\(\int \frac {(b d+2 c d x)^{10}}{(a+b x+c x^2)^3} \, dx\) [1177]

Optimal result

Integrand size = 24, antiderivative size = 160 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-252 c^2 \left (b^2-4 a c\right )^{5/2} d^{10} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \]

[Out]

252*c^2*(-4*a*c+b^2)^2*d^10*(2*c*x+b)+84*c^2*(-4*a*c+b^2)*d^10*(2*c*x+b)^3+252/5*c^2*d^10*(2*c*x+b)^5-1/2*d^10
*(2*c*x+b)^9/(c*x^2+b*x+a)^2-9*c*d^10*(2*c*x+b)^7/(c*x^2+b*x+a)-252*c^2*(-4*a*c+b^2)^(5/2)*d^10*arctanh((2*c*x
+b)/(-4*a*c+b^2)^(1/2))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {700, 706, 632, 212} \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=-252 c^2 d^{10} \left (b^2-4 a c\right )^{5/2} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+84 c^2 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^3+252 c^2 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\frac {252}{5} c^2 d^{10} (b+2 c x)^5 \]

[In]

Int[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

252*c^2*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x) + 84*c^2*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^3 + (252*c^2*d^10*(b + 2*c*x)
^5)/5 - (d^10*(b + 2*c*x)^9)/(2*(a + b*x + c*x^2)^2) - (9*c*d^10*(b + 2*c*x)^7)/(a + b*x + c*x^2) - 252*c^2*(b
^2 - 4*a*c)^(5/2)*d^10*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^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}+\left (9 c d^2\right ) \int \frac {(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 d^4\right ) \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx \\ & = \frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx \\ & = 84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx \\ & = 252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}+\left (126 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \int \frac {1}{a+b x+c x^2} \, dx \\ & = 252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-\left (252 c^2 \left (b^2-4 a c\right )^3 d^{10}\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right ) \\ & = 252 c^2 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)+84 c^2 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^3+\frac {252}{5} c^2 d^{10} (b+2 c x)^5-\frac {d^{10} (b+2 c x)^9}{2 \left (a+b x+c x^2\right )^2}-\frac {9 c d^{10} (b+2 c x)^7}{a+b x+c x^2}-252 c^2 \left (b^2-4 a c\right )^{5/2} d^{10} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.20 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=d^{10} \left (128 c^3 \left (5 b^4-30 a b^2 c+48 a^2 c^2\right ) x+128 b c^4 \left (5 b^2-12 a c\right ) x^2-256 c^5 \left (-3 b^2+4 a c\right ) x^3+512 b c^6 x^4+\frac {1024 c^7 x^5}{5}-\frac {\left (b^2-4 a c\right )^4 (b+2 c x)}{2 (a+x (b+c x))^2}+\frac {17 c \left (-b^2+4 a c\right )^3 (b+2 c x)}{a+x (b+c x)}-252 c^2 \left (-b^2+4 a c\right )^{5/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x]

[Out]

d^10*(128*c^3*(5*b^4 - 30*a*b^2*c + 48*a^2*c^2)*x + 128*b*c^4*(5*b^2 - 12*a*c)*x^2 - 256*c^5*(-3*b^2 + 4*a*c)*
x^3 + 512*b*c^6*x^4 + (1024*c^7*x^5)/5 - ((b^2 - 4*a*c)^4*(b + 2*c*x))/(2*(a + x*(b + c*x))^2) + (17*c*(-b^2 +
4*a*c)^3*(b + 2*c*x))/(a + x*(b + c*x)) - 252*c^2*(-b^2 + 4*a*c)^(5/2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]]
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(152)=304\).

