Integrand size = 24, antiderivative size = 122 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-2 \left (b^2-4 a c\right )^{7/2} d^8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \] Output:
2*(-4*a*c+b^2)^3*d^8*(2*c*x+b)+2/3*(-4*a*c+b^2)^2*d^8*(2*c*x+b)^3+2/5*(-4* a*c+b^2)*d^8*(2*c*x+b)^5+2/7*d^8*(2*c*x+b)^7-2*(-4*a*c+b^2)^(7/2)*d^8*arct anh((2*c*x+b)/(-4*a*c+b^2)^(1/2))
Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.54 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=d^8 \left (\frac {16}{105} c x \left (105 b^6+315 b^5 c x+420 b^3 c^2 x \left (-2 a+3 c x^2\right )+70 b^4 c \left (-9 a+11 c x^2\right )+840 b c^3 x \left (a^2-a c x^2+c^2 x^4\right )+112 b^2 c^2 \left (15 a^2-10 a c x^2+12 c^2 x^4\right )+16 c^3 \left (-105 a^3+35 a^2 c x^2-21 a c^2 x^4+15 c^3 x^6\right )\right )+2 \left (-b^2+4 a c\right )^{7/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2),x]
Output:
d^8*((16*c*x*(105*b^6 + 315*b^5*c*x + 420*b^3*c^2*x*(-2*a + 3*c*x^2) + 70* b^4*c*(-9*a + 11*c*x^2) + 840*b*c^3*x*(a^2 - a*c*x^2 + c^2*x^4) + 112*b^2* c^2*(15*a^2 - 10*a*c*x^2 + 12*c^2*x^4) + 16*c^3*(-105*a^3 + 35*a^2*c*x^2 - 21*a*c^2*x^4 + 15*c^3*x^6)))/105 + 2*(-b^2 + 4*a*c)^(7/2)*ArcTan[(b + 2*c *x)/Sqrt[-b^2 + 4*a*c]])
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1116, 27, 1116, 1116, 1116, 1083, 219}
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.
| \(\displaystyle \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle d^2 \left (b^2-4 a c\right ) \int \frac {d^6 (b+2 c x)^6}{c x^2+b x+a}dx+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \int \frac {(b+2 c x)^6}{c x^2+b x+a}dx+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^4}{c x^2+b x+a}dx+\frac {2}{5} (b+2 c x)^5\right )+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{c x^2+b x+a}dx+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{c x^2+b x+a}dx+2 (b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \left (b^2-4 a c\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )+\frac {2}{7} d^8 (b+2 c x)^7\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d^8 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )+\frac {2}{7} d^8 (b+2 c x)^7\) |
Input:
Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2),x]
Output:
(2*d^8*(b + 2*c*x)^7)/7 + (b^2 - 4*a*c)*d^8*((2*(b + 2*c*x)^5)/5 + (b^2 - 4*a*c)*((2*(b + 2*c*x)^3)/3 + (b^2 - 4*a*c)*(2*(b + 2*c*x) - 2*Sqrt[b^2 - 4*a*c]*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])))
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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) 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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(112)=224\).
Time = 0.96 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.02
| method | result | size |
| default | \(d^{8} \left (\frac {256 c^{7} x^{7}}{7}+128 b \,c^{6} x^{6}-\frac {256 a \,c^{6} x^{5}}{5}+\frac {1024 b^{2} c^{5} x^{5}}{5}-128 a b \,c^{5} x^{4}+192 b^{3} c^{4} x^{4}+\frac {256 a^{2} c^{5} x^{3}}{3}-\frac {512 a \,b^{2} c^{4} x^{3}}{3}+\frac {352 b^{4} c^{3} x^{3}}{3}+128 a^{2} b \,c^{4} x^{2}-128 a \,b^{3} c^{3} x^{2}+48 b^{5} c^{2} x^{2}-256 a^{3} c^{4} x +256 a^{2} b^{2} c^{3} x -96 a \,b^{4} c^{2} x +16 b^{6} c x +\frac {2 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\) | \(246\) |
| risch | \(\frac {256 c^{7} d^{8} x^{7}}{7}+128 c^{6} d^{8} x^{6} b -\frac {256 c^{6} d^{8} a \,x^{5}}{5}+\frac {1024 c^{5} d^{8} x^{5} b^{2}}{5}-128 c^{5} d^{8} a b \,x^{4}+192 c^{4} d^{8} b^{3} x^{4}-\frac {512 c^{4} d^{8} a \,b^{2} x^{3}}{3}+\frac {352 c^{3} d^{8} b^{4} x^{3}}{3}+\frac {256 c^{5} d^{8} x^{3} a^{2}}{3}+128 c^{4} d^{8} a^{2} b \,x^{2}-128 c^{3} d^{8} a \,b^{3} x^{2}+48 c^{2} d^{8} b^{5} x^{2}-256 c^{4} d^{8} a^{3} x +256 c^{3} d^{8} a^{2} b^{2} x -96 c^{2} d^{8} a \,b^{4} x +16 c \,d^{8} b^{6} x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}} d^{8} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b +256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )+\left (-4 a c +b^{2}\right )^{\frac {7}{2}} d^{8} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b +256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )\) | \(381\) |
Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
d^8*(256/7*c^7*x^7+128*b*c^6*x^6-256/5*a*c^6*x^5+1024/5*b^2*c^5*x^5-128*a* b*c^5*x^4+192*b^3*c^4*x^4+256/3*a^2*c^5*x^3-512/3*a*b^2*c^4*x^3+352/3*b^4* c^3*x^3+128*a^2*b*c^4*x^2-128*a*b^3*c^3*x^2+48*b^5*c^2*x^2-256*a^3*c^4*x+2 56*a^2*b^2*c^3*x-96*a*b^4*c^2*x+16*b^6*c*x+2*(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)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b ^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (112) = 224\).
