\(\int \frac {(b d+2 c d x)^9}{(a+b x+c x^2)^3} \, dx\) [1178]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+96 c^2 \left (b^2-4 a c\right )^2 d^9 \log \left (a+b x+c x^2\right ) \]

[Out]

96*c^2*(-4*a*c+b^2)*d^9*(2*c*x+b)^2+48*c^2*d^9*(2*c*x+b)^4-1/2*d^9*(2*c*x+b)^8/(c*x^2+b*x+a)^2-8*c*d^9*(2*c*x+
b)^6/(c*x^2+b*x+a)+96*c^2*(-4*a*c+b^2)^2*d^9*ln(c*x^2+b*x+a)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {700, 706, 642} \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4 \]

[In]

Int[(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x]

[Out]

96*c^2*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^2 + 48*c^2*d^9*(b + 2*c*x)^4 - (d^9*(b + 2*c*x)^8)/(2*(a + b*x + c*x^2)^2
) - (8*c*d^9*(b + 2*c*x)^6)/(a + b*x + c*x^2) + 96*c^2*(b^2 - 4*a*c)^2*d^9*Log[a + b*x + c*x^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+\left (8 c d^2\right ) \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx \\ & = -\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 d^4\right ) \int \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx \\ & = 48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx \\ & = 96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx \\ & = 96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+96 c^2 \left (b^2-4 a c\right )^2 d^9 \log \left (a+b x+c x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=d^9 \left (256 b c^3 \left (b^2-3 a c\right ) x-384 c^4 \left (-b^2+2 a c\right ) x^2+256 b c^5 x^3+128 c^6 x^4-\frac {\left (b^2-4 a c\right )^4}{2 (a+x (b+c x))^2}+\frac {16 c \left (-b^2+4 a c\right )^3}{a+x (b+c x)}+96 c^2 \left (b^2-4 a c\right )^2 \log (a+x (b+c x))\right ) \]

[In]

Integrate[(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x]

[Out]

d^9*(256*b*c^3*(b^2 - 3*a*c)*x - 384*c^4*(-b^2 + 2*a*c)*x^2 + 256*b*c^5*x^3 + 128*c^6*x^4 - (b^2 - 4*a*c)^4/(2
*(a + x*(b + c*x))^2) + (16*c*(-b^2 + 4*a*c)^3)/(a + x*(b + c*x)) + 96*c^2*(b^2 - 4*a*c)^2*Log[a + x*(b + c*x)
])