Time = 2.63 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.19

method	result	size
default	\(d^{10} \left (\frac {1024 c^{7} x^{5}}{5}+512 b \,c^{6} x^{4}-1024 x^{3} a \,c^{6}+768 x^{3} b^{2} c^{5}-1536 x^{2} a b \,c^{5}+640 x^{2} b^{3} c^{4}+6144 a^{2} c^{5} x -3840 a \,b^{2} c^{4} x +640 b^{4} c^{3} x -\frac {\left (-2176 a^{3} c^{6}+1632 a^{2} b^{2} c^{5}-408 a \,b^{4} c^{4}+34 b^{6} c^{3}\right ) x^{3}-51 b \,c^{2} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{2}-6 c \left (320 a^{4} c^{4}-48 a^{3} b^{2} c^{3}-84 a^{2} b^{4} c^{2}+31 a \,b^{6} c -3 b^{8}\right ) x -\frac {b \left (1920 a^{4} c^{4}-1376 a^{3} b^{2} c^{3}+312 a^{2} b^{4} c^{2}-18 a \,b^{6} c -b^{8}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {252 c^{2} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\)	\(351\)
risch	\(\frac {1024 c^{7} d^{10} x^{5}}{5}+512 c^{6} d^{10} b \,x^{4}-1024 c^{6} d^{10} x^{3} a +768 c^{5} d^{10} b^{2} x^{3}-1536 c^{5} d^{10} a b \,x^{2}+640 c^{4} d^{10} b^{3} x^{2}+6144 c^{5} d^{10} a^{2} x -3840 c^{4} d^{10} a \,b^{2} x +640 c^{3} d^{10} b^{4} x +\frac {\left (2176 a^{3} d^{10} c^{6}-1632 a^{2} b^{2} d^{10} c^{5}+408 a \,b^{4} d^{10} c^{4}-34 b^{6} d^{10} c^{3}\right ) x^{3}+51 b \,c^{2} d^{10} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{2}+6 c \,d^{10} \left (320 a^{4} c^{4}-48 a^{3} b^{2} c^{3}-84 a^{2} b^{4} c^{2}+31 a \,b^{6} c -3 b^{8}\right ) x +\frac {b \,d^{10} \left (1920 a^{4} c^{4}-1376 a^{3} b^{2} c^{3}+312 a^{2} b^{4} c^{2}-18 a \,b^{6} c -b^{8}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+126 c^{2} d^{10} \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 c^{3} a^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )-126 c^{2} d^{10} \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}} b -64 c^{3} a^{3}+48 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right )\)	\(476\)

[In]

int((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

d^10*(1024/5*c^7*x^5+512*b*c^6*x^4-1024*x^3*a*c^6+768*x^3*b^2*c^5-1536*x^2*a*b*c^5+640*x^2*b^3*c^4+6144*a^2*c^
5*x-3840*a*b^2*c^4*x+640*b^4*c^3*x-((-2176*a^3*c^6+1632*a^2*b^2*c^5-408*a*b^4*c^4+34*b^6*c^3)*x^3-51*b*c^2*(64
*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2-6*c*(320*a^4*c^4-48*a^3*b^2*c^3-84*a^2*b^4*c^2+31*a*b^6*c-3*b^8)*x
-1/2*b*(1920*a^4*c^4-1376*a^3*b^2*c^3+312*a^2*b^4*c^2-18*a*b^6*c-b^8))/(c*x^2+b*x+a)^2-252*c^2*(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 582 vs. \(2 (152) = 304\).

