3.14.73 \(\int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx\) [1373]
Optimal. Leaf size=231 \[ -\frac {b^7}{(b c-a d)^8 (a+b x)}-\frac {d}{7 (b c-a d)^2 (c+d x)^7}-\frac {b d}{3 (b c-a d)^3 (c+d x)^6}-\frac {3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac {b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac {5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac {3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac {7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9} \]
[Out]
-b^7/(-a*d+b*c)^8/(b*x+a)-1/7*d/(-a*d+b*c)^2/(d*x+c)^7-1/3*b*d/(-a*d+b*c)^3/(d*x+c)^6-3/5*b^2*d/(-a*d+b*c)^4/(
d*x+c)^5-b^3*d/(-a*d+b*c)^5/(d*x+c)^4-5/3*b^4*d/(-a*d+b*c)^6/(d*x+c)^3-3*b^5*d/(-a*d+b*c)^7/(d*x+c)^2-7*b^6*d/
(-a*d+b*c)^8/(d*x+c)-8*b^7*d*ln(b*x+a)/(-a*d+b*c)^9+8*b^7*d*ln(d*x+c)/(-a*d+b*c)^9
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Rubi [A]
time = 0.19, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46}
\begin {gather*} -\frac {b^7}{(a+b x) (b c-a d)^8}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9}-\frac {7 b^6 d}{(c+d x) (b c-a d)^8}-\frac {3 b^5 d}{(c+d x)^2 (b c-a d)^7}-\frac {5 b^4 d}{3 (c+d x)^3 (b c-a d)^6}-\frac {b^3 d}{(c+d x)^4 (b c-a d)^5}-\frac {3 b^2 d}{5 (c+d x)^5 (b c-a d)^4}-\frac {b d}{3 (c+d x)^6 (b c-a d)^3}-\frac {d}{7 (c+d x)^7 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)^2*(c + d*x)^8),x]
[Out]
-(b^7/((b*c - a*d)^8*(a + b*x))) - d/(7*(b*c - a*d)^2*(c + d*x)^7) - (b*d)/(3*(b*c - a*d)^3*(c + d*x)^6) - (3*
b^2*d)/(5*(b*c - a*d)^4*(c + d*x)^5) - (b^3*d)/((b*c - a*d)^5*(c + d*x)^4) - (5*b^4*d)/(3*(b*c - a*d)^6*(c + d
*x)^3) - (3*b^5*d)/((b*c - a*d)^7*(c + d*x)^2) - (7*b^6*d)/((b*c - a*d)^8*(c + d*x)) - (8*b^7*d*Log[a + b*x])/
(b*c - a*d)^9 + (8*b^7*d*Log[c + d*x])/(b*c - a*d)^9
Rule 46
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx &=\int \left (\frac {b^8}{(b c-a d)^8 (a+b x)^2}-\frac {8 b^8 d}{(b c-a d)^9 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^8}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)^7}+\frac {3 b^2 d^2}{(b c-a d)^4 (c+d x)^6}+\frac {4 b^3 d^2}{(b c-a d)^5 (c+d x)^5}+\frac {5 b^4 d^2}{(b c-a d)^6 (c+d x)^4}+\frac {6 b^5 d^2}{(b c-a d)^7 (c+d x)^3}+\frac {7 b^6 d^2}{(b c-a d)^8 (c+d x)^2}+\frac {8 b^7 d^2}{(b c-a d)^9 (c+d x)}\right ) \, dx\\ &=-\frac {b^7}{(b c-a d)^8 (a+b x)}-\frac {d}{7 (b c-a d)^2 (c+d x)^7}-\frac {b d}{3 (b c-a d)^3 (c+d x)^6}-\frac {3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac {b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac {5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac {3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac {7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 213, normalized size = 0.92 \begin {gather*} -\frac {\frac {105 b^7 (b c-a d)}{a+b x}-\frac {15 d (-b c+a d)^7}{(c+d x)^7}+\frac {35 b d (b c-a d)^6}{(c+d x)^6}+\frac {63 b^2 d (b c-a d)^5}{(c+d x)^5}+\frac {105 b^3 d (b c-a d)^4}{(c+d x)^4}+\frac {175 b^4 d (b c-a d)^3}{(c+d x)^3}+\frac {315 b^5 d (b c-a d)^2}{(c+d x)^2}+\frac {735 b^6 d (b c-a d)}{c+d x}+840 b^7 d \log (a+b x)-840 b^7 d \log (c+d x)}{105 (b c-a d)^9} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)^2*(c + d*x)^8),x]
[Out]
-1/105*((105*b^7*(b*c - a*d))/(a + b*x) - (15*d*(-(b*c) + a*d)^7)/(c + d*x)^7 + (35*b*d*(b*c - a*d)^6)/(c + d*
x)^6 + (63*b^2*d*(b*c - a*d)^5)/(c + d*x)^5 + (105*b^3*d*(b*c - a*d)^4)/(c + d*x)^4 + (175*b^4*d*(b*c - a*d)^3
)/(c + d*x)^3 + (315*b^5*d*(b*c - a*d)^2)/(c + d*x)^2 + (735*b^6*d*(b*c - a*d))/(c + d*x) + 840*b^7*d*Log[a +
b*x] - 840*b^7*d*Log[c + d*x])/(b*c - a*d)^9
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(2739\) vs. \(2(231)=462\).
time = 42.52, size = 2737, normalized size = 11.85
result too large to display
Antiderivative was successfully verified.
