Integrand size = 24, antiderivative size = 134 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac {140}{3} c^2 d^8 (b+2 c x)^3-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-140 c^2 \left (b^2-4 a c\right )^{3/2} d^8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \] Output:
140*c^2*(-4*a*c+b^2)*d^8*(2*c*x+b)+140/3*c^2*d^8*(2*c*x+b)^3-1/2*d^8*(2*c* x+b)^7/(c*x^2+b*x+a)^2-7*c*d^8*(2*c*x+b)^5/(c*x^2+b*x+a)-140*c^2*(-4*a*c+b ^2)^(3/2)*d^8*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))
Time = 0.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.06 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=d^8 \left (-256 c^3 \left (-b^2+3 a c\right ) x+128 b c^4 x^2+\frac {256 c^5 x^3}{3}-\frac {\left (b^2-4 a c\right )^3 (b+2 c x)}{2 (a+x (b+c x))^2}-\frac {13 c \left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+140 c^2 \left (-b^2+4 a c\right )^{3/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]
Output:
d^8*(-256*c^3*(-b^2 + 3*a*c)*x + 128*b*c^4*x^2 + (256*c^5*x^3)/3 - ((b^2 - 4*a*c)^3*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) - (13*c*(b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x)) + 140*c^2*(-b^2 + 4*a*c)^(3/2)*ArcTan[(b + 2*c *x)/Sqrt[-b^2 + 4*a*c]])
Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1110, 27, 1110, 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.
| \(\displaystyle \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle 7 c d^2 \int \frac {d^6 (b+2 c x)^6}{\left (c x^2+b x+a\right )^2}dx-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 7 c d^8 \int \frac {(b+2 c x)^6}{\left (c x^2+b x+a\right )^2}dx-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle 7 c d^8 \left (10 c \int \frac {(b+2 c x)^4}{c x^2+b x+a}dx-\frac {(b+2 c x)^5}{a+b x+c x^2}\right )-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 7 c d^8 \left (10 c \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{c x^2+b x+a}dx+\frac {2}{3} (b+2 c x)^3\right )-\frac {(b+2 c x)^5}{a+b x+c x^2}\right )-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 7 c d^8 \left (10 c \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{c x^2+b x+a}dx+2 (b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {(b+2 c x)^5}{a+b x+c x^2}\right )-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 7 c d^8 \left (10 c \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \left (b^2-4 a c\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {(b+2 c x)^5}{a+b x+c x^2}\right )-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 7 c d^8 \left (10 c \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {(b+2 c x)^5}{a+b x+c x^2}\right )-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}\) |
Input:
Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]
Output:
-1/2*(d^8*(b + 2*c*x)^7)/(a + b*x + c*x^2)^2 + 7*c*d^8*(-((b + 2*c*x)^5/(a + b*x + c*x^2)) + 10*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_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 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] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
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])
Time = 0.93 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.76
| method | result | size |
| default | \(d^{8} \left (\frac {256 c^{5} x^{3}}{3}+128 b \,c^{4} x^{2}-768 x \,c^{4} a +256 c^{3} b^{2} x +\frac {\left (-416 a^{2} c^{5}+208 a \,b^{2} c^{4}-26 b^{4} c^{3}\right ) x^{3}-39 b \,c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) x^{2}-2 c \left (176 a^{3} c^{3}+24 a^{2} b^{2} c^{2}-45 a \,b^{4} c +7 b^{6}\right ) x -\frac {b \left (352 a^{3} c^{3}-160 a^{2} b^{2} c^{2}+14 a \,b^{4} c +b^{6}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {140 c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\) | \(236\) |
| risch | \(\frac {256 d^{8} c^{5} x^{3}}{3}+128 d^{8} c^{4} b \,x^{2}-768 d^{8} c^{4} a x +256 d^{8} c^{3} b^{2} x +\frac {\left (-416 a^{2} d^{8} c^{5}+208 a \,b^{2} d^{8} c^{4}-26 b^{4} d^{8} c^{3}\right ) x^{3}-39 b \,d^{8} c^{2} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) x^{2}-2 d^{8} c \left (176 a^{3} c^{3}+24 a^{2} b^{2} c^{2}-45 a \,b^{4} c +7 b^{6}\right ) x -\frac {b \,d^{8} \left (352 a^{3} c^{3}-160 a^{2} b^{2} c^{2}+14 a \,b^{4} c +b^{6}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-70 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{8} c^{2} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )+70 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{8} c^{2} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )\) | \(335\) |
Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
d^8*(256/3*c^5*x^3+128*b*c^4*x^2-768*x*c^4*a+256*c^3*b^2*x+((-416*a^2*c^5+ 208*a*b^2*c^4-26*b^4*c^3)*x^3-39*b*c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-2*c* (176*a^3*c^3+24*a^2*b^2*c^2-45*a*b^4*c+7*b^6)*x-1/2*b*(352*a^3*c^3-160*a^2 *b^2*c^2+14*a*b^4*c+b^6))/(c*x^2+b*x+a)^2+140*c^2*(16*a^2*c^2-8*a*b^2*c+b^ 4)/(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. 424 vs. \(2 (126) = 252\).