Time = 0.36 (sec) , antiderivative size = 1185, normalized size of antiderivative = 7.41 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^10*x^7 + 5376*(5*b^3*c^6 - 4*a*
b*c^7)*d^10*x^6 + 5376*(5*b^4*c^5 - 10*a*b^2*c^6 + 8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a*b^3*c^5 + 12*a
^2*b*c^6)*d^10*x^4 + 20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 10*(51*b^7*
c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3*c^4 - 14016*a^3*b*c^5)*d^10*x^2 - 20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2*b^4*
c^3 + 2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2 + 1376*a^3*b^3*c^3 - 192
0*a^4*b*c^4)*d^10 + 1260*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*
c^5)*d^10*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d
^10*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^10)*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), 1/10*(2048*c^9*d^10*x^9 + 9216*b*c^8*d^10*x^8 + 1536*(13*b^2*c^7 - 4*a*c^8)*d^10*x^7 + 5376*(5*b^3*c^6 -
4*a*b*c^7)*d^10*x^6 + 5376*(5*b^4*c^5 - 10*a*b^2*c^6 + 8*a^2*c^7)*d^10*x^5 + 6400*(3*b^5*c^4 - 10*a*b^3*c^5 +
12*a^2*b*c^6)*d^10*x^4 + 20*(303*b^6*c^3 - 436*a*b^4*c^4 - 2736*a^2*b^2*c^5 + 6720*a^3*c^6)*d^10*x^3 - 10*(51
*b^7*c^2 - 1892*a*b^5*c^3 + 9488*a^2*b^3*c^4 - 14016*a^3*b*c^5)*d^10*x^2 - 20*(9*b^8*c - 93*a*b^6*c^2 - 68*a^2
*b^4*c^3 + 2064*a^3*b^2*c^4 - 4032*a^4*c^5)*d^10*x - 5*(b^9 + 18*a*b^7*c - 312*a^2*b^5*c^2 + 1376*a^3*b^3*c^3
- 1920*a^4*b*c^4)*d^10 - 2520*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^10*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a
^2*b*c^5)*d^10*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^10*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c
^4)*d^10*x + (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^10)*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. 660 vs. \(2 (160) = 320\).

Time = 6.02 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.12 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=512 b c^{6} d^{10} x^{4} + \frac {1024 c^{7} d^{10} x^{5}}{5} + 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} - 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} - 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}} \log {\left (x + \frac {2016 a^{2} b c^{4} d^{10} - 1008 a b^{3} c^{3} d^{10} + 126 b^{5} c^{2} d^{10} + 126 c^{2} d^{10} \sqrt {- \left (4 a c - b^{2}\right )^{5}}}{4032 a^{2} c^{5} d^{10} - 2016 a b^{2} c^{4} d^{10} + 252 b^{4} c^{3} d^{10}} \right )} + x^{3} \left (- 1024 a c^{6} d^{10} + 768 b^{2} c^{5} d^{10}\right ) + x^{2} \left (- 1536 a b c^{5} d^{10} + 640 b^{3} c^{4} d^{10}\right ) + x \left (6144 a^{2} c^{5} d^{10} - 3840 a b^{2} c^{4} d^{10} + 640 b^{4} c^{3} d^{10}\right ) + \frac {1920 a^{4} b c^{4} d^{10} - 1376 a^{3} b^{3} c^{3} d^{10} + 312 a^{2} b^{5} c^{2} d^{10} - 18 a b^{7} c d^{10} - b^{9} d^{10} + x^{3} \cdot \left (4352 a^{3} c^{6} d^{10} - 3264 a^{2} b^{2} c^{5} d^{10} + 816 a b^{4} c^{4} d^{10} - 68 b^{6} c^{3} d^{10}\right ) + x^{2} \cdot \left (6528 a^{3} b c^{5} d^{10} - 4896 a^{2} b^{3} c^{4} d^{10} + 1224 a b^{5} c^{3} d^{10} - 102 b^{7} c^{2} d^{10}\right ) + x \left (3840 a^{4} c^{5} d^{10} - 576 a^{3} b^{2} c^{4} d^{10} - 1008 a^{2} b^{4} c^{3} d^{10} + 372 a b^{6} c^{2} d^{10} - 36 b^{8} c d^{10}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \]

[In]

integrate((2*c*d*x+b*d)**10/(c*x**2+b*x+a)**3,x)

[Out]