[In]
mathics('Integrate[(a + b*x)^(-2)/(c + d*x)^8,x]')
[Out]
(8 b ^ 7 d (a ^ 9 c ^ 7 d ^ 8 - 8 a ^ 8 b c ^ 8 d ^ 7 + 28 a ^ 7 b ^ 2 c ^ 9 d ^ 6 - 56 a ^ 6 b ^ 3 c ^ 10 d ^
5 + 70 a ^ 5 b ^ 4 c ^ 11 d ^ 4 - 56 a ^ 4 b ^ 5 c ^ 12 d ^ 3 + 28 a ^ 3 b ^ 6 c ^ 13 d ^ 2 - 8 a ^ 2 b ^ 7 c
^ 14 d + a b ^ 8 c ^ 15 + b d ^ 7 x ^ 8 (a ^ 8 d ^ 8 - 8 a ^ 7 b c d ^ 7 + 28 a ^ 6 b ^ 2 c ^ 2 d ^ 6 - 56 a
^ 5 b ^ 3 c ^ 3 d ^ 5 + 70 a ^ 4 b ^ 4 c ^ 4 d ^ 4 - 56 a ^ 3 b ^ 5 c ^ 5 d ^ 3 + 28 a ^ 2 b ^ 6 c ^ 6 d ^ 2 -
8 a b ^ 7 c ^ 7 d + b ^ 8 c ^ 8) + c ^ 6 x (7 a ^ 9 d ^ 9 - 55 a ^ 8 b c d ^ 8 + 188 a ^ 7 b ^ 2 c ^ 2 d ^ 7
- 364 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 434 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 322 a ^ 4 b ^ 5 c ^ 5 d ^ 4 + 140 a ^ 3 b ^ 6 c
^ 6 d ^ 3 - 28 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - a b ^ 8 c ^ 8 d + b ^ 9 c ^ 9) + 7 c ^ 5 d x ^ 2 (3 a ^ 9 d ^ 9 - 23
a ^ 8 b c d ^ 8 + 76 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 140 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 154 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 98
a ^ 4 b ^ 5 c ^ 5 d ^ 4 + 28 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 4 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 5 a b ^ 8 c ^ 8 d + b ^ 9 c
^ 9) + c ^ 4 d ^ 2 x ^ 3 (35 a ^ 9 d ^ 9 - 259 a ^ 8 b c d ^ 8 + 812 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 1372 a ^ 6 b ^
3 c ^ 3 d ^ 6 + 1274 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 490 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 196 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 30
8 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 133 a b ^ 8 c ^ 8 d + 21 b ^ 9 c ^ 9) + c ^ 3 d ^ 3 x ^ 4 (35 a ^ 9 d ^ 9 - 245 a
^ 8 b c d ^ 8 + 700 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 980 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 490 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 490
a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 980 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 700 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 245 a b ^ 8 c ^ 8 d + 35
b ^ 9 c ^ 9) + c ^ 2 d ^ 4 x ^ 5 (21 a ^ 9 d ^ 9 - 133 a ^ 8 b c d ^ 8 + 308 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 196 a
^ 6 b ^ 3 c ^ 3 d ^ 6 - 490 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 1274 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 1372 a ^ 3 b ^ 6 c ^ 6 d
^ 3 + 812 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 259 a b ^ 8 c ^ 8 d + 35 b ^ 9 c ^ 9) + 7 c d ^ 5 x ^ 6 (a ^ 9 d ^ 9 - 5 a
^ 8 b c d ^ 8 + 4 a ^ 7 b ^ 2 c ^ 2 d ^ 7 + 28 a ^ 6 b ^ 3 c ^ 3 d ^ 6 - 98 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 154 a ^
4 b ^ 5 c ^ 5 d ^ 4 - 140 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 76 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 23 a b ^ 8 c ^ 8 d + 3 b ^ 9
c ^ 9) + d ^ 6 x ^ 7 (a ^ 9 d ^ 9 - a ^ 8 b c d ^ 8 - 28 a ^ 7 b ^ 2 c ^ 2 d ^ 7 + 140 a ^ 6 b ^ 3 c ^ 3 d ^
6 - 322 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 434 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 364 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 188 a ^ 2 b ^ 7
c ^ 7 d ^ 2 - 55 a b ^ 8 c ^ 8 d + 7 b ^ 9 c ^ 9)) (Log[(a + b x) / b] - Log[(c + d x) / d]) + (-15 a ^ 7 d ^
7 + 125 a ^ 6 b c d ^ 6 - 463 a ^ 5 b ^ 2 c ^ 2 d ^ 5 + 1007 a ^ 4 b ^ 3 c ^ 3 d ^ 4 - 1443 a ^ 3 b ^ 4 c ^ 4
d ^ 3 + 1497 a ^ 2 b ^ 5 c ^ 5 d ^ 2 - 1443 a b ^ 6 c ^ 6 d - 105 b ^ 7 c ^ 7 - 2 b d x (-10 a ^ 6 d ^ 6 + 88
a ^ 5 b c d ^ 5 - 353 a ^ 4 b ^ 2 c ^ 2 d ^ 4 + 872 a ^ 3 b ^ 3 c ^ 3 d ^ 3 - 1578 a ^ 2 b ^ 4 c ^ 4 d ^ 2 + 2
832 a b ^ 5 c ^ 5 d + 1089 b ^ 6 c ^ 6) + b ^ 2 d ^ 2 x ^ 2 (-28 a ^ 5 d ^ 5 + 266 a ^ 4 b c d ^ 4 - 1204 a ^
3 b ^ 2 c ^ 2 d ^ 3 + 3696 a ^ 2 b ^ 3 c ^ 3 d ^ 2 - 11004 a b ^ 4 c ^ 4 d - 9366 b ^ 5 c ^ 5) + b ^ 3 d ^ 3 x
^ 3 (42 a ^ 4 d ^ 4 - 448 a ^ 3 b c d ^ 3 + 2492 a ^ 2 b ^ 2 c ^ 2 d ^ 2 - 12208 a b ^ 3 c ^ 3 d - 19278 b ^
4 c ^ 4) + b ^ 4 d ^ 4 x ^ 4 (-70 a ^ 3 d ^ 3 + 910 a ^ 2 b c d ^ 2 - 7910 a b ^ 2 c ^ 2 d - 22330 b ^ 3 c ^ 3
) + b ^ 5 d ^ 5 x ^ 5 (140 a ^ 2 d ^ 2 - 2800 a b c d - 14980 b ^ 2 c ^ 2) + b ^ 6 d ^ 6 x ^ 6 (-420 a d - 546