Time = 0.09 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.39 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\left [\frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d^{8} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x, \frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 2 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c} d^{8} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x\right ] \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[256/7*c^7*d^8*x^7 + 128*b*c^6*d^8*x^6 + 256/5*(4*b^2*c^5 - a*c^6)*d^8*x^5 + 64*(3*b^3*c^4 - 2*a*b*c^5)*d^8*x^4 + 32/3*(11*b^4*c^3 - 16*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 16*(3*b^5*c^2 - 8*a*b^3*c^3 + 8*a^2*b*c^4)*d^8*x^2 - (b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(b^2 - 4*a*c)*d^8*log ((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^ 2 + b*x + a)) + 16*(b^6*c - 6*a*b^4*c^2 + 16*a^2*b^2*c^3 - 16*a^3*c^4)*d^8 *x, 256/7*c^7*d^8*x^7 + 128*b*c^6*d^8*x^6 + 256/5*(4*b^2*c^5 - a*c^6)*d^8* x^5 + 64*(3*b^3*c^4 - 2*a*b*c^5)*d^8*x^4 + 32/3*(11*b^4*c^3 - 16*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 16*(3*b^5*c^2 - 8*a*b^3*c^3 + 8*a^2*b*c^4)*d^8*x^2 - 2*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)*d ^8*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 16*(b^6*c - 6*a *b^4*c^2 + 16*a^2*b^2*c^3 - 16*a^3*c^4)*d^8*x]
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (124) = 248\).
Time = 0.53 (sec) , antiderivative size = 502, normalized size of antiderivative = 4.11 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=128 b c^{6} d^{8} x^{6} + \frac {256 c^{7} d^{8} x^{7}}{7} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + x^{5} \left (- \frac {256 a c^{6} d^{8}}{5} + \frac {1024 b^{2} c^{5} d^{8}}{5}\right ) + x^{4} \left (- 128 a b c^{5} d^{8} + 192 b^{3} c^{4} d^{8}\right ) + x^{3} \cdot \left (\frac {256 a^{2} c^{5} d^{8}}{3} - \frac {512 a b^{2} c^{4} d^{8}}{3} + \frac {352 b^{4} c^{3} d^{8}}{3}\right ) + x^{2} \cdot \left (128 a^{2} b c^{4} d^{8} - 128 a b^{3} c^{3} d^{8} + 48 b^{5} c^{2} d^{8}\right ) + x \left (- 256 a^{3} c^{4} d^{8} + 256 a^{2} b^{2} c^{3} d^{8} - 96 a b^{4} c^{2} d^{8} + 16 b^{6} c d^{8}\right ) \] Input:
integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a),x)
Output:
128*b*c**6*d**8*x**6 + 256*c**7*d**8*x**7/7 - d**8*sqrt(-(4*a*c - b**2)**7 )*log(x + (64*a**3*b*c**3*d**8 - 48*a**2*b**3*c**2*d**8 + 12*a*b**5*c*d**8 - b**7*d**8 - d**8*sqrt(-(4*a*c - b**2)**7))/(128*a**3*c**4*d**8 - 96*a** 2*b**2*c**3*d**8 + 24*a*b**4*c**2*d**8 - 2*b**6*c*d**8)) + d**8*sqrt(-(4*a *c - b**2)**7)*log(x + (64*a**3*b*c**3*d**8 - 48*a**2*b**3*c**2*d**8 + 12* a*b**5*c*d**8 - b**7*d**8 + d**8*sqrt(-(4*a*c - b**2)**7))/(128*a**3*c**4* d**8 - 96*a**2*b**2*c**3*d**8 + 24*a*b**4*c**2*d**8 - 2*b**6*c*d**8)) + x* *5*(-256*a*c**6*d**8/5 + 1024*b**2*c**5*d**8/5) + x**4*(-128*a*b*c**5*d**8 + 192*b**3*c**4*d**8) + x**3*(256*a**2*c**5*d**8/3 - 512*a*b**2*c**4*d**8 /3 + 352*b**4*c**3*d**8/3) + x**2*(128*a**2*b*c**4*d**8 - 128*a*b**3*c**3* d**8 + 48*b**5*c**2*d**8) + x*(-256*a**3*c**4*d**8 + 256*a**2*b**2*c**3*d* *8 - 96*a*b**4*c**2*d**8 + 16*b**6*c*d**8)
Exception generated. \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
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. 313 vs. \(2 (112) = 224\).