512*b*c**6*d**10*x**4 + 1024*c**7*d**10*x**5/5 + 126*c**2*d**10*sqrt(-(4*a*c - b**2)**5)*log(x + (2016*a**2*b*
c**4*d**10 - 1008*a*b**3*c**3*d**10 + 126*b**5*c**2*d**10 - 126*c**2*d**10*sqrt(-(4*a*c - b**2)**5))/(4032*a**
2*c**5*d**10 - 2016*a*b**2*c**4*d**10 + 252*b**4*c**3*d**10)) - 126*c**2*d**10*sqrt(-(4*a*c - b**2)**5)*log(x
+ (2016*a**2*b*c**4*d**10 - 1008*a*b**3*c**3*d**10 + 126*b**5*c**2*d**10 + 126*c**2*d**10*sqrt(-(4*a*c - b**2)
**5))/(4032*a**2*c**5*d**10 - 2016*a*b**2*c**4*d**10 + 252*b**4*c**3*d**10)) + x**3*(-1024*a*c**6*d**10 + 768*
b**2*c**5*d**10) + x**2*(-1536*a*b*c**5*d**10 + 640*b**3*c**4*d**10) + x*(6144*a**2*c**5*d**10 - 3840*a*b**2*c
**4*d**10 + 640*b**4*c**3*d**10) + (1920*a**4*b*c**4*d**10 - 1376*a**3*b**3*c**3*d**10 + 312*a**2*b**5*c**2*d*
*10 - 18*a*b**7*c*d**10 - b**9*d**10 + x**3*(4352*a**3*c**6*d**10 - 3264*a**2*b**2*c**5*d**10 + 816*a*b**4*c**
4*d**10 - 68*b**6*c**3*d**10) + x**2*(6528*a**3*b*c**5*d**10 - 4896*a**2*b**3*c**4*d**10 + 1224*a*b**5*c**3*d*
*10 - 102*b**7*c**2*d**10) + x*(3840*a**4*c**5*d**10 - 576*a**3*b**2*c**4*d**10 - 1008*a**2*b**4*c**3*d**10 +
372*a*b**6*c**2*d**10 - 36*b**8*c*d**10))/(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. \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((2*c*d*x+b*d)^10/(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. 460 vs. \(2 (152) = 304\).

Time = 0.31 (sec) , antiderivative size = 460, normalized size of antiderivative = 2.88 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {252 \, {\left (b^{6} c^{2} d^{10} - 12 \, a b^{4} c^{3} d^{10} + 48 \, a^{2} b^{2} c^{4} d^{10} - 64 \, a^{3} c^{5} d^{10}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {68 \, b^{6} c^{3} d^{10} x^{3} - 816 \, a b^{4} c^{4} d^{10} x^{3} + 3264 \, a^{2} b^{2} c^{5} d^{10} x^{3} - 4352 \, a^{3} c^{6} d^{10} x^{3} + 102 \, b^{7} c^{2} d^{10} x^{2} - 1224 \, a b^{5} c^{3} d^{10} x^{2} + 4896 \, a^{2} b^{3} c^{4} d^{10} x^{2} - 6528 \, a^{3} b c^{5} d^{10} x^{2} + 36 \, b^{8} c d^{10} x - 372 \, a b^{6} c^{2} d^{10} x + 1008 \, a^{2} b^{4} c^{3} d^{10} x + 576 \, a^{3} b^{2} c^{4} d^{10} x - 3840 \, a^{4} c^{5} d^{10} x + b^{9} d^{10} + 18 \, a b^{7} c d^{10} - 312 \, a^{2} b^{5} c^{2} d^{10} + 1376 \, a^{3} b^{3} c^{3} d^{10} - 1920 \, a^{4} b c^{4} d^{10}}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (8 \, c^{22} d^{10} x^{5} + 20 \, b c^{21} d^{10} x^{4} + 30 \, b^{2} c^{20} d^{10} x^{3} - 40 \, a c^{21} d^{10} x^{3} + 25 \, b^{3} c^{19} d^{10} x^{2} - 60 \, a b c^{20} d^{10} x^{2} + 25 \, b^{4} c^{18} d^{10} x - 150 \, a b^{2} c^{19} d^{10} x + 240 \, a^{2} c^{20} d^{10} x\right )}}{5 \, c^{15}} \]