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.85

method	result	size
default	\(d^{9} \left (128 c^{6} x^{4}+256 b \,c^{5} x^{3}-768 a \,c^{5} x^{2}+384 b^{2} c^{4} x^{2}-768 a b \,c^{4} x +256 b^{3} c^{3} x +\frac {16 c^{2} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{2}+16 b c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x +896 a^{4} c^{4}-640 a^{3} b^{2} c^{3}+144 a^{2} b^{4} c^{2}-8 a \,b^{6} c -\frac {b^{8}}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+96 c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\)	\(228\)
risch	\(128 c^{6} d^{9} x^{4}+256 b \,c^{5} d^{9} x^{3}-768 a \,c^{5} d^{9} x^{2}+384 b^{2} c^{4} d^{9} x^{2}-768 a b \,c^{4} d^{9} x +256 b^{3} c^{3} d^{9} x +1152 a^{2} d^{9} c^{4}-768 a \,b^{2} d^{9} c^{3}+128 b^{4} d^{9} c^{2}+\frac {16 c^{2} d^{9} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{2}+16 b c \,d^{9} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x +\frac {d^{9} \left (1792 a^{4} c^{4}-1280 a^{3} b^{2} c^{3}+288 a^{2} b^{4} c^{2}-16 a \,b^{6} c -b^{8}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+1536 \ln \left (c \,x^{2}+b x +a \right ) a^{2} c^{4} d^{9}-768 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{2} c^{3} d^{9}+96 \ln \left (c \,x^{2}+b x +a \right ) b^{4} c^{2} d^{9}\)	\(320\)
norman	\(\frac {\left (-512 a \,d^{9} c^{7}+1024 b^{2} d^{9} c^{6}\right ) x^{6}+\frac {\left (3072 a^{3} c^{7} d^{9}+1536 a^{2} b^{2} c^{6} d^{9}-64 a \,b^{4} c^{5} d^{9}-912 b^{6} c^{4} d^{9}\right ) x^{2}}{c^{2}}+\frac {4608 a^{4} c^{6} d^{9}+768 a^{3} b^{2} c^{5} d^{9}-1504 a^{2} b^{4} c^{4} d^{9}-16 a \,b^{6} c^{3} d^{9}-b^{8} c^{2} d^{9}}{2 c^{2}}+128 d^{9} c^{8} x^{8}+512 b \,d^{9} c^{7} x^{7}+\frac {2 b \left (1280 a \,b^{2} c^{5} d^{9}-768 b^{4} c^{4} d^{9}\right ) x^{3}}{c}+\frac {2 b \left (1536 a^{3} c^{6} d^{9}+768 a^{2} b^{2} c^{5} d^{9}-800 a \,b^{4} c^{4} d^{9}-8 b^{6} c^{3} d^{9}\right ) x}{c^{2}}-256 b \,c^{5} d^{9} \left (6 a c -5 b^{2}\right ) x^{5}}{\left (c \,x^{2}+b x +a \right )^{2}}+\left (1536 a^{2} d^{9} c^{4}-768 a \,b^{2} d^{9} c^{3}+96 b^{4} d^{9} c^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )\)	\(346\)
parallelrisch	\(\frac {-1536 \ln \left (c \,x^{2}+b x +a \right ) x^{4} a \,b^{2} c^{7} d^{9}+6144 \ln \left (c \,x^{2}+b x +a \right ) x^{3} a^{2} b \,c^{7} d^{9}-3072 \ln \left (c \,x^{2}+b x +a \right ) x^{3} a \,b^{3} c^{6} d^{9}-1152 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a \,b^{4} c^{5} d^{9}+6144 \ln \left (c \,x^{2}+b x +a \right ) x \,a^{3} b \,c^{6} d^{9}-3072 \ln \left (c \,x^{2}+b x +a \right ) x \,a^{2} b^{3} c^{5} d^{9}+384 \ln \left (c \,x^{2}+b x +a \right ) x a \,b^{5} c^{4} d^{9}+4608 a^{4} c^{6} d^{9}+1024 b \,d^{9} c^{9} x^{7}-1024 x^{6} a \,c^{9} d^{9}+2048 x^{6} b^{2} c^{8} d^{9}+2560 x^{5} b^{3} c^{7} d^{9}-32 x \,b^{7} c^{3} d^{9}+6144 x^{2} a^{3} c^{7} d^{9}-1824 x^{2} b^{6} c^{4} d^{9}-3072 x^{3} b^{5} c^{5} d^{9}+3072 \ln \left (c \,x^{2}+b x +a \right ) a^{4} c^{6} d^{9}-b^{8} c^{2} d^{9}-16 a \,b^{6} c^{3} d^{9}+768 a^{3} b^{2} c^{5} d^{9}-1504 a^{2} b^{4} c^{4} d^{9}+256 d^{9} c^{10} x^{8}-3072 x^{5} a b \,c^{8} d^{9}+6144 x \,a^{3} b \,c^{6} d^{9}+3072 x \,a^{2} b^{3} c^{5} d^{9}-3200 x a \,b^{5} c^{4} d^{9}+3072 x^{2} a^{2} b^{2} c^{6} d^{9}-128 x^{2} a \,b^{4} c^{5} d^{9}+5120 x^{3} a \,b^{3} c^{6} d^{9}+192 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b^{4} c^{4} d^{9}+3072 \ln \left (c \,x^{2}+b x +a \right ) x^{4} a^{2} c^{8} d^{9}+192 \ln \left (c \,x^{2}+b x +a \right ) x^{4} b^{4} c^{6} d^{9}+384 \ln \left (c \,x^{2}+b x +a \right ) x^{3} b^{5} c^{5} d^{9}+6144 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a^{3} c^{7} d^{9}+192 \ln \left (c \,x^{2}+b x +a \right ) x^{2} b^{6} c^{4} d^{9}-1536 \ln \left (c \,x^{2}+b x +a \right ) a^{3} b^{2} c^{5} d^{9}}{2 c^{2} \left (c \,x^{2}+b x +a \right )^{2}}\)	\(674\)