0 b c) - 840 b ^ 7 d ^ 7 x ^ 7) (a d - b c) ^ 9 / 105) / ((a d - b c) ^ 9 (a ^ 9 c ^ 7 d ^ 8 - 8 a ^ 8 b c ^ 8
d ^ 7 + 28 a ^ 7 b ^ 2 c ^ 9 d ^ 6 - 56 a ^ 6 b ^ 3 c ^ 10 d ^ 5 + 70 a ^ 5 b ^ 4 c ^ 11 d ^ 4 - 56 a ^ 4 b ^
5 c ^ 12 d ^ 3 + 28 a ^ 3 b ^ 6 c ^ 13 d ^ 2 - 8 a ^ 2 b ^ 7 c ^ 14 d + a b ^ 8 c ^ 15 + b d ^ 7 x ^ 8 (a ^ 8
d ^ 8 - 8 a ^ 7 b c d ^ 7 + 28 a ^ 6 b ^ 2 c ^ 2 d ^ 6 - 56 a ^ 5 b ^ 3 c ^ 3 d ^ 5 + 70 a ^ 4 b ^ 4 c ^ 4 d
^ 4 - 56 a ^ 3 b ^ 5 c ^ 5 d ^ 3 + 28 a ^ 2 b ^ 6 c ^ 6 d ^ 2 - 8 a b ^ 7 c ^ 7 d + b ^ 8 c ^ 8) + c ^ 6 x (7
a ^ 9 d ^ 9 - 55 a ^ 8 b c d ^ 8 + 188 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 364 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 434 a ^ 5 b ^ 4
c ^ 4 d ^ 5 - 322 a ^ 4 b ^ 5 c ^ 5 d ^ 4 + 140 a ^ 3 b ^ 6 c ^ 6 d ^ 3 - 28 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - a b ^
8 c ^ 8 d + b ^ 9 c ^ 9) + 7 c ^ 5 d x ^ 2 (3 a ^ 9 d ^ 9 - 23 a ^ 8 b c d ^ 8 + 76 a ^ 7 b ^ 2 c ^ 2 d ^ 7 -
140 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 154 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 98 a ^ 4 b ^ 5 c ^ 5 d ^ 4 + 28 a ^ 3 b ^ 6 c ^ 6
d ^ 3 + 4 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 5 a b ^ 8 c ^ 8 d + b ^ 9 c ^ 9) + 7 c ^ 4 d ^ 2 x ^ 3 (5 a ^ 9 d ^ 9 - 37
a ^ 8 b c d ^ 8 + 116 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 196 a ^ 6 b ^ 3 c ^ 3 d ^ 6 + 182 a ^ 5 b ^ 4 c ^ 4 d ^ 5 - 7
0 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 28 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 44 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 19 a b ^ 8 c ^ 8 d + 3 b
^ 9 c ^ 9) + 35 c ^ 3 d ^ 3 x ^ 4 (a ^ 9 d ^ 9 - 7 a ^ 8 b c d ^ 8 + 20 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 28 a ^ 6 b
^ 3 c ^ 3 d ^ 6 + 14 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 14 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 28 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 20 a
^ 2 b ^ 7 c ^ 7 d ^ 2 - 7 a b ^ 8 c ^ 8 d + b ^ 9 c ^ 9) + 7 c ^ 2 d ^ 4 x ^ 5 (3 a ^ 9 d ^ 9 - 19 a ^ 8 b c d
^ 8 + 44 a ^ 7 b ^ 2 c ^ 2 d ^ 7 - 28 a ^ 6 b ^ 3 c ^ 3 d ^ 6 - 70 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 182 a ^ 4 b ^ 5
c ^ 5 d ^ 4 - 196 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 116 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 37 a b ^ 8 c ^ 8 d + 5 b ^ 9 c ^ 9)
+ 7 c d ^ 5 x ^ 6 (a ^ 9 d ^ 9 - 5 a ^ 8 b c d ^ 8 + 4 a ^ 7 b ^ 2 c ^ 2 d ^ 7 + 28 a ^ 6 b ^ 3 c ^ 3 d ^ 6 -
98 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 154 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 140 a ^ 3 b ^ 6 c ^ 6 d ^ 3 + 76 a ^ 2 b ^ 7 c ^ 7
d ^ 2 - 23 a b ^ 8 c ^ 8 d + 3 b ^ 9 c ^ 9) + d ^ 6 x ^ 7 (a ^ 9 d ^ 9 - a ^ 8 b c d ^ 8 - 28 a ^ 7 b ^ 2 c ^
2 d ^ 7 + 140 a ^ 6 b ^ 3 c ^ 3 d ^ 6 - 322 a ^ 5 b ^ 4 c ^ 4 d ^ 5 + 434 a ^ 4 b ^ 5 c ^ 5 d ^ 4 - 364 a ^ 3
b ^ 6 c ^ 6 d ^ 3 + 188 a ^ 2 b ^ 7 c ^ 7 d ^ 2 - 55 a b ^ 8 c ^ 8 d + 7 b ^ 9 c ^ 9)))
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Maple [A]
time = 0.20, size = 223, normalized size = 0.97
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {d}{7 \left (a d -b c \right )^{2} \left (d x +c \right )^{7}}-\frac {8 d \,b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{9}}-\frac {7 d \,b^{6}}{\left (a d -b c \right )^{8} \left (d x +c \right )}+\frac {3 d \,b^{5}}{\left (a d -b c \right )^{7} \left (d x +c \right )^{2}}-\frac {5 d \,b^{4}}{3 \left (a d -b c \right )^{6} \left (d x +c \right )^{3}}+\frac {d \,b^{3}}{\left (a d -b c \right )^{5} \left (d x +c \right )^{4}}-\frac {3 d \,b^{2}}{5 \left (a d -b c \right )^{4} \left (d x +c \right )^{5}}+\frac {d b}{3 \left (a d -b c \right )^{3} \left (d x +c \right )^{6}}-\frac {b^{7}}{\left (a d -b c \right )^{8} \left (b x +a \right )}+\frac {8 d \,b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{9}}\) |
\(223\) |
| risch |
