\(\int \frac {(b d+2 c d x)^8}{(a+b x+c x^2)^3} \, dx\) [1179]

Optimal result

Integrand size = 24, antiderivative size = 134 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac {140}{3} c^2 d^8 (b+2 c x)^3-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac {7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-140 c^2 \left (b^2-4 a c\right )^{3/2} d^8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \]

[Out]

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))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {700, 706, 632, 212} \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-\frac {7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac {140}{3} c^2 d^8 (b+2 c x)^3 \]

[In]

Int[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]

[Out]

140*c^2*(b^2 - 4*a*c)*d^8*(b + 2*c*x) + (140*c^2*d^8*(b + 2*c*x)^3)/3 - (d^8*(b + 2*c*x)^7)/(2*(a + b*x + c*x^
2)^2) - (7*c*d^8*(b + 2*c*x)^5)/(a + b*x + c*x^2) - 140*c^2*(b^2 - 4*a*c)^(3/2)*d^8*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 700

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*(d + e*x)^(m - 1)*(
(a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2
*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && 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])

Rubi steps \begin{align*} \text {integral}& = -\frac {d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\left (7 c d^2\right ) \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\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}+\left (70 c^2 d^4\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx \\ & = \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}+\left (70 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, 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}+\left (70 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {1}{a+b x+c x^2} \, 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}-\left (140 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right ) \\ & = 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 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.06 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=d^8 \left (-256 c^3 \left (-b^2+3 a c\right ) x+128 b c^4 x^2+\frac {256 c^5 x^3}{3}-\frac {\left (b^2-4 a c\right )^3 (b+2 c x)}{2 (a+x (b+c x))^2}-\frac {13 c \left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+140 c^2 \left (-b^2+4 a c\right )^{3/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x]

[Out]

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]])

Maple [A] (verified)

Time = 2.67 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.76

method	result	size
default	\(d^{8} \left (\frac {256 c^{5} x^{3}}{3}+128 b \,c^{4} x^{2}-768 a \,c^{4} x +256 b^{2} x \,c^{3}+\frac {\left (-416 a^{2} c^{5}+208 a \,b^{2} c^{4}-26 b^{4} c^{3}\right ) x^{3}-39 c^{2} b \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{2}-2 c \left (176 c^{3} a^{3}+24 a^{2} b^{2} c^{2}-45 a \,b^{4} c +7 b^{6}\right ) x -\frac {b \left (352 c^{3} a^{3}-160 a^{2} b^{2} c^{2}+14 a \,b^{4} c +b^{6}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {140 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\)	\(236\)
risch	\(\frac {256 c^{5} d^{8} x^{3}}{3}+128 c^{4} d^{8} b \,x^{2}-768 c^{4} d^{8} a x +256 c^{3} d^{8} b^{2} x +\frac {\left (-416 a^{2} d^{8} c^{5}+208 a \,b^{2} d^{8} c^{4}-26 b^{4} d^{8} c^{3}\right ) x^{3}-39 b \,c^{2} d^{8} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) x^{2}-2 c \,d^{8} \left (176 c^{3} a^{3}+24 a^{2} b^{2} c^{2}-45 a \,b^{4} c +7 b^{6}\right ) x -\frac {b \,d^{8} \left (352 c^{3} a^{3}-160 a^{2} b^{2} c^{2}+14 a \,b^{4} c +b^{6}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-70 c^{2} d^{8} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )+70 c^{2} d^{8} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )\)	\(335\)

[In]

int((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

d^8*(256/3*c^5*x^3+128*b*c^4*x^2-768*a*c^4*x+256*b^2*x*c^3+((-416*a^2*c^5+208*a*b^2*c^4-26*b^4*c^3)*x^3-39*c^2
*b*(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (126) = 252\).

Time = 0.28 (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=\left [\frac {512 \, c^{7} d^{8} x^{7} + 1792 \, b c^{6} d^{8} x^{6} + 3584 \, {\left (b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 256 \, {\left (15 \, b^{3} c^{4} - 26 \, a b c^{5}\right )} d^{8} x^{4} + 4 \, {\left (345 \, b^{4} c^{3} + 312 \, a b^{2} c^{4} - 2800 \, a^{2} c^{5}\right )} d^{8} x^{3} - 6 \, {\left (39 \, b^{5} c^{2} - 824 \, a b^{3} c^{3} + 2032 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 12 \, {\left (7 \, b^{6} c - 45 \, a b^{4} c^{2} - 104 \, a^{2} b^{2} c^{3} + 560 \, a^{3} c^{4}\right )} d^{8} x - 3 \, {\left (b^{7} + 14 \, a b^{5} c - 160 \, a^{2} b^{3} c^{2} + 352 \, a^{3} b c^{3}\right )} d^{8} - 420 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{8} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{8} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{8} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{8} x + {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d^{8}\right )} \sqrt {b^{2} - 4 \, a c} \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 )}{6 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, \frac {512 \, c^{7} d^{8} x^{7} + 1792 \, b c^{6} d^{8} x^{6} + 3584 \, {\left (b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 256 \, {\left (15 \, b^{3} c^{4} - 26 \, a b c^{5}\right )} d^{8} x^{4} + 4 \, {\left (345 \, b^{4} c^{3} + 312 \, a b^{2} c^{4} - 2800 \, a^{2} c^{5}\right )} d^{8} x^{3} - 6 \, {\left (39 \, b^{5} c^{2} - 824 \, a b^{3} c^{3} + 2032 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 12 \, {\left (7 \, b^{6} c - 45 \, a b^{4} c^{2} - 104 \, a^{2} b^{2} c^{3} + 560 \, a^{3} c^{4}\right )} d^{8} x - 3 \, {\left (b^{7} + 14 \, a b^{5} c - 160 \, a^{2} b^{3} c^{2} + 352 \, a^{3} b c^{3}\right )} d^{8} - 840 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{8} x^{4} + 2 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{8} x^{3} + {\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{8} x^{2} + 2 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{8} x + {\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d^{8}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{6 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \]