[In]

int((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

d^9*(128*c^6*x^4+256*b*c^5*x^3-768*a*c^5*x^2+384*b^2*c^4*x^2-768*a*b*c^4*x+256*b^3*c^3*x+(16*c^2*(64*a^3*c^3-4
8*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2+16*b*c*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+896*a^4*c^4-640*a^3*b^2*
c^3+144*a^2*b^4*c^2-8*a*b^6*c-1/2*b^8)/(c*x^2+b*x+a)^2+96*c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^2+b*x+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (121) = 242\).

Time = 0.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.98 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=\frac {256 \, c^{8} d^{9} x^{8} + 1024 \, b c^{7} d^{9} x^{7} + 1024 \, {\left (2 \, b^{2} c^{6} - a c^{7}\right )} d^{9} x^{6} + 512 \, {\left (5 \, b^{3} c^{5} - 6 \, a b c^{6}\right )} d^{9} x^{5} + 256 \, {\left (7 \, b^{4} c^{4} - 8 \, a b^{2} c^{5} - 11 \, a^{2} c^{6}\right )} d^{9} x^{4} + 512 \, {\left (b^{5} c^{3} + 2 \, a b^{3} c^{4} - 11 \, a^{2} b c^{5}\right )} d^{9} x^{3} - 32 \, {\left (b^{6} c^{2} - 44 \, a b^{4} c^{3} + 120 \, a^{2} b^{2} c^{4} - 16 \, a^{3} c^{5}\right )} d^{9} x^{2} - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b c^{4}\right )} d^{9} x - {\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9} + 192 \, {\left ({\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{9} x^{4} + 2 \, {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{9} x^{3} + {\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} d^{9} x^{2} + 2 \, {\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} d^{9} x + {\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} d^{9}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(256*c^8*d^9*x^8 + 1024*b*c^7*d^9*x^7 + 1024*(2*b^2*c^6 - a*c^7)*d^9*x^6 + 512*(5*b^3*c^5 - 6*a*b*c^6)*d^9
*x^5 + 256*(7*b^4*c^4 - 8*a*b^2*c^5 - 11*a^2*c^6)*d^9*x^4 + 512*(b^5*c^3 + 2*a*b^3*c^4 - 11*a^2*b*c^5)*d^9*x^3
- 32*(b^6*c^2 - 44*a*b^4*c^3 + 120*a^2*b^2*c^4 - 16*a^3*c^5)*d^9*x^2 - 32*(b^7*c - 12*a*b^5*c^2 + 32*a^2*b^3*
c^3 - 16*a^3*b*c^4)*d^9*x - (b^8 + 16*a*b^6*c - 288*a^2*b^4*c^2 + 1280*a^3*b^2*c^3 - 1792*a^4*c^4)*d^9 + 192*(
(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^9*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^9*x^3 + (b^6*c^2 - 6
*a*b^4*c^3 + 32*a^3*c^5)*d^9*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^9*x + (a^2*b^4*c^2 - 8*a^3*b
^2*c^3 + 16*a^4*c^4)*d^9)*log(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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (122) = 244\).