[In]

integrate((2*c*d*x+b*d)^10/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

252*(b^6*c^2*d^10 - 12*a*b^4*c^3*d^10 + 48*a^2*b^2*c^4*d^10 - 64*a^3*c^5*d^10)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(68*b^6*c^3*d^10*x^3 - 816*a*b^4*c^4*d^10*x^3 + 3264*a^2*b^2*c^5*d^10*x^3 - 4
352*a^3*c^6*d^10*x^3 + 102*b^7*c^2*d^10*x^2 - 1224*a*b^5*c^3*d^10*x^2 + 4896*a^2*b^3*c^4*d^10*x^2 - 6528*a^3*b
*c^5*d^10*x^2 + 36*b^8*c*d^10*x - 372*a*b^6*c^2*d^10*x + 1008*a^2*b^4*c^3*d^10*x + 576*a^3*b^2*c^4*d^10*x - 38
40*a^4*c^5*d^10*x + b^9*d^10 + 18*a*b^7*c*d^10 - 312*a^2*b^5*c^2*d^10 + 1376*a^3*b^3*c^3*d^10 - 1920*a^4*b*c^4
*d^10)/(c*x^2 + b*x + a)^2 + 128/5*(8*c^22*d^10*x^5 + 20*b*c^21*d^10*x^4 + 30*b^2*c^20*d^10*x^3 - 40*a*c^21*d^
10*x^3 + 25*b^3*c^19*d^10*x^2 - 60*a*b*c^20*d^10*x^2 + 25*b^4*c^18*d^10*x - 150*a*b^2*c^19*d^10*x + 240*a^2*c^
20*d^10*x)/c^15

Mupad [B] (verification not implemented)

Time = 9.61 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.34 \[ \int \frac {(b d+2 c d x)^{10}}{\left (a+b x+c x^2\right )^3} \, dx=x\,\left (\frac {3\,b\,\left (1024\,c^4\,d^{10}\,\left (b^3+6\,a\,c\,b\right )-15360\,b^3\,c^4\,d^{10}-\frac {3\,b\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )}{c}+6144\,b\,c^4\,d^{10}\,\left (b^2+a\,c\right )\right )}{c}+\frac {3\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )\,\left (b^2+a\,c\right )}{c^2}+13440\,b^4\,c^3\,d^{10}-3072\,a\,c^4\,d^{10}\,\left (b^2+a\,c\right )-2048\,b\,c^3\,d^{10}\,\left (b^3+6\,a\,c\,b\right )\right )-x^2\,\left (512\,c^4\,d^{10}\,\left (b^3+6\,a\,c\,b\right )-7680\,b^3\,c^4\,d^{10}-\frac {3\,b\,\left (3072\,c^5\,d^{10}\,\left (b^2+a\,c\right )-5376\,b^2\,c^5\,d^{10}\right )}{2\,c}+3072\,b\,c^4\,d^{10}\,\left (b^2+a\,c\right )\right )-x^3\,\left (1024\,c^5\,d^{10}\,\left (b^2+a\,c\right )-1792\,b^2\,c^5\,d^{10}\right )-\frac {x^2\,\left (-3264\,a^3\,b\,c^5\,d^{10}+2448\,a^2\,b^3\,c^4\,d^{10}-612\,a\,b^5\,c^3\,d^{10}+51\,b^7\,c^2\,d^{10}\right )-x^3\,\left (2176\,a^3\,c^6\,d^{10}-1632\,a^2\,b^2\,c^5\,d^{10}+408\,a\,b^4\,c^4\,d^{10}-34\,b^6\,c^3\,d^{10}\right )+\frac {b^9\,d^{10}}{2}+x\,\left (-1920\,a^4\,c^5\,d^{10}+288\,a^3\,b^2\,c^4\,d^{10}+504\,a^2\,b^4\,c^3\,d^{10}-186\,a\,b^6\,c^2\,d^{10}+18\,b^8\,c\,d^{10}\right )-960\,a^4\,b\,c^4\,d^{10}-156\,a^2\,b^5\,c^2\,d^{10}+688\,a^3\,b^3\,c^3\,d^{10}+9\,a\,b^7\,c\,d^{10}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {1024\,c^7\,d^{10}\,x^5}{5}+512\,b\,c^6\,d^{10}\,x^4-252\,c^2\,d^{10}\,\mathrm {atan}\left (\frac {126\,b\,c^2\,d^{10}\,{\left (4\,a\,c-b^2\right )}^{5/2}+252\,c^3\,d^{10}\,x\,{\left (4\,a\,c-b^2\right )}^{5/2}}{8064\,a^3\,c^5\,d^{10}-6048\,a^2\,b^2\,c^4\,d^{10}+1512\,a\,b^4\,c^3\,d^{10}-126\,b^6\,c^2\,d^{10}}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2} \]

