Integrand size = 24, antiderivative size = 126 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac {28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac {28}{5} c d^8 (b+2 c x)^5-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}-28 c \left (b^2-4 a c\right )^{5/2} d^8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \] Output:
28*c*(-4*a*c+b^2)^2*d^8*(2*c*x+b)+28/3*c*(-4*a*c+b^2)*d^8*(2*c*x+b)^3+28/5 *c*d^8*(2*c*x+b)^5-d^8*(2*c*x+b)^7/(c*x^2+b*x+a)-28*c*(-4*a*c+b^2)^(5/2)*d ^8*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))
Time = 0.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.23 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=d^8 \left (32 c^2 \left (3 b^4-16 a b^2 c+24 a^2 c^2\right ) x+128 b c^3 \left (b^2-2 a c\right ) x^2-\frac {512}{3} c^4 \left (-b^2+a c\right ) x^3+128 b c^5 x^4+\frac {256 c^6 x^5}{5}-\frac {\left (b^2-4 a c\right )^3 (b+2 c x)}{a+x (b+c x)}-28 c \left (-b^2+4 a c\right )^{5/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^2,x]
Output:
d^8*(32*c^2*(3*b^4 - 16*a*b^2*c + 24*a^2*c^2)*x + 128*b*c^3*(b^2 - 2*a*c)* x^2 - (512*c^4*(-b^2 + a*c)*x^3)/3 + 128*b*c^5*x^4 + (256*c^6*x^5)/5 - ((b ^2 - 4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x)) - 28*c*(-b^2 + 4*a*c)^(5/2)*A rcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
Time = 0.31 (sec) , antiderivative size = 119, 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 = {1110, 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.
| \(\displaystyle \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle 14 c d^2 \int \frac {d^6 (b+2 c x)^6}{c x^2+b x+a}dx-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 14 c d^8 \int \frac {(b+2 c x)^6}{c x^2+b x+a}dx-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 14 c d^8 \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^4}{c x^2+b x+a}dx+\frac {2}{5} (b+2 c x)^5\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 14 c d^8 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{c x^2+b x+a}dx+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle 14 c d^8 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{c x^2+b x+a}dx+2 (b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 14 c d^8 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \left (b^2-4 a c\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 14 c d^8 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )+\frac {2}{3} (b+2 c x)^3\right )+\frac {2}{5} (b+2 c x)^5\right )-\frac {d^8 (b+2 c x)^7}{a+b x+c x^2}\) |
Input:
Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^2,x]
Output:
-((d^8*(b + 2*c*x)^7)/(a + b*x + c*x^2)) + 14*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*Sqr t[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])
Leaf count of result is larger than twice the leaf count of optimal. \(239\) vs. \(2(118)=236\).
Time = 0.89 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.90
| method | result | size |
| default | \(d^{8} \left (\frac {256 x^{5} c^{6}}{5}+128 x^{4} b \,c^{5}-\frac {512 a \,c^{5} x^{3}}{3}+\frac {512 b^{2} c^{4} x^{3}}{3}-256 a b \,c^{4} x^{2}+128 b^{3} c^{3} x^{2}+768 a^{2} c^{4} x -512 a \,b^{2} c^{3} x +96 c^{2} x \,b^{4}-\frac {-2 c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x -64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}}{c \,x^{2}+b x +a}-\frac {28 c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\) | \(240\) |
| risch | \(\frac {256 c^{6} d^{8} x^{5}}{5}+128 c^{5} d^{8} b \,x^{4}-\frac {512 c^{5} d^{8} a \,x^{3}}{3}+\frac {512 c^{4} d^{8} b^{2} x^{3}}{3}-256 c^{4} d^{8} a b \,x^{2}+128 c^{3} d^{8} b^{3} x^{2}+768 c^{4} d^{8} a^{2} x -512 c^{3} d^{8} a \,b^{2} x +96 c^{2} d^{8} b^{4} x +\frac {2 c \,d^{8} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x +64 a^{3} b \,c^{3} d^{8}-48 a^{2} b^{3} c^{2} d^{8}+12 a \,b^{5} c \,d^{8}-b^{7} d^{8}}{c \,x^{2}+b x +a}+14 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{8} c \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )-14 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{8} c \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )\) | \(359\) |
Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
d^8*(256/5*x^5*c^6+128*x^4*b*c^5-512/3*a*c^5*x^3+512/3*b^2*c^4*x^3-256*a*b *c^4*x^2+128*b^3*c^3*x^2+768*a^2*c^4*x-512*a*b^2*c^3*x+96*c^2*x*b^4-(-2*c* (64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x-64*a^3*b*c^3+48*a^2*b^3*c^2-1 2*a*b^5*c+b^7)/(c*x^2+b*x+a)-28*c*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^ 6)/(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. 366 vs. \(2 (118) = 236\).