\(\frac {-\frac {8 b^{7} d^{7} x^{7}}{a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}-\frac {4 d^{6} \left (a d +13 b c \right ) b^{6} x^{6}}{a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}+\frac {4 d^{5} b^{5} \left (a^{2} d^{2}-20 a b c d -107 b^{2} c^{2}\right ) x^{5}}{3 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}-\frac {2 b^{4} d^{4} \left (a^{3} d^{3}-13 a^{2} b c \,d^{2}+113 a \,b^{2} c^{2} d +319 b^{3} c^{3}\right ) x^{4}}{3 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {2 b^{3} d^{3} \left (3 a^{4} d^{4}-32 a^{3} b c \,d^{3}+178 a^{2} b^{2} c^{2} d^{2}-872 a \,b^{3} c^{3} d -1377 b^{4} c^{4}\right ) x^{3}}{15 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}-\frac {2 b^{2} d^{2} \left (2 a^{5} d^{5}-19 a^{4} b c \,d^{4}+86 a^{3} b^{2} c^{2} d^{3}-264 a^{2} b^{3} c^{3} d^{2}+786 a \,b^{4} c^{4} d +669 b^{5} c^{5}\right ) x^{2}}{15 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {2 \left (10 a^{6} d^{6}-88 a^{5} b c \,d^{5}+353 a^{4} b^{2} c^{2} d^{4}-872 a^{3} b^{3} c^{3} d^{3}+1578 a^{2} b^{4} c^{4} d^{2}-2832 a \,b^{5} c^{5} d -1089 b^{6} c^{6}\right ) b d x}{105 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}-\frac {15 a^{7} d^{7}-125 a^{6} b c \,d^{6}+463 a^{5} b^{2} c^{2} d^{5}-1007 a^{4} b^{3} c^{3} d^{4}+1443 a^{3} b^{4} c^{4} d^{3}-1497 a^{2} b^{5} c^{5} d^{2}+1443 a \,b^{6} c^{6} d +105 b^{7} c^{7}}{105 \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{7}}+\frac {8 b^{7} d \ln \left (-b x -a \right )}{a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} d^{3} c^{6}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}}-\frac {8 b^{7} d \ln \left (d x +c \right )}{a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} d^{3} c^{6}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}}\) |
\(1572\) |
| norman |
\(\frac {-\frac {8 b^{7} d^{7} x^{7}}{a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}}+\frac {\left (-4 a \,b^{7} d^{9}-52 b^{8} c \,d^{8}\right ) x^{6}}{d^{2} b \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {-15 a^{7} b \,d^{14}+125 a^{6} b^{2} c \,d^{13}-463 a^{5} b^{3} c^{2} d^{12}+1007 a^{4} b^{4} c^{3} d^{11}-1443 a^{3} b^{5} c^{4} d^{10}+1497 a^{2} b^{6} c^{5} d^{9}-1443 a \,b^{7} c^{6} d^{8}-105 c^{7} b^{8} d^{7}}{105 b \,d^{7} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {\left (-4 a^{5} b^{3} d^{13}+38 a^{4} b^{4} c \,d^{12}-172 a^{3} b^{5} c^{2} d^{11}+528 a^{2} b^{6} c^{3} d^{10}-1572 a \,b^{7} c^{4} d^{9}-1338 b^{8} c^{5} d^{8}\right ) x^{2}}{15 d^{6} b \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {\left (6 a^{4} b^{4} d^{12}-64 a^{3} b^{5} c \,d^{11}+356 a^{2} b^{6} c^{2} d^{10}-1744 a \,b^{7} c^{3} d^{9}-2754 b^{8} c^{4} d^{8}\right ) x^{3}}{15 d^{5} b \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {\left (4 b^{6} a^{2} d^{10}-80 a \,b^{7} c \,d^{9}-428 b^{8} c^{2} d^{8}\right ) x^{5}}{3 b \,d^{3} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {\left (-2 a^{3} b^{5} d^{11}+26 a^{2} b^{6} c \,d^{10}-226 a \,b^{7} c^{2} d^{9}-638 b^{8} c^{3} d^{8}\right ) x^{4}}{3 b \,d^{4} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}+\frac {\left (20 a^{6} b^{2} d^{14}-176 a^{5} b^{3} c \,d^{13}+706 a^{4} b^{4} c^{2} d^{12}-1744 a^{3} b^{5} c^{3} d^{11}+3156 a^{2} b^{6} c^{4} d^{10}-5664 a \,b^{7} c^{5} d^{9}-2178 b^{8} c^{6} d^{8}\right ) x}{105 b \,d^{7} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{7}}+\frac {8 b^{7} d \ln \left (b x +a \right )}{a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} d^{3} c^{6}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}}-\frac {8 b^{7} d \ln \left (d x +c \right )}{a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} d^{3} c^{6}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}}\) |
\(1649\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)^2/(d*x+c)^8,x,method=_RETURNVERBOSE)
[Out]
-1/7*d/(a*d-b*c)^2/(d*x+c)^7-8*d/(a*d-b*c)^9*b^7*ln(d*x+c)-7*d/(a*d-b*c)^8*b^6/(d*x+c)+3*d/(a*d-b*c)^7*b^5/(d*
x+c)^2-5/3*d/(a*d-b*c)^6*b^4/(d*x+c)^3+d/(a*d-b*c)^5*b^3/(d*x+c)^4-3/5*d/(a*d-b*c)^4*b^2/(d*x+c)^5+1/3*d/(a*d-
b*c)^3*b/(d*x+c)^6-b^7/(a*d-b*c)^8/(b*x+a)+8*d/(a*d-b*c)^9*b^7*ln(b*x+a)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1881 vs.
\(2 (223) = 446\).
time = 0.46, size = 1881, normalized size = 8.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="maxima")
[Out]
-8*b^7*d*log(b*x + a)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4
- 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) + 8*b^7*d*log(d*x