Time = 0.10 (sec) , antiderivative size = 868, normalized size of antiderivative = 6.48 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="fricas")
Output:
[1/6*(512*c^7*d^8*x^7 + 1792*b*c^6*d^8*x^6 + 3584*(b^2*c^5 - a*c^6)*d^8*x^ 5 + 256*(15*b^3*c^4 - 26*a*b*c^5)*d^8*x^4 + 4*(345*b^4*c^3 + 312*a*b^2*c^4 - 2800*a^2*c^5)*d^8*x^3 - 6*(39*b^5*c^2 - 824*a*b^3*c^3 + 2032*a^2*b*c^4) *d^8*x^2 - 12*(7*b^6*c - 45*a*b^4*c^2 - 104*a^2*b^2*c^3 + 560*a^3*c^4)*d^8 *x - 3*(b^7 + 14*a*b^5*c - 160*a^2*b^3*c^2 + 352*a^3*b*c^3)*d^8 - 420*((b^ 2*c^4 - 4*a*c^5)*d^8*x^4 + 2*(b^3*c^3 - 4*a*b*c^4)*d^8*x^3 + (b^4*c^2 - 2* a*b^2*c^3 - 8*a^2*c^4)*d^8*x^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d^8*x + (a^2* b^2*c^2 - 4*a^3*c^3)*d^8)*sqrt(b^2 - 4*a*c)*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)))/(c^2*x^4 + 2 *b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), 1/6*(512*c^7*d^8*x^7 + 1792 *b*c^6*d^8*x^6 + 3584*(b^2*c^5 - a*c^6)*d^8*x^5 + 256*(15*b^3*c^4 - 26*a*b *c^5)*d^8*x^4 + 4*(345*b^4*c^3 + 312*a*b^2*c^4 - 2800*a^2*c^5)*d^8*x^3 - 6 *(39*b^5*c^2 - 824*a*b^3*c^3 + 2032*a^2*b*c^4)*d^8*x^2 - 12*(7*b^6*c - 45* a*b^4*c^2 - 104*a^2*b^2*c^3 + 560*a^3*c^4)*d^8*x - 3*(b^7 + 14*a*b^5*c - 1 60*a^2*b^3*c^2 + 352*a^3*b*c^3)*d^8 - 840*((b^2*c^4 - 4*a*c^5)*d^8*x^4 + 2 *(b^3*c^3 - 4*a*b*c^4)*d^8*x^3 + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*d^8*x ^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*d^8*x + (a^2*b^2*c^2 - 4*a^3*c^3)*d^8)*sq rt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(c ^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)]
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (133) = 266\).
