 32*a^2*b^6*c^3 - 11776*a^3*b^4*c^4 + 77824*a^4*b^2*c^5 - 131072*a^5 *c^6 - 3840*(b^2*c^9 - 4*a*c^10)*x^8 - 15360*(b^3*c^8 - 4*a*b*c^9)*x^7 - 6 40*(43*b^4*c^7 - 176*a*b^2*c^8 + 16*a^2*c^9)*x^6 - 1920*(15*b^5*c^6 - 64*a *b^3*c^7 + 16*a^2*b*c^8)*x^5 - 128*(119*b^6*c^5 - 303*a*b^4*c^6 - 1188*a^2 *b^2*c^7 + 1984*a^3*c^8)*x^4 - 128*(3*b^7*c^4 + 434*a*b^5*c^5 - 2776*a^2*b ^3*c^6 + 3968*a^3*b*c^7)*x^3 + 24*(97*b^8*c^3 - 1600*a*b^6*c^4 + 6128*a^2* b^4*c^5 - 1536*a^3*b^2*c^6 - 14336*a^4*c^7)*x^2 + 8*(35*b^9*c^2 - 48*a*b^7 *c^3 - 4464*a^2*b^5*c^4 + 27136*a^3*b^3*c^5 - 43008*a^4*b*c^6)*x)*sqrt(c*x ^2 + b*x + a))/(1024*(b^6*c^14 - 12*a*b^4*c^15 + 48*a^2*b^2*c^16 - 64*a^3* c^17)*d^11*x^10 + 5120*(b^7*c^13 - 12*a*b^5*c^14 + 48*a^2*b^3*c^15 - 64*a^ 3*b*c^16)*d^11*x^9 + 11520*(b^8*c^12 - 12*a*b^6*c^13 + 48*a^2*b^4*c^14 - 6 4*a^3*b^2*c^15)*d^11*x^8 + 15360*(b^9*c^11 - 12*a*b^7*c^12 + 48*a^2*b^5*c^ 13 - 64*a^3*b^3*c^14)*d^11*x^7 + 13440*(b^10*c^10 - 12*a*b^8*c^11 + 48*a^2 *b^6*c^12 - 64*a^3*b^4*c^13)*d^11*x^6 + 8064*(b^11*c^9 - 12*a*b^9*c^10 + 4 8*a^2*b^7*c^11 - 64*a^3*b^5*c^12)*d^11*x^5 + 3360*(b^12*c^8 - 12*a*b^10...
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{11} + 22 b^{10} c x + 220 b^{9} c^{2} x^{2} + 1320 b^{8} c^{3} x^{3} + 5280 b^{7} c^{4} x^{4} + 14784 b^{6} c^{5} x^{5} + 29568 b^{5} c^{6} x^{6} + 42240 b^{4} c^{7} x^{7} + 42240 b^{3} c^{8} x^{8} + 28160 b^{2} c^{9} x^{9} + 11264 b c^{10} x^{10} + 2048 c^{11} x^{11}}\, dx}{d^{11}} \]
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c** 2*x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 281 60*b**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral (b**2*x**2*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x* *2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29 568*b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b **2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(c** 2*x**4*sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568* b**5*c**6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2* c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(2*a*b*x *sqrt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320* b**8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c **6*x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x **9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(2*a*c*x**2*sq rt(a + b*x + c*x**2)/(b**11 + 22*b**10*c*x + 220*b**9*c**2*x**2 + 1320*b** 8*c**3*x**3 + 5280*b**7*c**4*x**4 + 14784*b**6*c**5*x**5 + 29568*b**5*c**6 *x**6 + 42240*b**4*c**7*x**7 + 42240*b**3*c**8*x**8 + 28160*b**2*c**9*x**9 + 11264*b*c**10*x**10 + 2048*c**11*x**11), x) + Integral(2*b*c*x**3*sq...
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\text {Exception raised: ValueError} \]
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. 3431 vs. \(2 (207) = 414\).
Time = 1.09 (sec) , antiderivative size = 3431, normalized size of antiderivative = 14.36 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\text {Too large to display} \]
3/8192*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt( b^2*c - 4*a*c^2))/((b^4*c^3*d^11 - 8*a*b^2*c^4*d^11 + 16*a^2*c^5*d^11)*sqr t(b^2*c - 4*a*c^2)) - 1/40960*(7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^19 *c^(19/2) + 72960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^18*b*c^9 + 346880*(s qrt(c)*x - sqrt(c*x^2 + b*x + a))^17*b^2*c^(17/2) - 74240*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))^17*a*c^(19/2) + 1088000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*b^3*c^8 - 631040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^16*a*b*c^9 + 2380544*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*b^4*c^(15/2) - 1636352*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^15*a*b^2*c^(17/2) - 1775616*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*a^2*c^(19/2) + 3536000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*b^5*c^7 + 348160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*a*b^ 3*c^8 - 13317120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*a^2*b*c^9 + 318656 0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b^6*c^(13/2) + 11265280*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))^13*a*b^4*c^(15/2) - 42624000*(sqrt(c)*x - sqrt( c*x^2 + b*x + a))^13*a^2*b^2*c^(17/2) - 5314560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*a^3*c^(19/2) + 840320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12* b^7*c^6 + 29660800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*b^5*c^7 - 7507 9680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a^2*b^3*c^8 - 34544640*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^12*a^3*b*c^9 - 2029440*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))^11*b^8*c^(11/2) + 42554880*(sqrt(c)*x - sqrt(c*x^2 + b*x ...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{11}} \,d x \]