+ c)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^
4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) - 1/105*(840*b^7*d^7*x^7 + 105*b^7*
c^7 + 1443*a*b^6*c^6*d - 1497*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*
d^5 - 125*a^6*b*c*d^6 + 15*a^7*d^7 + 420*(13*b^7*c*d^6 + a*b^6*d^7)*x^6 + 140*(107*b^7*c^2*d^5 + 20*a*b^6*c*d^
6 - a^2*b^5*d^7)*x^5 + 70*(319*b^7*c^3*d^4 + 113*a*b^6*c^2*d^5 - 13*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 14*(137
7*b^7*c^4*d^3 + 872*a*b^6*c^3*d^4 - 178*a^2*b^5*c^2*d^5 + 32*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 14*(669*b^7*
c^5*d^2 + 786*a*b^6*c^4*d^3 - 264*a^2*b^5*c^3*d^4 + 86*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 2*a^5*b^2*d^7)*x^2
+ 2*(1089*b^7*c^6*d + 2832*a*b^6*c^5*d^2 - 1578*a^2*b^5*c^4*d^3 + 872*a^3*b^4*c^3*d^4 - 353*a^4*b^3*c^2*d^5 +
88*a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x)/(a*b^8*c^15 - 8*a^2*b^7*c^14*d + 28*a^3*b^6*c^13*d^2 - 56*a^4*b^5*c^12*d^
3 + 70*a^5*b^4*c^11*d^4 - 56*a^6*b^3*c^10*d^5 + 28*a^7*b^2*c^9*d^6 - 8*a^8*b*c^8*d^7 + a^9*c^7*d^8 + (b^9*c^8*
d^7 - 8*a*b^8*c^7*d^8 + 28*a^2*b^7*c^6*d^9 - 56*a^3*b^6*c^5*d^10 + 70*a^4*b^5*c^4*d^11 - 56*a^5*b^4*c^3*d^12 +
28*a^6*b^3*c^2*d^13 - 8*a^7*b^2*c*d^14 + a^8*b*d^15)*x^8 + (7*b^9*c^9*d^6 - 55*a*b^8*c^8*d^7 + 188*a^2*b^7*c^
7*d^8 - 364*a^3*b^6*c^6*d^9 + 434*a^4*b^5*c^5*d^10 - 322*a^5*b^4*c^4*d^11 + 140*a^6*b^3*c^3*d^12 - 28*a^7*b^2*
c^2*d^13 - a^8*b*c*d^14 + a^9*d^15)*x^7 + 7*(3*b^9*c^10*d^5 - 23*a*b^8*c^9*d^6 + 76*a^2*b^7*c^8*d^7 - 140*a^3*
b^6*c^7*d^8 + 154*a^4*b^5*c^6*d^9 - 98*a^5*b^4*c^5*d^10 + 28*a^6*b^3*c^4*d^11 + 4*a^7*b^2*c^3*d^12 - 5*a^8*b*c
^2*d^13 + a^9*c*d^14)*x^6 + 7*(5*b^9*c^11*d^4 - 37*a*b^8*c^10*d^5 + 116*a^2*b^7*c^9*d^6 - 196*a^3*b^6*c^8*d^7
+ 182*a^4*b^5*c^7*d^8 - 70*a^5*b^4*c^6*d^9 - 28*a^6*b^3*c^5*d^10 + 44*a^7*b^2*c^4*d^11 - 19*a^8*b*c^3*d^12 + 3
*a^9*c^2*d^13)*x^5 + 35*(b^9*c^12*d^3 - 7*a*b^8*c^11*d^4 + 20*a^2*b^7*c^10*d^5 - 28*a^3*b^6*c^9*d^6 + 14*a^4*b
^5*c^8*d^7 + 14*a^5*b^4*c^7*d^8 - 28*a^6*b^3*c^6*d^9 + 20*a^7*b^2*c^5*d^10 - 7*a^8*b*c^4*d^11 + a^9*c^3*d^12)*
x^4 + 7*(3*b^9*c^13*d^2 - 19*a*b^8*c^12*d^3 + 44*a^2*b^7*c^11*d^4 - 28*a^3*b^6*c^10*d^5 - 70*a^4*b^5*c^9*d^6 +
182*a^5*b^4*c^8*d^7 - 196*a^6*b^3*c^7*d^8 + 116*a^7*b^2*c^6*d^9 - 37*a^8*b*c^5*d^10 + 5*a^9*c^4*d^11)*x^3 + 7
*(b^9*c^14*d - 5*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 + 28*a^3*b^6*c^11*d^4 - 98*a^4*b^5*c^10*d^5 + 154*a^5*b^4
*c^9*d^6 - 140*a^6*b^3*c^8*d^7 + 76*a^7*b^2*c^7*d^8 - 23*a^8*b*c^6*d^9 + 3*a^9*c^5*d^10)*x^2 + (b^9*c^15 - a*b
^8*c^14*d - 28*a^2*b^7*c^13*d^2 + 140*a^3*b^6*c^12*d^3 - 322*a^4*b^5*c^11*d^4 + 434*a^5*b^4*c^10*d^5 - 364*a^6
*b^3*c^9*d^6 + 188*a^7*b^2*c^8*d^7 - 55*a^8*b*c^7*d^8 + 7*a^9*c^6*d^9)*x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2264 vs.
\(2 (223) = 446\).
time = 0.35, size = 2264, normalized size = 9.80
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="fricas")
[Out]
-1/105*(105*b^8*c^8 + 1338*a*b^7*c^7*d - 2940*a^2*b^6*c^6*d^2 + 2940*a^3*b^5*c^5*d^3 - 2450*a^4*b^4*c^4*d^4 +
1470*a^5*b^3*c^3*d^5 - 588*a^6*b^2*c^2*d^6 + 140*a^7*b*c*d^7 - 15*a^8*d^8 + 840*(b^8*c*d^7 - a*b^7*d^8)*x^7 +
420*(13*b^8*c^2*d^6 - 12*a*b^7*c*d^7 - a^2*b^6*d^8)*x^6 + 140*(107*b^8*c^3*d^5 - 87*a*b^7*c^2*d^6 - 21*a^2*b^6
*c*d^7 + a^3*b^5*d^8)*x^5 + 70*(319*b^8*c^4*d^4 - 206*a*b^7*c^3*d^5 - 126*a^2*b^6*c^2*d^6 + 14*a^3*b^5*c*d^7 -
a^4*b^4*d^8)*x^4 + 14*(1377*b^8*c^5*d^3 - 505*a*b^7*c^4*d^4 - 1050*a^2*b^6*c^3*d^5 + 210*a^3*b^5*c^2*d^6 - 35
*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 14*(669*b^8*c^6*d^2 + 117*a*b^7*c^5*d^3 - 1050*a^2*b^6*c^4*d^4 + 350*a^3
*b^5*c^3*d^5 - 105*a^4*b^4*c^2*d^6 + 21*a^5*b^3*c*d^7 - 2*a^6*b^2*d^8)*x^2 + 2*(1089*b^8*c^7*d + 1743*a*b^7*c^
6*d^2 - 4410*a^2*b^6*c^5*d^3 + 2450*a^3*b^5*c^4*d^4 - 1225*a^4*b^4*c^3*d^5 + 441*a^5*b^3*c^2*d^6 - 98*a^6*b^2*
c*d^7 + 10*a^7*b*d^8)*x + 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 +
a*b^7*c*d^7)*x^6 + 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c
^5*d^3 + 5*a*b^7*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(b
*x + a) - 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 + a*b^7*c*d^7)*x^6