Time = 2.49 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.50 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=128 b c^{4} d^{8} x^{2} + \frac {256 c^{5} d^{8} x^{3}}{3} - 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} - 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} + 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + x \left (- 768 a c^{4} d^{8} + 256 b^{2} c^{3} d^{8}\right ) + \frac {- 352 a^{3} b c^{3} d^{8} + 160 a^{2} b^{3} c^{2} d^{8} - 14 a b^{5} c d^{8} - b^{7} d^{8} + x^{3} \left (- 832 a^{2} c^{5} d^{8} + 416 a b^{2} c^{4} d^{8} - 52 b^{4} c^{3} d^{8}\right ) + x^{2} \left (- 1248 a^{2} b c^{4} d^{8} + 624 a b^{3} c^{3} d^{8} - 78 b^{5} c^{2} d^{8}\right ) + x \left (- 704 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 180 a b^{4} c^{2} d^{8} - 28 b^{6} c d^{8}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \] Input:
integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**3,x)
Output:
128*b*c**4*d**8*x**2 + 256*c**5*d**8*x**3/3 - 70*c**2*d**8*sqrt(-(4*a*c - b**2)**3)*log(x + (280*a*b*c**3*d**8 - 70*b**3*c**2*d**8 - 70*c**2*d**8*sq rt(-(4*a*c - b**2)**3))/(560*a*c**4*d**8 - 140*b**2*c**3*d**8)) + 70*c**2* d**8*sqrt(-(4*a*c - b**2)**3)*log(x + (280*a*b*c**3*d**8 - 70*b**3*c**2*d* *8 + 70*c**2*d**8*sqrt(-(4*a*c - b**2)**3))/(560*a*c**4*d**8 - 140*b**2*c* *3*d**8)) + x*(-768*a*c**4*d**8 + 256*b**2*c**3*d**8) + (-352*a**3*b*c**3* d**8 + 160*a**2*b**3*c**2*d**8 - 14*a*b**5*c*d**8 - b**7*d**8 + x**3*(-832 *a**2*c**5*d**8 + 416*a*b**2*c**4*d**8 - 52*b**4*c**3*d**8) + x**2*(-1248* a**2*b*c**4*d**8 + 624*a*b**3*c**3*d**8 - 78*b**5*c**2*d**8) + x*(-704*a** 3*c**4*d**8 - 96*a**2*b**2*c**3*d**8 + 180*a*b**4*c**2*d**8 - 28*b**6*c*d* *8))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**2*(4*a*c + 2*b**2))
Exception generated. \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,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. 315 vs. \(2 (126) = 252\).
Time = 0.14 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.35 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\frac {140 \, {\left (b^{4} c^{2} d^{8} - 8 \, a b^{2} c^{3} d^{8} + 16 \, a^{2} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {52 \, b^{4} c^{3} d^{8} x^{3} - 416 \, a b^{2} c^{4} d^{8} x^{3} + 832 \, a^{2} c^{5} d^{8} x^{3} + 78 \, b^{5} c^{2} d^{8} x^{2} - 624 \, a b^{3} c^{3} d^{8} x^{2} + 1248 \, a^{2} b c^{4} d^{8} x^{2} + 28 \, b^{6} c d^{8} x - 180 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x + 704 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} + 14 \, a b^{5} c d^{8} - 160 \, a^{2} b^{3} c^{2} d^{8} + 352 \, a^{3} b c^{3} d^{8}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (2 \, c^{14} d^{8} x^{3} + 3 \, b c^{13} d^{8} x^{2} + 6 \, b^{2} c^{12} d^{8} x - 18 \, a c^{13} d^{8} x\right )}}{3 \, c^{9}} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="giac")
Output:
140*(b^4*c^2*d^8 - 8*a*b^2*c^3*d^8 + 16*a^2*c^4*d^8)*arctan((2*c*x + b)/sq rt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(52*b^4*c^3*d^8*x^3 - 416*a*b^2 *c^4*d^8*x^3 + 832*a^2*c^5*d^8*x^3 + 78*b^5*c^2*d^8*x^2 - 624*a*b^3*c^3*d^ 8*x^2 + 1248*a^2*b*c^4*d^8*x^2 + 28*b^6*c*d^8*x - 180*a*b^4*c^2*d^8*x + 96 *a^2*b^2*c^3*d^8*x + 704*a^3*c^4*d^8*x + b^7*d^8 + 14*a*b^5*c*d^8 - 160*a^ 2*b^3*c^2*d^8 + 352*a^3*b*c^3*d^8)/(c*x^2 + b*x + a)^2 + 128/3*(2*c^14*d^8 *x^3 + 3*b*c^13*d^8*x^2 + 6*b^2*c^12*d^8*x - 18*a*c^13*d^8*x)/c^9