Time = 22.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.60 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=256 b c^{5} d^{9} x^{3} + 128 c^{6} d^{9} x^{4} + 96 c^{2} d^{9} \left (4 a c - b^{2}\right )^{2} \log {\left (a + b x + c x^{2} \right )} + x^{2} \left (- 768 a c^{5} d^{9} + 384 b^{2} c^{4} d^{9}\right ) + x \left (- 768 a b c^{4} d^{9} + 256 b^{3} c^{3} d^{9}\right ) + \frac {1792 a^{4} c^{4} d^{9} - 1280 a^{3} b^{2} c^{3} d^{9} + 288 a^{2} b^{4} c^{2} d^{9} - 16 a b^{6} c d^{9} - b^{8} d^{9} + x^{2} \cdot \left (2048 a^{3} c^{5} d^{9} - 1536 a^{2} b^{2} c^{4} d^{9} + 384 a b^{4} c^{3} d^{9} - 32 b^{6} c^{2} d^{9}\right ) + x \left (2048 a^{3} b c^{4} d^{9} - 1536 a^{2} b^{3} c^{3} d^{9} + 384 a b^{5} c^{2} d^{9} - 32 b^{7} c d^{9}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \]

[In]

integrate((2*c*d*x+b*d)**9/(c*x**2+b*x+a)**3,x)

[Out]

256*b*c**5*d**9*x**3 + 128*c**6*d**9*x**4 + 96*c**2*d**9*(4*a*c - b**2)**2*log(a + b*x + c*x**2) + x**2*(-768*
a*c**5*d**9 + 384*b**2*c**4*d**9) + x*(-768*a*b*c**4*d**9 + 256*b**3*c**3*d**9) + (1792*a**4*c**4*d**9 - 1280*
a**3*b**2*c**3*d**9 + 288*a**2*b**4*c**2*d**9 - 16*a*b**6*c*d**9 - b**8*d**9 + x**2*(2048*a**3*c**5*d**9 - 153
6*a**2*b**2*c**4*d**9 + 384*a*b**4*c**3*d**9 - 32*b**6*c**2*d**9) + x*(2048*a**3*b*c**4*d**9 - 1536*a**2*b**3*
c**3*d**9 + 384*a*b**5*c**2*d**9 - 32*b**7*c*d**9))/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (121) = 242\).

Time = 0.20 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.26 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=128 \, c^{6} d^{9} x^{4} + 256 \, b c^{5} d^{9} x^{3} + 384 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{9} x^{2} + 256 \, {\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{9} x + 96 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{9} \log \left (c x^{2} + b x + a\right ) - \frac {32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{9} x^{2} + 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{9} x + {\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

128*c^6*d^9*x^4 + 256*b*c^5*d^9*x^3 + 384*(b^2*c^4 - 2*a*c^5)*d^9*x^2 + 256*(b^3*c^3 - 3*a*b*c^4)*d^9*x + 96*(
b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^9*log(c*x^2 + b*x + a) - 1/2*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^
4 - 64*a^3*c^5)*d^9*x^2 + 32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d^9*x + (b^8 + 16*a*b^6*c
- 288*a^2*b^4*c^2 + 1280*a^3*b^2*c^3 - 1792*a^4*c^4)*d^9)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 +
a^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (121) = 242\).

Time = 0.33 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.43 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=96 \, {\left (b^{4} c^{2} d^{9} - 8 \, a b^{2} c^{3} d^{9} + 16 \, a^{2} c^{4} d^{9}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{8} d^{9} + 16 \, a b^{6} c d^{9} - 288 \, a^{2} b^{4} c^{2} d^{9} + 1280 \, a^{3} b^{2} c^{3} d^{9} - 1792 \, a^{4} c^{4} d^{9} + 32 \, {\left (b^{6} c^{2} d^{9} - 12 \, a b^{4} c^{3} d^{9} + 48 \, a^{2} b^{2} c^{4} d^{9} - 64 \, a^{3} c^{5} d^{9}\right )} x^{2} + 32 \, {\left (b^{7} c d^{9} - 12 \, a b^{5} c^{2} d^{9} + 48 \, a^{2} b^{3} c^{3} d^{9} - 64 \, a^{3} b c^{4} d^{9}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (c^{18} d^{9} x^{4} + 2 \, b c^{17} d^{9} x^{3} + 3 \, b^{2} c^{16} d^{9} x^{2} - 6 \, a c^{17} d^{9} x^{2} + 2 \, b^{3} c^{15} d^{9} x - 6 \, a b c^{16} d^{9} x\right )}}{c^{12}} \]