+ 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c^5*d^3 + 5*a*b^7
*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(d*x + c))/(a*b^9*
c^16 - 9*a^2*b^8*c^15*d + 36*a^3*b^7*c^14*d^2 - 84*a^4*b^6*c^13*d^3 + 126*a^5*b^5*c^12*d^4 - 126*a^6*b^4*c^11*
d^5 + 84*a^7*b^3*c^10*d^6 - 36*a^8*b^2*c^9*d^7 + 9*a^9*b*c^8*d^8 - a^10*c^7*d^9 + (b^10*c^9*d^7 - 9*a*b^9*c^8*
d^8 + 36*a^2*b^8*c^7*d^9 - 84*a^3*b^7*c^6*d^10 + 126*a^4*b^6*c^5*d^11 - 126*a^5*b^5*c^4*d^12 + 84*a^6*b^4*c^3*
d^13 - 36*a^7*b^3*c^2*d^14 + 9*a^8*b^2*c*d^15 - a^9*b*d^16)*x^8 + (7*b^10*c^10*d^6 - 62*a*b^9*c^9*d^7 + 243*a^
2*b^8*c^8*d^8 - 552*a^3*b^7*c^7*d^9 + 798*a^4*b^6*c^6*d^10 - 756*a^5*b^5*c^5*d^11 + 462*a^6*b^4*c^4*d^12 - 168
*a^7*b^3*c^3*d^13 + 27*a^8*b^2*c^2*d^14 + 2*a^9*b*c*d^15 - a^10*d^16)*x^7 + 7*(3*b^10*c^11*d^5 - 26*a*b^9*c^10
*d^6 + 99*a^2*b^8*c^9*d^7 - 216*a^3*b^7*c^8*d^8 + 294*a^4*b^6*c^7*d^9 - 252*a^5*b^5*c^6*d^10 + 126*a^6*b^4*c^5
*d^11 - 24*a^7*b^3*c^4*d^12 - 9*a^8*b^2*c^3*d^13 + 6*a^9*b*c^2*d^14 - a^10*c*d^15)*x^6 + 7*(5*b^10*c^12*d^4 -
42*a*b^9*c^11*d^5 + 153*a^2*b^8*c^10*d^6 - 312*a^3*b^7*c^9*d^7 + 378*a^4*b^6*c^8*d^8 - 252*a^5*b^5*c^7*d^9 + 4
2*a^6*b^4*c^6*d^10 + 72*a^7*b^3*c^5*d^11 - 63*a^8*b^2*c^4*d^12 + 22*a^9*b*c^3*d^13 - 3*a^10*c^2*d^14)*x^5 + 35
*(b^10*c^13*d^3 - 8*a*b^9*c^12*d^4 + 27*a^2*b^8*c^11*d^5 - 48*a^3*b^7*c^10*d^6 + 42*a^4*b^6*c^9*d^7 - 42*a^6*b
^4*c^7*d^9 + 48*a^7*b^3*c^6*d^10 - 27*a^8*b^2*c^5*d^11 + 8*a^9*b*c^4*d^12 - a^10*c^3*d^13)*x^4 + 7*(3*b^10*c^1
4*d^2 - 22*a*b^9*c^13*d^3 + 63*a^2*b^8*c^12*d^4 - 72*a^3*b^7*c^11*d^5 - 42*a^4*b^6*c^10*d^6 + 252*a^5*b^5*c^9*
d^7 - 378*a^6*b^4*c^8*d^8 + 312*a^7*b^3*c^7*d^9 - 153*a^8*b^2*c^6*d^10 + 42*a^9*b*c^5*d^11 - 5*a^10*c^4*d^12)*
x^3 + 7*(b^10*c^15*d - 6*a*b^9*c^14*d^2 + 9*a^2*b^8*c^13*d^3 + 24*a^3*b^7*c^12*d^4 - 126*a^4*b^6*c^11*d^5 + 25
2*a^5*b^5*c^10*d^6 - 294*a^6*b^4*c^9*d^7 + 216*a^7*b^3*c^8*d^8 - 99*a^8*b^2*c^7*d^9 + 26*a^9*b*c^6*d^10 - 3*a^
10*c^5*d^11)*x^2 + (b^10*c^16 - 2*a*b^9*c^15*d - 27*a^2*b^8*c^14*d^2 + 168*a^3*b^7*c^13*d^3 - 462*a^4*b^6*c^12
*d^4 + 756*a^5*b^5*c^11*d^5 - 798*a^6*b^4*c^10*d^6 + 552*a^7*b^3*c^9*d^7 - 243*a^8*b^2*c^8*d^8 + 62*a^9*b*c^7*
d^9 - 7*a^10*c^6*d^10)*x)
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2336 vs.
\(2 (209) = 418\).
time = 25.48, size = 2336, normalized size = 10.11
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)**2/(d*x+c)**8,x)
[Out]
-8*b**7*d*log(x + (-8*a**10*b**7*d**11/(a*d - b*c)**9 + 80*a**9*b**8*c*d**10/(a*d - b*c)**9 - 360*a**8*b**9*c*
*2*d**9/(a*d - b*c)**9 + 960*a**7*b**10*c**3*d**8/(a*d - b*c)**9 - 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 +
2016*a**5*b**12*c**5*d**6/(a*d - b*c)**9 - 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 + 960*a**3*b**14*c**7*d**4
/(a*d - b*c)**9 - 360*a**2*b**15*c**8*d**3/(a*d - b*c)**9 + 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**
2 - 8*b**17*c**10*d/(a*d - b*c)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 + 8*b**7*d*log(x + (8*a**10*b*
*7*d**11/(a*d - b*c)**9 - 80*a**9*b**8*c*d**10/(a*d - b*c)**9 + 360*a**8*b**9*c**2*d**9/(a*d - b*c)**9 - 960*a
**7*b**10*c**3*d**8/(a*d - b*c)**9 + 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 - 2016*a**5*b**12*c**5*d**6/(a*d
- b*c)**9 + 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 - 960*a**3*b**14*c**7*d**4/(a*d - b*c)**9 + 360*a**2*b**
15*c**8*d**3/(a*d - b*c)**9 - 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**2 + 8*b**17*c**10*d/(a*d - b*c
)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 + (-15*a**7*d**7 + 125*a**6*b*c*d**6 - 463*a**5*b**2*c**2*d*
*5 + 1007*a**4*b**3*c**3*d**4 - 1443*a**3*b**4*c**4*d**3 + 1497*a**2*b**5*c**5*d**2 - 1443*a*b**6*c**6*d - 105
*b**7*c**7 - 840*b**7*d**7*x**7 + x**6*(-420*a*b**6*d**7 - 5460*b**7*c*d**6) + x**5*(140*a**2*b**5*d**7 - 2800
*a*b**6*c*d**6 - 14980*b**7*c**2*d**5) + x**4*(-70*a**3*b**4*d**7 + 910*a**2*b**5*c*d**6 - 7910*a*b**6*c**2*d*
*5 - 22330*b**7*c**3*d**4) + x**3*(42*a**4*b**3*d**7 - 448*a**3*b**4*c*d**6 + 2492*a**2*b**5*c**2*d**5 - 12208
*a*b**6*c**3*d**4 - 19278*b**7*c**4*d**3) + x**2*(-28*a**5*b**2*d**7 + 266*a**4*b**3*c*d**6 - 1204*a**3*b**4*c