Time = 0.09 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.75 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\frac {256\,c^5\,d^8\,x^3}{3}-\frac {x^2\,\left (624\,a^2\,b\,c^4\,d^8-312\,a\,b^3\,c^3\,d^8+39\,b^5\,c^2\,d^8\right )+x\,\left (352\,a^3\,c^4\,d^8+48\,a^2\,b^2\,c^3\,d^8-90\,a\,b^4\,c^2\,d^8+14\,b^6\,c\,d^8\right )+\frac {b^7\,d^8}{2}+x^3\,\left (416\,a^2\,c^5\,d^8-208\,a\,b^2\,c^4\,d^8+26\,b^4\,c^3\,d^8\right )+176\,a^3\,b\,c^3\,d^8-80\,a^2\,b^3\,c^2\,d^8+7\,a\,b^5\,c\,d^8}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-x\,\left (768\,c^3\,d^8\,\left (b^2+a\,c\right )-1024\,b^2\,c^3\,d^8\right )+128\,b\,c^4\,d^8\,x^2+140\,c^2\,d^8\,\mathrm {atan}\left (\frac {70\,b\,c^2\,d^8\,{\left (4\,a\,c-b^2\right )}^{3/2}+140\,c^3\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{3/2}}{1120\,a^2\,c^4\,d^8-560\,a\,b^2\,c^3\,d^8+70\,b^4\,c^2\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{3/2} \] Input:
int((b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x)
Output:
(256*c^5*d^8*x^3)/3 - (x^2*(39*b^5*c^2*d^8 - 312*a*b^3*c^3*d^8 + 624*a^2*b *c^4*d^8) + x*(14*b^6*c*d^8 + 352*a^3*c^4*d^8 - 90*a*b^4*c^2*d^8 + 48*a^2* b^2*c^3*d^8) + (b^7*d^8)/2 + x^3*(416*a^2*c^5*d^8 + 26*b^4*c^3*d^8 - 208*a *b^2*c^4*d^8) + 176*a^3*b*c^3*d^8 - 80*a^2*b^3*c^2*d^8 + 7*a*b^5*c*d^8)/(x ^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) - x*(768*c^3*d^8*( a*c + b^2) - 1024*b^2*c^3*d^8) + 128*b*c^4*d^8*x^2 + 140*c^2*d^8*atan((70* b*c^2*d^8*(4*a*c - b^2)^(3/2) + 140*c^3*d^8*x*(4*a*c - b^2)^(3/2))/(1120*a ^2*c^4*d^8 + 70*b^4*c^2*d^8 - 560*a*b^2*c^3*d^8))*(4*a*c - b^2)^(3/2)
Time = 0.20 (sec) , antiderivative size = 713, normalized size of antiderivative = 5.32 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\frac {d^{8} \left (-1680 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c^{3} x^{3}-840 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{4} x^{4}+5600 a^{2} c^{6} x^{4}+1792 b^{2} c^{6} x^{6}+5600 a^{4} c^{4}-84 b^{7} c x -924 b^{6} c^{2} x^{2}+3150 b^{4} c^{4} x^{4}+3584 b^{3} c^{5} x^{5}+512 b \,c^{7} x^{7}-1680 a^{3} b^{2} c^{3}+11200 a^{3} c^{5} x^{2}-210 a^{2} b^{4} c^{2}-42 a \,b^{6} c -3 b^{8}+6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{4} x^{3}+3360 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{5} x^{4}+4480 a^{3} b \,c^{4} x -840 a \,b^{5} c^{2} x -7840 a^{2} b^{2} c^{4} x^{2}+2940 a \,b^{4} c^{3} x^{2}-7280 a \,b^{2} c^{5} x^{4}-3584 a b \,c^{6} x^{5}+3360 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{3} b \,c^{3}-840 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{3} c^{2}-840 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{5} c^{2} x^{2}+6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{3} x +6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{4} x^{2}-1680 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c^{2} x +1680 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c^{3} x^{2}\right )}{6 b \left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x)
Output:
(d**8*(3360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b *c**3 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b **3*c**2 + 6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* *2*b**2*c**3*x + 6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 2))*a**2*b*c**4*x**2 - 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c**2*x + 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*a*b**3*c**3*x**2 + 6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s qrt(4*a*c - b**2))*a*b**2*c**4*x**3 + 3360*sqrt(4*a*c - b**2)*atan((b + 2* c*x)/sqrt(4*a*c - b**2))*a*b*c**5*x**4 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*c**2*x**2 - 1680*sqrt(4*a*c - b**2)*atan(( b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c**3*x**3 - 840*sqrt(4*a*c - b**2)*ata n((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**4*x**4 + 5600*a**4*c**4 - 1680*a **3*b**2*c**3 + 4480*a**3*b*c**4*x + 11200*a**3*c**5*x**2 - 210*a**2*b**4* c**2 - 7840*a**2*b**2*c**4*x**2 + 5600*a**2*c**6*x**4 - 42*a*b**6*c - 840* a*b**5*c**2*x + 2940*a*b**4*c**3*x**2 - 7280*a*b**2*c**5*x**4 - 3584*a*b*c **6*x**5 - 3*b**8 - 84*b**7*c*x - 924*b**6*c**2*x**2 + 3150*b**4*c**4*x**4 + 3584*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 512*b*c**7*x**7))/(6*b*(a** 2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4))