Integrand size = 26, antiderivative size = 197 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=-\frac {5 \sqrt {a+b x+c x^2}}{2048 c^3 d^9 (b+2 c x)^4}+\frac {5 \sqrt {a+b x+c x^2}}{4096 c^3 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{384 c^2 d^9 (b+2 c x)^6}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}+\frac {5 \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{8192 c^{7/2} \left (b^2-4 a c\right )^{3/2} d^9} \]
-5/384*(c*x^2+b*x+a)^(3/2)/c^2/d^9/(2*c*x+b)^6-1/16*(c*x^2+b*x+a)^(5/2)/c/ d^9/(2*c*x+b)^8+5/8192*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)^(3/2)/d^9-5/2048*(c*x^2+b*x+a)^(1/2)/c^3/d^9/(2 *c*x+b)^4+5/4096*(c*x^2+b*x+a)^(1/2)/c^3/(-4*a*c+b^2)/d^9/(2*c*x+b)^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.31 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\frac {2 (a+x (b+c x))^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},5,\frac {9}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{7 \left (b^2-4 a c\right )^5 d^9} \]
(2*(a + x*(b + c*x))^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(7*(b^2 - 4*a*c)^5*d^9)
Time = 0.37 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1108, 27, 1108, 1108, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{d^7 (b+2 c x)^7}dx}{32 c d^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(b+2 c x)^7}dx}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {\sqrt {c x^2+b x+a}}{(b+2 c x)^5}dx}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\int \frac {1}{(b+2 c x)^3 \sqrt {c x^2+b x+a}}dx}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 1117 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{2 \left (b^2-4 a c\right )}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 1112 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {2 c \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}}{b^2-4 a c}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 \left (\frac {\frac {\frac {\arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {c} \left (b^2-4 a c\right )^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) (b+2 c x)^2}}{16 c}-\frac {\sqrt {a+b x+c x^2}}{8 c (b+2 c x)^4}}{8 c}-\frac {\left (a+b x+c x^2\right )^{3/2}}{12 c (b+2 c x)^6}\right )}{32 c d^9}-\frac {\left (a+b x+c x^2\right )^{5/2}}{16 c d^9 (b+2 c x)^8}\) |
-1/16*(a + b*x + c*x^2)^(5/2)/(c*d^9*(b + 2*c*x)^8) + (5*(-1/12*(a + b*x + c*x^2)^(3/2)/(c*(b + 2*c*x)^6) + (-1/8*Sqrt[a + b*x + c*x^2]/(c*(b + 2*c* x)^4) + (Sqrt[a + b*x + c*x^2]/((b^2 - 4*a*c)*(b + 2*c*x)^2) + ArcTan[(2*S qrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]]/(2*Sqrt[c]*(b^2 - 4*a*c)^ (3/2)))/(16*c))/(8*c)))/(32*c*d^9)
3.13.30.3.1 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 = 13.19 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.22
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {5 \left (2 c x +b \right )^{8} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )}{2048}+\sqrt {4 c^{2} a -b^{2} c}\, \left (\frac {5 c^{6} x^{6}}{16}+\left (\frac {15}{16} b \,x^{5}+\frac {59}{24} a \,x^{4}\right ) c^{5}+\left (\frac {107}{192} b^{2} x^{4}+\frac {17}{6} a^{2} x^{2}+\frac {59}{12} a b \,x^{3}\right ) c^{4}+\left (a^{3}-\frac {43}{96} b^{3} x^{3}+\frac {109}{48} a \,b^{2} x^{2}+\frac {17}{6} a^{2} b x \right ) c^{3}-\frac {b^{2} \left (\frac {347}{32} b^{2} x^{2}+\frac {9}{2} a b x +a^{2}\right ) c^{2}}{24}-\frac {5 b^{4} \left (\frac {11 b x}{2}+a \right ) c}{384}-\frac {5 b^{6}}{1024}\right ) \sqrt {c \,x^{2}+b x +a}}{16 \sqrt {4 c^{2} a -b^{2} c}\, d^{9} \left (2 c x +b \right )^{8} c^{3} \left (-\frac {b^{2}}{4}+a c \right )}\) | \(241\) |
| default | \(\frac {-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{2 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{8}}-\frac {c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{6}}+\frac {2 c^{2} \left (-\frac {c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4}}+\frac {3 c^{2} \left (-\frac {2 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {10 c^{2} \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 c}\right )}{4 a c -b^{2}}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}\right )}{2 \left (4 a c -b^{2}\right )}}{512 d^{9} c^{9}}\) | \(537\) |
-1/16/(4*a*c^2-b^2*c)^(1/2)*(-5/2048*(2*c*x+b)^8*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)*(5/16*c^6*x^6+(15/16*b *x^5+59/24*a*x^4)*c^5+(107/192*b^2*x^4+17/6*a^2*x^2+59/12*a*b*x^3)*c^4+(a^ 3-43/96*b^3*x^3+109/48*a*b^2*x^2+17/6*a^2*b*x)*c^3-1/24*b^2*(347/32*b^2*x^ 2+9/2*a*b*x+a^2)*c^2-5/384*b^4*(11/2*b*x+a)*c-5/1024*b^6)*(c*x^2+b*x+a)^(1 /2))/d^9/(2*c*x+b)^8/c^3/(-1/4*b^2+a*c)
Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (169) = 338\).