**2*d**5 + 3696*a**2*b**5*c**3*d**4 - 11004*a*b**6*c**4*d**3 - 9366*b**7*c**5*d**2) + x*(20*a**6*b*d**7 - 176*
a**5*b**2*c*d**6 + 706*a**4*b**3*c**2*d**5 - 1744*a**3*b**4*c**3*d**4 + 3156*a**2*b**5*c**4*d**3 - 5664*a*b**6
*c**5*d**2 - 2178*b**7*c**6*d))/(105*a**9*c**7*d**8 - 840*a**8*b*c**8*d**7 + 2940*a**7*b**2*c**9*d**6 - 5880*a
**6*b**3*c**10*d**5 + 7350*a**5*b**4*c**11*d**4 - 5880*a**4*b**5*c**12*d**3 + 2940*a**3*b**6*c**13*d**2 - 840*
a**2*b**7*c**14*d + 105*a*b**8*c**15 + x**8*(105*a**8*b*d**15 - 840*a**7*b**2*c*d**14 + 2940*a**6*b**3*c**2*d*
*13 - 5880*a**5*b**4*c**3*d**12 + 7350*a**4*b**5*c**4*d**11 - 5880*a**3*b**6*c**5*d**10 + 2940*a**2*b**7*c**6*
d**9 - 840*a*b**8*c**7*d**8 + 105*b**9*c**8*d**7) + x**7*(105*a**9*d**15 - 105*a**8*b*c*d**14 - 2940*a**7*b**2
*c**2*d**13 + 14700*a**6*b**3*c**3*d**12 - 33810*a**5*b**4*c**4*d**11 + 45570*a**4*b**5*c**5*d**10 - 38220*a**
3*b**6*c**6*d**9 + 19740*a**2*b**7*c**7*d**8 - 5775*a*b**8*c**8*d**7 + 735*b**9*c**9*d**6) + x**6*(735*a**9*c*
d**14 - 3675*a**8*b*c**2*d**13 + 2940*a**7*b**2*c**3*d**12 + 20580*a**6*b**3*c**4*d**11 - 72030*a**5*b**4*c**5
*d**10 + 113190*a**4*b**5*c**6*d**9 - 102900*a**3*b**6*c**7*d**8 + 55860*a**2*b**7*c**8*d**7 - 16905*a*b**8*c*
*9*d**6 + 2205*b**9*c**10*d**5) + x**5*(2205*a**9*c**2*d**13 - 13965*a**8*b*c**3*d**12 + 32340*a**7*b**2*c**4*
d**11 - 20580*a**6*b**3*c**5*d**10 - 51450*a**5*b**4*c**6*d**9 + 133770*a**4*b**5*c**7*d**8 - 144060*a**3*b**6
*c**8*d**7 + 85260*a**2*b**7*c**9*d**6 - 27195*a*b**8*c**10*d**5 + 3675*b**9*c**11*d**4) + x**4*(3675*a**9*c**
3*d**12 - 25725*a**8*b*c**4*d**11 + 73500*a**7*b**2*c**5*d**10 - 102900*a**6*b**3*c**6*d**9 + 51450*a**5*b**4*
c**7*d**8 + 51450*a**4*b**5*c**8*d**7 - 102900*a**3*b**6*c**9*d**6 + 73500*a**2*b**7*c**10*d**5 - 25725*a*b**8
*c**11*d**4 + 3675*b**9*c**12*d**3) + x**3*(3675*a**9*c**4*d**11 - 27195*a**8*b*c**5*d**10 + 85260*a**7*b**2*c
**6*d**9 - 144060*a**6*b**3*c**7*d**8 + 133770*a**5*b**4*c**8*d**7 - 51450*a**4*b**5*c**9*d**6 - 20580*a**3*b*
*6*c**10*d**5 + 32340*a**2*b**7*c**11*d**4 - 13965*a*b**8*c**12*d**3 + 2205*b**9*c**13*d**2) + x**2*(2205*a**9
*c**5*d**10 - 16905*a**8*b*c**6*d**9 + 55860*a**7*b**2*c**7*d**8 - 102900*a**6*b**3*c**8*d**7 + 113190*a**5*b*
*4*c**9*d**6 - 72030*a**4*b**5*c**10*d**5 + 20580*a**3*b**6*c**11*d**4 + 2940*a**2*b**7*c**12*d**3 - 3675*a*b*
*8*c**13*d**2 + 735*b**9*c**14*d) + x*(735*a**9*c**6*d**9 - 5775*a**8*b*c**7*d**8 + 19740*a**7*b**2*c**8*d**7
- 38220*a**6*b**3*c**9*d**6 + 45570*a**5*b**4*c**10*d**5 - 33810*a**4*b**5*c**11*d**4 + 14700*a**3*b**6*c**12*
d**3 - 2940*a**2*b**7*c**13*d**2 - 105*a*b**8*c**14*d + 105*b**9*c**15))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 866 vs.
\(2 (223) = 446\).
time = 0.01, size = 894, normalized size = 3.87 \begin {gather*} -\frac {8 b^{8} d \ln \left |x b+a\right |}{b^{10} c^{9}-9 b^{9} a d c^{8}+36 b^{8} a^{2} d^{2} c^{7}-84 b^{7} a^{3} d^{3} c^{6}+126 b^{6} a^{4} d^{4} c^{5}-126 b^{5} a^{5} d^{5} c^{4}+84 b^{4} a^{6} d^{6} c^{3}-36 b^{3} a^{7} d^{7} c^{2}+9 b^{2} a^{8} d^{8} c-b a^{9} d^{9}}+\frac {8 b^{7} d^{2} \ln \left |x d+c\right |}{b^{9} d c^{9}-9 b^{8} a d^{2} c^{8}+36 b^{7} a^{2} d^{3} c^{7}-84 b^{6} a^{3} d^{4} c^{6}+126 b^{5} a^{4} d^{5} c^{5}-126 b^{4} a^{5} d^{6} c^{4}+84 b^{3} a^{6} d^{7} c^{3}-36 b^{2} a^{7} d^{8} c^{2}+9 b a^{8} d^{9} c-a^{9} d^{10}}+\frac {\frac {1}{105} \left (\left (-840 b^{8} d^{7} c+840 b^{7} d^{8} a\right ) x^{7}+\left (-5460 b^{8} d^{6} c^{2}+5040 b^{7} d^{7} c a+420 b^{6} d^{8} a^{2}\right ) x^{6}+\left (-14980 b^{8} d^{5} c^{3}+12180 b^{7} d^{6} c^{2} a+2940 b^{6} d^{7} c a^{2}-140 b^{5} d^{8} a^{3}\right ) x^{5}+\left (-22330 b^{8} d^{4} c^{4}+14420 b^{7} d^{5} c^{3} a+8820 b^{6} d^{6} c^{2} a^{2}-980 b^{5} d^{7} c a^{3}+70 b^{4} d^{8} a^{4}\right ) x^{4}+\left (-19278 b^{8} d^{3} c^{5}+7070 b^{7} d^{4} c^{4} a+14700 b^{6} d^{5} c^{3} a^{2}-2940 b^{5} d^{6} c^{2} a^{3}+490 b^{4} d^{7} c a^{4}-42 b^{3} d^{8} a^{5}\right ) x^{3}+\left (-9366 b^{8} d^{2} c^{6}-1638 b^{7} d^{3} c^{5} a+14700 b^{6} d^{4} c^{4} a^{2}-4900 b^{5} d^{5} c^{3} a^{3}+1470 b^{4} d^{6} c^{2} a^{4}-294 b^{3} d^{7} c a^{5}+28 b^{2} d^{8} a^{6}\right ) x^{2}+\left (-2178 b^{8} d c^{7}-3486 b^{7} d^{2} c^{6} a+8820 b^{6} d^{3} c^{5} a^{2}-4900 b^{5} d^{4} c^{4} a^{3}+2450 b^{4} d^{5} c^{3} a^{4}-882 b^{3} d^{6} c^{2} a^{5}+196 b^{2} d^{7} c a^{6}-20 b d^{8} a^{7}\right ) x-105 b^{8} c^{8}-1338 b^{7} d c^{7} a+2940 b^{6} d^{2} c^{6} a^{2}-2940 b^{5} d^{3} c^{5} a^{3}+2450 b^{4} d^{4} c^{4} a^{4}-1470 b^{3} d^{5} c^{3} a^{5}+588 b^{2} d^{6} c^{2} a^{6}-140 b d^{7} c a^{7}+15 d^{8} a^{8}\right )}{\left (b c-d a\right )^{9} \left (x d+c\right )^{7} \left (x b+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x)