Time = 0.10 (sec) , antiderivative size = 753, normalized size of antiderivative = 5.98 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[1/15*(768*c^7*d^8*x^7 + 2688*b*c^6*d^8*x^6 + 896*(5*b^2*c^5 - 2*a*c^6)*d^ 8*x^5 + 4480*(b^3*c^4 - a*b*c^5)*d^8*x^4 + 1120*(3*b^4*c^3 - 8*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 480*(3*b^5*c^2 - 12*a*b^3*c^3 + 16*a^2*b*c^4)*d^8*x^ 2 - 30*(b^6*c - 60*a*b^4*c^2 + 304*a^2*b^2*c^3 - 448*a^3*c^4)*d^8*x - 15*( b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*d^8 + 210*((b^4*c^2 - 8* a*b^2*c^3 + 16*a^2*c^4)*d^8*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d^8 *x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*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*x^2 + b*x + a), 1/15*(768*c^7*d^8*x^7 + 2688*b*c^6*d^8*x^6 + 896*(5*b^2*c^5 - 2*a*c^6)*d^8*x^5 + 4480*(b^3*c^4 - a*b*c^5)*d^8*x^4 + 11 20*(3*b^4*c^3 - 8*a*b^2*c^4 + 8*a^2*c^5)*d^8*x^3 + 480*(3*b^5*c^2 - 12*a*b ^3*c^3 + 16*a^2*b*c^4)*d^8*x^2 - 30*(b^6*c - 60*a*b^4*c^2 + 304*a^2*b^2*c^ 3 - 448*a^3*c^4)*d^8*x - 15*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b* c^3)*d^8 - 420*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^8*x^2 + (b^5*c - 8* a*b^3*c^2 + 16*a^2*b*c^3)*d^8*x + (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*d ^8)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c )))/(c*x^2 + b*x + a)]
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (126) = 252\).
Time = 1.25 (sec) , antiderivative size = 476, normalized size of antiderivative = 3.78 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=128 b c^{5} d^{8} x^{4} + \frac {256 c^{6} d^{8} x^{5}}{5} + 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} - 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} - 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} + 14 c d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} + x^{3} \left (- \frac {512 a c^{5} d^{8}}{3} + \frac {512 b^{2} c^{4} d^{8}}{3}\right ) + x^{2} \left (- 256 a b c^{4} d^{8} + 128 b^{3} c^{3} d^{8}\right ) + x \left (768 a^{2} c^{4} d^{8} - 512 a b^{2} c^{3} d^{8} + 96 b^{4} c^{2} d^{8}\right ) + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + x \left (128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}\right )}{a + b x + c x^{2}} \] Input:
integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**2,x)
Output:
128*b*c**5*d**8*x**4 + 256*c**6*d**8*x**5/5 + 14*c*d**8*sqrt(-(4*a*c - b** 2)**5)*log(x + (224*a**2*b*c**3*d**8 - 112*a*b**3*c**2*d**8 + 14*b**5*c*d* *8 - 14*c*d**8*sqrt(-(4*a*c - b**2)**5))/(448*a**2*c**4*d**8 - 224*a*b**2* c**3*d**8 + 28*b**4*c**2*d**8)) - 14*c*d**8*sqrt(-(4*a*c - b**2)**5)*log(x + (224*a**2*b*c**3*d**8 - 112*a*b**3*c**2*d**8 + 14*b**5*c*d**8 + 14*c*d* *8*sqrt(-(4*a*c - b**2)**5))/(448*a**2*c**4*d**8 - 224*a*b**2*c**3*d**8 + 28*b**4*c**2*d**8)) + x**3*(-512*a*c**5*d**8/3 + 512*b**2*c**4*d**8/3) + x **2*(-256*a*b*c**4*d**8 + 128*b**3*c**3*d**8) + x*(768*a**2*c**4*d**8 - 51 2*a*b**2*c**3*d**8 + 96*b**4*c**2*d**8) + (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 + x*(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))/(a + b*x + c*x **2)
Exception generated. \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^2,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. 308 vs. \(2 (118) = 236\).