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[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 - 160*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)*sqrt(-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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (133) = 266\).

Time = 2.96 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.50 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=128 b c^{4} d^{8} x^{2} + \frac {256 c^{5} d^{8} x^{3}}{3} - 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} - 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} + 70 c^{2} d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + x \left (- 768 a c^{4} d^{8} + 256 b^{2} c^{3} d^{8}\right ) + \frac {- 352 a^{3} b c^{3} d^{8} + 160 a^{2} b^{3} c^{2} d^{8} - 14 a b^{5} c d^{8} - b^{7} d^{8} + x^{3} \left (- 832 a^{2} c^{5} d^{8} + 416 a b^{2} c^{4} d^{8} - 52 b^{4} c^{3} d^{8}\right ) + x^{2} \left (- 1248 a^{2} b c^{4} d^{8} + 624 a b^{3} c^{3} d^{8} - 78 b^{5} c^{2} d^{8}\right ) + x \left (- 704 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 180 a b^{4} c^{2} d^{8} - 28 b^{6} c d^{8}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \]

[In]

integrate((2*c*d*x+b*d)**8/(c*x**2+b*x+a)**3,x)

[Out]

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*sqrt(-(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*(-7
04*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))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (126) = 252\).

Time = 0.32 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.35 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\frac {140 \, {\left (b^{4} c^{2} d^{8} - 8 \, a b^{2} c^{3} d^{8} + 16 \, a^{2} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {52 \, b^{4} c^{3} d^{8} x^{3} - 416 \, a b^{2} c^{4} d^{8} x^{3} + 832 \, a^{2} c^{5} d^{8} x^{3} + 78 \, b^{5} c^{2} d^{8} x^{2} - 624 \, a b^{3} c^{3} d^{8} x^{2} + 1248 \, a^{2} b c^{4} d^{8} x^{2} + 28 \, b^{6} c d^{8} x - 180 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x + 704 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} + 14 \, a b^{5} c d^{8} - 160 \, a^{2} b^{3} c^{2} d^{8} + 352 \, a^{3} b c^{3} d^{8}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (2 \, c^{14} d^{8} x^{3} + 3 \, b c^{13} d^{8} x^{2} + 6 \, b^{2} c^{12} d^{8} x - 18 \, a c^{13} d^{8} x\right )}}{3 \, c^{9}} \]

[In]

integrate((2*c*d*x+b*d)^8/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

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)/sqrt(-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

Mupad [B] (verification not implemented)

Time = 9.58 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.75 \[ \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^3} \, dx=\frac {256\,c^5\,d^8\,x^3}{3}-\frac {x^2\,\left (624\,a^2\,b\,c^4\,d^8-312\,a\,b^3\,c^3\,d^8+39\,b^5\,c^2\,d^8\right )+x\,\left (352\,a^3\,c^4\,d^8+48\,a^2\,b^2\,c^3\,d^8-90\,a\,b^4\,c^2\,d^8+14\,b^6\,c\,d^8\right )+\frac {b^7\,d^8}{2}+x^3\,\left (416\,a^2\,c^5\,d^8-208\,a\,b^2\,c^4\,d^8+26\,b^4\,c^3\,d^8\right )+176\,a^3\,b\,c^3\,d^8-80\,a^2\,b^3\,c^2\,d^8+7\,a\,b^5\,c\,d^8}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-x\,\left (768\,c^3\,d^8\,\left (b^2+a\,c\right )-1024\,b^2\,c^3\,d^8\right )+128\,b\,c^4\,d^8\,x^2+140\,c^2\,d^8\,\mathrm {atan}\left (\frac {70\,b\,c^2\,d^8\,{\left (4\,a\,c-b^2\right )}^{3/2}+140\,c^3\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{3/2}}{1120\,a^2\,c^4\,d^8-560\,a\,b^2\,c^3\,d^8+70\,b^4\,c^2\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{3/2} \]

[In]

int((b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3,x)

[Out]

(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 - 56
0*a*b^2*c^3*d^8))*(4*a*c - b^2)^(3/2)