[Out]
-8*b^8*d*log(abs(b*x + a))/(b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c
^5*d^4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9) + 8*b^7*
d^2*log(abs(d*x + c))/(b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5*c^5
*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 + 9*a^8*b*c*d^9 - a^9*d^10) - 1/105*(105*
b^8*c^8 + 1338*a*b^7*c^7*d - 2940*a^2*b^6*c^6*d^2 + 2940*a^3*b^5*c^5*d^3 - 2450*a^4*b^4*c^4*d^4 + 1470*a^5*b^3
*c^3*d^5 - 588*a^6*b^2*c^2*d^6 + 140*a^7*b*c*d^7 - 15*a^8*d^8 + 840*(b^8*c*d^7 - a*b^7*d^8)*x^7 + 420*(13*b^8*
c^2*d^6 - 12*a*b^7*c*d^7 - a^2*b^6*d^8)*x^6 + 140*(107*b^8*c^3*d^5 - 87*a*b^7*c^2*d^6 - 21*a^2*b^6*c*d^7 + a^3
*b^5*d^8)*x^5 + 70*(319*b^8*c^4*d^4 - 206*a*b^7*c^3*d^5 - 126*a^2*b^6*c^2*d^6 + 14*a^3*b^5*c*d^7 - a^4*b^4*d^8
)*x^4 + 14*(1377*b^8*c^5*d^3 - 505*a*b^7*c^4*d^4 - 1050*a^2*b^6*c^3*d^5 + 210*a^3*b^5*c^2*d^6 - 35*a^4*b^4*c*d
^7 + 3*a^5*b^3*d^8)*x^3 + 14*(669*b^8*c^6*d^2 + 117*a*b^7*c^5*d^3 - 1050*a^2*b^6*c^4*d^4 + 350*a^3*b^5*c^3*d^5
- 105*a^4*b^4*c^2*d^6 + 21*a^5*b^3*c*d^7 - 2*a^6*b^2*d^8)*x^2 + 2*(1089*b^8*c^7*d + 1743*a*b^7*c^6*d^2 - 4410
*a^2*b^6*c^5*d^3 + 2450*a^3*b^5*c^4*d^4 - 1225*a^4*b^4*c^3*d^5 + 441*a^5*b^3*c^2*d^6 - 98*a^6*b^2*c*d^7 + 10*a
^7*b*d^8)*x)/((b*c - a*d)^9*(b*x + a)*(d*x + c)^7)
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Mupad [B]
time = 1.39, size = 1738, normalized size = 7.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*x)^2*(c + d*x)^8),x)
[Out]
(16*b^7*d*atanh((a^9*d^9 + b^9*c^9 + 20*a^2*b^7*c^7*d^2 - 28*a^3*b^6*c^6*d^3 + 14*a^4*b^5*c^5*d^4 + 14*a^5*b^4
*c^4*d^5 - 28*a^6*b^3*c^3*d^6 + 20*a^7*b^2*c^2*d^7 - 7*a*b^8*c^8*d - 7*a^8*b*c*d^8)/(a*d - b*c)^9 + (2*b*d*x*(
a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6
*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/(a*d - b*c)^9))/(a*d - b*c)^9 - ((15*a^7*d^7 + 105*b^7*c^7 - 14
97*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*d^5 + 1443*a*b^6*c^6*d - 12
5*a^6*b*c*d^6)/(105*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5
*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (4*b^5*x^5*(107*b^2*c^2*d^5 - a^2*d^7 +
20*a*b*c*d^6))/(3*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b
^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b^2*x^2*(2*a^5*d^7 + 669*b^5*c^5*d^2 +
786*a*b^4*c^4*d^3 - 264*a^2*b^3*c^3*d^4 + 86*a^3*b^2*c^2*d^5 - 19*a^4*b*c*d^6))/(15*(a^8*d^8 + b^8*c^8 + 28*a^
2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^
7*d - 8*a^7*b*c*d^7)) + (2*b^4*x^4*(a^3*d^7 + 319*b^3*c^3*d^4 + 113*a*b^2*c^2*d^5 - 13*a^2*b*c*d^6))/(3*(a^8*d
^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*
c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b*x*(1089*b^6*c^6*d - 10*a^6*d^7 + 2832*a*b^5*c^5*d^2 - 1578*a^
2*b^4*c^4*d^3 + 872*a^3*b^3*c^3*d^4 - 353*a^4*b^2*c^2*d^5 + 88*a^5*b*c*d^6))/(105*(a^8*d^8 + b^8*c^8 + 28*a^2*
b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*
d - 8*a^7*b*c*d^7)) + (8*b^7*d^7*x^7)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^
4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (4*b^6*x^6*(a*d^7 + 13*
b*c*d^6))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d
^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (2*b^3*x^3*(1377*b^4*c^4*d^3 - 3*a^4*d^7 + 872*a*b^
3*c^3*d^4 - 178*a^2*b^2*c^2*d^5 + 32*a^3*b*c*d^6))/(15*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^
5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)))/(x^7*(
a*d^7 + 7*b*c*d^6) + x^3*(35*a*c^4*d^3 + 21*b*c^5*d^2) + x^5*(21*a*c^2*d^5 + 35*b*c^3*d^4) + x^4*(35*a*c^3*d^4
+ 35*b*c^4*d^3) + a*c^7 + x*(b*c^7 + 7*a*c^6*d) + x^2*(21*a*c^5*d^2 + 7*b*c^6*d) + x^6*(21*b*c^2*d^5 + 7*a*c*
d^6) + b*d^7*x^8)
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