Time = 0.15 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.44 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=\frac {28 \, {\left (b^{6} c d^{8} - 12 \, a b^{4} c^{2} d^{8} + 48 \, a^{2} b^{2} c^{3} d^{8} - 64 \, a^{3} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{6} c d^{8} x - 24 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x - 128 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} - 12 \, a b^{5} c d^{8} + 48 \, a^{2} b^{3} c^{2} d^{8} - 64 \, a^{3} b c^{3} d^{8}}{c x^{2} + b x + a} + \frac {32 \, {\left (24 \, c^{16} d^{8} x^{5} + 60 \, b c^{15} d^{8} x^{4} + 80 \, b^{2} c^{14} d^{8} x^{3} - 80 \, a c^{15} d^{8} x^{3} + 60 \, b^{3} c^{13} d^{8} x^{2} - 120 \, a b c^{14} d^{8} x^{2} + 45 \, b^{4} c^{12} d^{8} x - 240 \, a b^{2} c^{13} d^{8} x + 360 \, a^{2} c^{14} d^{8} x\right )}}{15 \, c^{10}} \] Input:
integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
28*(b^6*c*d^8 - 12*a*b^4*c^2*d^8 + 48*a^2*b^2*c^3*d^8 - 64*a^3*c^4*d^8)*ar ctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - (2*b^6*c*d^8*x - 24*a*b^4*c^2*d^8*x + 96*a^2*b^2*c^3*d^8*x - 128*a^3*c^4*d^8*x + b^7*d^8 - 12*a*b^5*c*d^8 + 48*a^2*b^3*c^2*d^8 - 64*a^3*b*c^3*d^8)/(c*x^2 + b*x + a) + 32/15*(24*c^16*d^8*x^5 + 60*b*c^15*d^8*x^4 + 80*b^2*c^14*d^8*x^3 - 80*a *c^15*d^8*x^3 + 60*b^3*c^13*d^8*x^2 - 120*a*b*c^14*d^8*x^2 + 45*b^4*c^12*d ^8*x - 240*a*b^2*c^13*d^8*x + 360*a^2*c^14*d^8*x)/c^10
Time = 5.59 (sec) , antiderivative size = 508, normalized size of antiderivative = 4.03 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=x^2\,\left (896\,b^3\,c^3\,d^8+\frac {b\,\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )}{c}-256\,b\,c^3\,d^8\,\left (b^2+2\,a\,c\right )-256\,a\,b\,c^4\,d^8\right )-\frac {x\,\left (-128\,a^3\,c^4\,d^8+96\,a^2\,b^2\,c^3\,d^8-24\,a\,b^4\,c^2\,d^8+2\,b^6\,c\,d^8\right )+b^7\,d^8-64\,a^3\,b\,c^3\,d^8+48\,a^2\,b^3\,c^2\,d^8-12\,a\,b^5\,c\,d^8}{c\,x^2+b\,x+a}-x^3\,\left (\frac {256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )}{3}-256\,b^2\,c^4\,d^8\right )-x\,\left (\frac {2\,b\,\left (1792\,b^3\,c^3\,d^8+\frac {2\,b\,\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )}{c}-512\,b\,c^3\,d^8\,\left (b^2+2\,a\,c\right )-512\,a\,b\,c^4\,d^8\right )}{c}-\frac {\left (256\,c^4\,d^8\,\left (b^2+2\,a\,c\right )-768\,b^2\,c^4\,d^8\right )\,\left (b^2+2\,a\,c\right )}{c^2}+256\,a^2\,c^4\,d^8-1120\,b^4\,c^2\,d^8+1024\,a\,b^2\,c^3\,d^8\right )+\frac {256\,c^6\,d^8\,x^5}{5}+28\,c\,d^8\,\mathrm {atan}\left (\frac {28\,c^2\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}+14\,b\,c\,d^8\,{\left (4\,a\,c-b^2\right )}^{5/2}}{-896\,a^3\,c^4\,d^8+672\,a^2\,b^2\,c^3\,d^8-168\,a\,b^4\,c^2\,d^8+14\,b^6\,c\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}+128\,b\,c^5\,d^8\,x^4 \] Input:
int((b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^2,x)
Output:
x^2*(896*b^3*c^3*d^8 + (b*(256*c^4*d^8*(2*a*c + b^2) - 768*b^2*c^4*d^8))/c - 256*b*c^3*d^8*(2*a*c + b^2) - 256*a*b*c^4*d^8) - (x*(2*b^6*c*d^8 - 128* a^3*c^4*d^8 - 24*a*b^4*c^2*d^8 + 96*a^2*b^2*c^3*d^8) + b^7*d^8 - 64*a^3*b* c^3*d^8 + 48*a^2*b^3*c^2*d^8 - 12*a*b^5*c*d^8)/(a + b*x + c*x^2) - x^3*((2 56*c^4*d^8*(2*a*c + b^2))/3 - 256*b^2*c^4*d^8) - x*((2*b*(1792*b^3*c^3*d^8 + (2*b*(256*c^4*d^8*(2*a*c + b^2) - 768*b^2*c^4*d^8))/c - 512*b*c^3*d^8*( 2*a*c + b^2) - 512*a*b*c^4*d^8))/c - ((256*c^4*d^8*(2*a*c + b^2) - 768*b^2 *c^4*d^8)*(2*a*c + b^2))/c^2 + 256*a^2*c^4*d^8 - 1120*b^4*c^2*d^8 + 1024*a *b^2*c^3*d^8) + (256*c^6*d^8*x^5)/5 + 28*c*d^8*atan((28*c^2*d^8*x*(4*a*c - b^2)^(5/2) + 14*b*c*d^8*(4*a*c - b^2)^(5/2))/(14*b^6*c*d^8 - 896*a^3*c^4* d^8 - 168*a*b^4*c^2*d^8 + 672*a^2*b^2*c^3*d^8))*(4*a*c - b^2)^(5/2) + 128* b*c^5*d^8*x^4
Time = 0.21 (sec) , antiderivative size = 589, normalized size of antiderivative = 4.67 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx=\frac {d^{8} \left (2688 b^{2} c^{6} x^{6}+3360 b^{5} c^{3} x^{3}-13440 a^{4} c^{4}+1470 b^{6} c^{2} x^{2}+4480 b^{4} c^{4} x^{4}+4480 b^{3} c^{5} x^{5}+768 b \,c^{7} x^{7}+10080 a^{3} b^{2} c^{3}-13440 a^{3} c^{5} x^{2}-2520 a^{2} b^{4} c^{2}+210 a \,b^{6} c -15 b^{8}+16800 a^{2} b^{2} c^{4} x^{2}+8960 a^{2} b \,c^{5} x^{3}-7560 a \,b^{4} c^{3} x^{2}-8960 a \,b^{3} c^{4} x^{3}-4480 a \,b^{2} c^{5} x^{4}-1792 a b \,c^{6} x^{5}-6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{3} b \,c^{3}+3360 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{3} c^{2}-420 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{5} c -420 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{6} c x -420 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{5} c^{2} x^{2}-6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{3} x -6720 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{4} x^{2}+3360 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c^{2} x +3360 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c^{3} x^{2}\right )}{15 b \left (c \,x^{2}+b x +a \right )} \] Input:
int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^2,x)
Output:
(d**8*( - 6720*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a** 3*b*c**3 + 3360*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 - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*a*b**5*c + 3360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a* c - b**2))*a*b**4*c**2*x + 3360*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 *a*c - b**2))*a*b**3*c**3*x**2 - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s qrt(4*a*c - b**2))*b**6*c*x - 420*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt (4*a*c - b**2))*b**5*c**2*x**2 - 13440*a**4*c**4 + 10080*a**3*b**2*c**3 - 13440*a**3*c**5*x**2 - 2520*a**2*b**4*c**2 + 16800*a**2*b**2*c**4*x**2 + 8 960*a**2*b*c**5*x**3 + 210*a*b**6*c - 7560*a*b**4*c**3*x**2 - 8960*a*b**3* c**4*x**3 - 4480*a*b**2*c**5*x**4 - 1792*a*b*c**6*x**5 - 15*b**8 + 1470*b* *6*c**2*x**2 + 3360*b**5*c**3*x**3 + 4480*b**4*c**4*x**4 + 4480*b**3*c**5* x**5 + 2688*b**2*c**6*x**6 + 768*b*c**7*x**7))/(15*b*(a + b*x + c*x**2))