Time = 9.38 (sec) , antiderivative size = 1448, normalized size of antiderivative = 7.35 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \]
[1/49152*(15*(256*c^8*x^8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c ^5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c^3*x^3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*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)*sqrt(c*x^2 + b*x + a))/(4*c^2*x^2 + 4*b*c*x + b^2) ) - 4*(15*b^8*c - 20*a*b^6*c^2 - 32*a^2*b^4*c^3 - 3584*a^3*b^2*c^4 + 12288 *a^4*c^5 - 960*(b^2*c^7 - 4*a*c^8)*x^6 - 2880*(b^3*c^6 - 4*a*b*c^7)*x^5 - 16*(107*b^4*c^5 + 44*a*b^2*c^6 - 1888*a^2*c^7)*x^4 + 32*(43*b^5*c^4 - 644* a*b^3*c^5 + 1888*a^2*b*c^6)*x^3 + 4*(347*b^6*c^3 - 3132*a*b^4*c^4 + 4800*a ^2*b^2*c^5 + 8704*a^3*c^6)*x^2 + 4*(55*b^7*c^2 - 76*a*b^5*c^3 - 2752*a^2*b ^3*c^4 + 8704*a^3*b*c^5)*x)*sqrt(c*x^2 + b*x + a))/(256*(b^4*c^12 - 8*a*b^ 2*c^13 + 16*a^2*c^14)*d^9*x^8 + 1024*(b^5*c^11 - 8*a*b^3*c^12 + 16*a^2*b*c ^13)*d^9*x^7 + 1792*(b^6*c^10 - 8*a*b^4*c^11 + 16*a^2*b^2*c^12)*d^9*x^6 + 1792*(b^7*c^9 - 8*a*b^5*c^10 + 16*a^2*b^3*c^11)*d^9*x^5 + 1120*(b^8*c^8 - 8*a*b^6*c^9 + 16*a^2*b^4*c^10)*d^9*x^4 + 448*(b^9*c^7 - 8*a*b^7*c^8 + 16*a ^2*b^5*c^9)*d^9*x^3 + 112*(b^10*c^6 - 8*a*b^8*c^7 + 16*a^2*b^6*c^8)*d^9*x^ 2 + 16*(b^11*c^5 - 8*a*b^9*c^6 + 16*a^2*b^7*c^7)*d^9*x + (b^12*c^4 - 8*a*b ^10*c^5 + 16*a^2*b^8*c^6)*d^9), -1/24576*(15*(256*c^8*x^8 + 1024*b*c^7*x^7 + 1792*b^2*c^6*x^6 + 1792*b^3*c^5*x^5 + 1120*b^4*c^4*x^4 + 448*b^5*c^3*x^ 3 + 112*b^6*c^2*x^2 + 16*b^7*c*x + b^8)*sqrt(b^2*c - 4*a*c^2)*arctan(1/2*s qrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2...
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{9} + 18 b^{8} c x + 144 b^{7} c^{2} x^{2} + 672 b^{6} c^{3} x^{3} + 2016 b^{5} c^{4} x^{4} + 4032 b^{4} c^{5} x^{5} + 5376 b^{3} c^{6} x^{6} + 4608 b^{2} c^{7} x^{7} + 2304 b c^{8} x^{8} + 512 c^{9} x^{9}}\, dx}{d^{9}} \]
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2* x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 53 76*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9 ), x) + Integral(b**2*x**2*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 14 4*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c* *5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 5 12*c**9*x**9), x) + Integral(c**2*x**4*sqrt(a + b*x + c*x**2)/(b**9 + 18*b **8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c **8*x**8 + 512*c**9*x**9), x) + Integral(2*a*b*x*sqrt(a + b*x + c*x**2)/(b **9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c* *4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b* *2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9), x) + Integral(2*b*c*x**3 *sqrt(a + b*x + c*x**2)/(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b** 6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x **6 + 4608*b**2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9), x))/d**9
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\text {Exception raised: ValueError} \]
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. 2227 vs. \(2 (169) = 338\).
Time = 0.60 (sec) , antiderivative size = 2227, normalized size of antiderivative = 11.30 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\text {Too large to display} \]
5/4096*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^2*c^3*d^9 - 4*a*c^4*d^9)*sqrt(b^2*c - 4*a*c^2)) - 1/ 12288*(1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^15*c^(15/2) + 14400*(sqrt( c)*x - sqrt(c*x^2 + b*x + a))^14*b*c^7 + 37696*(sqrt(c)*x - sqrt(c*x^2 + b *x + a))^13*b^2*c^(13/2) + 50816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*a* c^(15/2) + 26624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^3*c^6 + 330304*( sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*a*b*c^7 - 76768*(sqrt(c)*x - sqrt(c* x^2 + b*x + a))^11*b^4*c^(11/2) + 933632*(sqrt(c)*x - sqrt(c*x^2 + b*x + a ))^11*a*b^2*c^(13/2) + 114560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a^2*c ^(15/2) - 234608*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^5*c^5 + 1501632* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a*b^3*c^6 + 630080*(sqrt(c)*x - sqr t(c*x^2 + b*x + a))^10*a^2*b*c^7 - 322640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^6*c^(9/2) + 1525600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^4*c^ (11/2) + 1405760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c^(13/2) + 225920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^(15/2) - 282624*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))^8*b^7*c^4 + 1052976*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^5*c^5 + 1600320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2 *b^3*c^6 + 1016640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*b*c^7 - 17557 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^8*c^(7/2) + 548224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^6*c^(9/2) + 922560*(sqrt(c)*x - sqrt(c*x^2...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^9} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^9} \,d x \]