[In]

integrate((2*c*d*x+b*d)^9/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

96*(b^4*c^2*d^9 - 8*a*b^2*c^3*d^9 + 16*a^2*c^4*d^9)*log(c*x^2 + b*x + a) - 1/2*(b^8*d^9 + 16*a*b^6*c*d^9 - 288
*a^2*b^4*c^2*d^9 + 1280*a^3*b^2*c^3*d^9 - 1792*a^4*c^4*d^9 + 32*(b^6*c^2*d^9 - 12*a*b^4*c^3*d^9 + 48*a^2*b^2*c
^4*d^9 - 64*a^3*c^5*d^9)*x^2 + 32*(b^7*c*d^9 - 12*a*b^5*c^2*d^9 + 48*a^2*b^3*c^3*d^9 - 64*a^3*b*c^4*d^9)*x)/(c
*x^2 + b*x + a)^2 + 128*(c^18*d^9*x^4 + 2*b*c^17*d^9*x^3 + 3*b^2*c^16*d^9*x^2 - 6*a*c^17*d^9*x^2 + 2*b^3*c^15*
d^9*x - 6*a*b*c^16*d^9*x)/c^12

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 385, normalized size of antiderivative = 3.13 \[ \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx=\ln \left (c\,x^2+b\,x+a\right )\,\left (1536\,a^2\,c^4\,d^9-768\,a\,b^2\,c^3\,d^9+96\,b^4\,c^2\,d^9\right )-x^2\,\left (768\,c^4\,d^9\,\left (b^2+a\,c\right )-1152\,b^2\,c^4\,d^9\right )-\frac {\frac {b^8\,d^9}{2}-x^2\,\left (1024\,a^3\,c^5\,d^9-768\,a^2\,b^2\,c^4\,d^9+192\,a\,b^4\,c^3\,d^9-16\,b^6\,c^2\,d^9\right )-896\,a^4\,c^4\,d^9+16\,b\,x\,\left (-64\,a^3\,c^4\,d^9+48\,a^2\,b^2\,c^3\,d^9-12\,a\,b^4\,c^2\,d^9+b^6\,c\,d^9\right )-144\,a^2\,b^4\,c^2\,d^9+640\,a^3\,b^2\,c^3\,d^9+8\,a\,b^6\,c\,d^9}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-x\,\left (512\,c^3\,d^9\,\left (b^3+6\,a\,c\,b\right )-5376\,b^3\,c^3\,d^9-\frac {3\,b\,\left (1536\,c^4\,d^9\,\left (b^2+a\,c\right )-2304\,b^2\,c^4\,d^9\right )}{c}+2304\,b\,c^3\,d^9\,\left (b^2+a\,c\right )\right )+128\,c^6\,d^9\,x^4+256\,b\,c^5\,d^9\,x^3 \]

[In]

int((b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3,x)

[Out]

log(a + b*x + c*x^2)*(1536*a^2*c^4*d^9 + 96*b^4*c^2*d^9 - 768*a*b^2*c^3*d^9) - x^2*(768*c^4*d^9*(a*c + b^2) -
1152*b^2*c^4*d^9) - ((b^8*d^9)/2 - x^2*(1024*a^3*c^5*d^9 - 16*b^6*c^2*d^9 + 192*a*b^4*c^3*d^9 - 768*a^2*b^2*c^
4*d^9) - 896*a^4*c^4*d^9 + 16*b*x*(b^6*c*d^9 - 64*a^3*c^4*d^9 - 12*a*b^4*c^2*d^9 + 48*a^2*b^2*c^3*d^9) - 144*a
^2*b^4*c^2*d^9 + 640*a^3*b^2*c^3*d^9 + 8*a*b^6*c*d^9)/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3
) - x*(512*c^3*d^9*(b^3 + 6*a*b*c) - 5376*b^3*c^3*d^9 - (3*b*(1536*c^4*d^9*(a*c + b^2) - 2304*b^2*c^4*d^9))/c
+ 2304*b*c^3*d^9*(a*c + b^2)) + 128*c^6*d^9*x^4 + 256*b*c^5*d^9*x^3