Time = 0.15 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.57 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\frac {2 \, {\left (b^{8} d^{8} - 16 \, a b^{6} c d^{8} + 96 \, a^{2} b^{4} c^{2} d^{8} - 256 \, a^{3} b^{2} c^{3} d^{8} + 256 \, a^{4} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} + \frac {16 \, {\left (240 \, c^{14} d^{8} x^{7} + 840 \, b c^{13} d^{8} x^{6} + 1344 \, b^{2} c^{12} d^{8} x^{5} - 336 \, a c^{13} d^{8} x^{5} + 1260 \, b^{3} c^{11} d^{8} x^{4} - 840 \, a b c^{12} d^{8} x^{4} + 770 \, b^{4} c^{10} d^{8} x^{3} - 1120 \, a b^{2} c^{11} d^{8} x^{3} + 560 \, a^{2} c^{12} d^{8} x^{3} + 315 \, b^{5} c^{9} d^{8} x^{2} - 840 \, a b^{3} c^{10} d^{8} x^{2} + 840 \, a^{2} b c^{11} d^{8} x^{2} + 105 \, b^{6} c^{8} d^{8} x - 630 \, a b^{4} c^{9} d^{8} x + 1680 \, a^{2} b^{2} c^{10} d^{8} x - 1680 \, a^{3} c^{11} d^{8} x\right )}}{105 \, c^{7}} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a),x, algorithm="giac")
Output:
2*(b^8*d^8 - 16*a*b^6*c*d^8 + 96*a^2*b^4*c^2*d^8 - 256*a^3*b^2*c^3*d^8 + 2 56*a^4*c^4*d^8)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) + 16/105*(240*c^14*d^8*x^7 + 840*b*c^13*d^8*x^6 + 1344*b^2*c^12*d^8*x^5 - 336*a*c^13*d^8*x^5 + 1260*b^3*c^11*d^8*x^4 - 840*a*b*c^12*d^8*x^4 + 770*b^ 4*c^10*d^8*x^3 - 1120*a*b^2*c^11*d^8*x^3 + 560*a^2*c^12*d^8*x^3 + 315*b^5* c^9*d^8*x^2 - 840*a*b^3*c^10*d^8*x^2 + 840*a^2*b*c^11*d^8*x^2 + 105*b^6*c^ 8*d^8*x - 630*a*b^4*c^9*d^8*x + 1680*a^2*b^2*c^10*d^8*x - 1680*a^3*c^11*d^ 8*x)/c^7
Time = 0.06 (sec) , antiderivative size = 761, normalized size of antiderivative = 6.24 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=x^3\,\left (\frac {1120\,b^4\,c^3\,d^8}{3}-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{3\,c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{3\,c}\right )-x^5\,\left (\frac {256\,a\,c^6\,d^8}{5}-\frac {1024\,b^2\,c^5\,d^8}{5}\right )-x^2\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{2\,c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{2\,c}-224\,b^5\,c^2\,d^8\right )+x^4\,\left (448\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{4\,c}-192\,a\,b\,c^5\,d^8\right )+x\,\left (112\,b^6\,c\,d^8-\frac {a\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}+\frac {b\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}-448\,b^5\,c^2\,d^8\right )}{c}\right )+2\,d^8\,\mathrm {atan}\left (\frac {b\,d^8\,{\left (4\,a\,c-b^2\right )}^{7/2}+2\,c\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{7/2}}{256\,a^4\,c^4\,d^8-256\,a^3\,b^2\,c^3\,d^8+96\,a^2\,b^4\,c^2\,d^8-16\,a\,b^6\,c\,d^8+b^8\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{7/2}+\frac {256\,c^7\,d^8\,x^7}{7}+128\,b\,c^6\,d^8\,x^6 \] Input:
int((b*d + 2*c*d*x)^8/(a + b*x + c*x^2),x)
Output:
x^3*((1120*b^4*c^3*d^8)/3 - (b*(1792*b^3*c^4*d^8 + (b*(256*a*c^6*d^8 - 102 4*b^2*c^5*d^8))/c - 768*a*b*c^5*d^8))/(3*c) + (a*(256*a*c^6*d^8 - 1024*b^2 *c^5*d^8))/(3*c)) - x^5*((256*a*c^6*d^8)/5 - (1024*b^2*c^5*d^8)/5) - x^2*( (a*(1792*b^3*c^4*d^8 + (b*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/c - 768*a*b* c^5*d^8))/(2*c) + (b*(1120*b^4*c^3*d^8 - (b*(1792*b^3*c^4*d^8 + (b*(256*a* c^6*d^8 - 1024*b^2*c^5*d^8))/c - 768*a*b*c^5*d^8))/c + (a*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/c))/(2*c) - 224*b^5*c^2*d^8) + x^4*(448*b^3*c^4*d^8 + (b*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/(4*c) - 192*a*b*c^5*d^8) + x*(112*b ^6*c*d^8 - (a*(1120*b^4*c^3*d^8 - (b*(1792*b^3*c^4*d^8 + (b*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/c - 768*a*b*c^5*d^8))/c + (a*(256*a*c^6*d^8 - 1024*b ^2*c^5*d^8))/c))/c + (b*((a*(1792*b^3*c^4*d^8 + (b*(256*a*c^6*d^8 - 1024*b ^2*c^5*d^8))/c - 768*a*b*c^5*d^8))/c + (b*(1120*b^4*c^3*d^8 - (b*(1792*b^3 *c^4*d^8 + (b*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/c - 768*a*b*c^5*d^8))/c + (a*(256*a*c^6*d^8 - 1024*b^2*c^5*d^8))/c))/c - 448*b^5*c^2*d^8))/c) + 2* d^8*atan((b*d^8*(4*a*c - b^2)^(7/2) + 2*c*d^8*x*(4*a*c - b^2)^(7/2))/(b^8* d^8 + 256*a^4*c^4*d^8 + 96*a^2*b^4*c^2*d^8 - 256*a^3*b^2*c^3*d^8 - 16*a*b^ 6*c*d^8))*(4*a*c - b^2)^(7/2) + (256*c^7*d^8*x^7)/7 + 128*b*c^6*d^8*x^6
Time = 0.19 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.70 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\frac {2 d^{8} \left (6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{3} c^{3}-5040 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{2}+1260 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c -105 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{6}-13440 a^{3} c^{4} x +13440 a^{2} b^{2} c^{3} x +6720 a^{2} b \,c^{4} x^{2}+4480 a^{2} c^{5} x^{3}-5040 a \,b^{4} c^{2} x -6720 a \,b^{3} c^{3} x^{2}-8960 a \,b^{2} c^{4} x^{3}-6720 a b \,c^{5} x^{4}-2688 a \,c^{6} x^{5}+840 b^{6} c x +2520 b^{5} c^{2} x^{2}+6160 b^{4} c^{3} x^{3}+10080 b^{3} c^{4} x^{4}+10752 b^{2} c^{5} x^{5}+6720 b \,c^{6} x^{6}+1920 c^{7} x^{7}\right )}{105} \] Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a),x)
Output:
(2*d**8*(6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3 *c**3 - 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2* b**2*c**2 + 1260*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a *b**4*c - 105*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**6 - 13440*a**3*c**4*x + 13440*a**2*b**2*c**3*x + 6720*a**2*b*c**4*x**2 + 44 80*a**2*c**5*x**3 - 5040*a*b**4*c**2*x - 6720*a*b**3*c**3*x**2 - 8960*a*b* *2*c**4*x**3 - 6720*a*b*c**5*x**4 - 2688*a*c**6*x**5 + 840*b**6*c*x + 2520 *b**5*c**2*x**2 + 6160*b**4*c**3*x**3 + 10080*b**3*c**4*x**4 + 10752*b**2* c**5*x**5 + 6720*b*c**6*x**6 + 1920*c**7*x**7))/105