[In]

int((b*d + 2*c*d*x)^10/(a + b*x + c*x^2)^3,x)

[Out]

x*((3*b*(1024*c^4*d^10*(b^3 + 6*a*b*c) - 15360*b^3*c^4*d^10 - (3*b*(3072*c^5*d^10*(a*c + b^2) - 5376*b^2*c^5*d
^10))/c + 6144*b*c^4*d^10*(a*c + b^2)))/c + (3*(3072*c^5*d^10*(a*c + b^2) - 5376*b^2*c^5*d^10)*(a*c + b^2))/c^
2 + 13440*b^4*c^3*d^10 - 3072*a*c^4*d^10*(a*c + b^2) - 2048*b*c^3*d^10*(b^3 + 6*a*b*c)) - x^2*(512*c^4*d^10*(b
^3 + 6*a*b*c) - 7680*b^3*c^4*d^10 - (3*b*(3072*c^5*d^10*(a*c + b^2) - 5376*b^2*c^5*d^10))/(2*c) + 3072*b*c^4*d
^10*(a*c + b^2)) - x^3*(1024*c^5*d^10*(a*c + b^2) - 1792*b^2*c^5*d^10) - (x^2*(51*b^7*c^2*d^10 - 612*a*b^5*c^3
*d^10 - 3264*a^3*b*c^5*d^10 + 2448*a^2*b^3*c^4*d^10) - x^3*(2176*a^3*c^6*d^10 - 34*b^6*c^3*d^10 + 408*a*b^4*c^
4*d^10 - 1632*a^2*b^2*c^5*d^10) + (b^9*d^10)/2 + x*(18*b^8*c*d^10 - 1920*a^4*c^5*d^10 - 186*a*b^6*c^2*d^10 + 5
04*a^2*b^4*c^3*d^10 + 288*a^3*b^2*c^4*d^10) - 960*a^4*b*c^4*d^10 - 156*a^2*b^5*c^2*d^10 + 688*a^3*b^3*c^3*d^10
+ 9*a*b^7*c*d^10)/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + (1024*c^7*d^10*x^5)/5 + 512*b*c
^6*d^10*x^4 - 252*c^2*d^10*atan((126*b*c^2*d^10*(4*a*c - b^2)^(5/2) + 252*c^3*d^10*x*(4*a*c - b^2)^(5/2))/(806
4*a^3*c^5*d^10 - 126*b^6*c^2*d^10 + 1512*a*b^4*c^3*d^10 - 6048*a^2*b^2*c^4*d^10))*(4*a*c - b^2)^(5/2)