3.14.74 \(\int \frac {1}{(a+b x)^3 (c+d x)^8} \, dx\) [1374]
Optimal. Leaf size=276 \[ -\frac {b^7}{2 (b c-a d)^8 (a+b x)^2}+\frac {8 b^7 d}{(b c-a d)^9 (a+b x)}+\frac {d^2}{7 (b c-a d)^3 (c+d x)^7}+\frac {b d^2}{2 (b c-a d)^4 (c+d x)^6}+\frac {6 b^2 d^2}{5 (b c-a d)^5 (c+d x)^5}+\frac {5 b^3 d^2}{2 (b c-a d)^6 (c+d x)^4}+\frac {5 b^4 d^2}{(b c-a d)^7 (c+d x)^3}+\frac {21 b^5 d^2}{2 (b c-a d)^8 (c+d x)^2}+\frac {28 b^6 d^2}{(b c-a d)^9 (c+d x)}+\frac {36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac {36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}} \]
[Out]
-1/2*b^7/(-a*d+b*c)^8/(b*x+a)^2+8*b^7*d/(-a*d+b*c)^9/(b*x+a)+1/7*d^2/(-a*d+b*c)^3/(d*x+c)^7+1/2*b*d^2/(-a*d+b*
c)^4/(d*x+c)^6+6/5*b^2*d^2/(-a*d+b*c)^5/(d*x+c)^5+5/2*b^3*d^2/(-a*d+b*c)^6/(d*x+c)^4+5*b^4*d^2/(-a*d+b*c)^7/(d
*x+c)^3+21/2*b^5*d^2/(-a*d+b*c)^8/(d*x+c)^2+28*b^6*d^2/(-a*d+b*c)^9/(d*x+c)+36*b^7*d^2*ln(b*x+a)/(-a*d+b*c)^10
-36*b^7*d^2*ln(d*x+c)/(-a*d+b*c)^10
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 276, 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 {36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac {36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}+\frac {8 b^7 d}{(a+b x) (b c-a d)^9}-\frac {b^7}{2 (a+b x)^2 (b c-a d)^8}+\frac {28 b^6 d^2}{(c+d x) (b c-a d)^9}+\frac {21 b^5 d^2}{2 (c+d x)^2 (b c-a d)^8}+\frac {5 b^4 d^2}{(c+d x)^3 (b c-a d)^7}+\frac {5 b^3 d^2}{2 (c+d x)^4 (b c-a d)^6}+\frac {6 b^2 d^2}{5 (c+d x)^5 (b c-a d)^5}+\frac {b d^2}{2 (c+d x)^6 (b c-a d)^4}+\frac {d^2}{7 (c+d x)^7 (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)^3*(c + d*x)^8),x]
[Out]
-1/2*b^7/((b*c - a*d)^8*(a + b*x)^2) + (8*b^7*d)/((b*c - a*d)^9*(a + b*x)) + d^2/(7*(b*c - a*d)^3*(c + d*x)^7)
+ (b*d^2)/(2*(b*c - a*d)^4*(c + d*x)^6) + (6*b^2*d^2)/(5*(b*c - a*d)^5*(c + d*x)^5) + (5*b^3*d^2)/(2*(b*c - a
*d)^6*(c + d*x)^4) + (5*b^4*d^2)/((b*c - a*d)^7*(c + d*x)^3) + (21*b^5*d^2)/(2*(b*c - a*d)^8*(c + d*x)^2) + (2
8*b^6*d^2)/((b*c - a*d)^9*(c + d*x)) + (36*b^7*d^2*Log[a + b*x])/(b*c - a*d)^10 - (36*b^7*d^2*Log[c + d*x])/(b
*c - a*d)^10
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)^3 (c+d x)^8} \, dx &=\int \left (\frac {b^8}{(b c-a d)^8 (a+b x)^3}-\frac {8 b^8 d}{(b c-a d)^9 (a+b x)^2}+\frac {36 b^8 d^2}{(b c-a d)^{10} (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)^8}-\frac {3 b d^3}{(b c-a d)^4 (c+d x)^7}-\frac {6 b^2 d^3}{(b c-a d)^5 (c+d x)^6}-\frac {10 b^3 d^3}{(b c-a d)^6 (c+d x)^5}-\frac {15 b^4 d^3}{(b c-a d)^7 (c+d x)^4}-\frac {21 b^5 d^3}{(b c-a d)^8 (c+d x)^3}-\frac {28 b^6 d^3}{(b c-a d)^9 (c+d x)^2}-\frac {36 b^7 d^3}{(b c-a d)^{10} (c+d x)}\right ) \, dx\\ &=-\frac {b^7}{2 (b c-a d)^8 (a+b x)^2}+\frac {8 b^7 d}{(b c-a d)^9 (a+b x)}+\frac {d^2}{7 (b c-a d)^3 (c+d x)^7}+\frac {b d^2}{2 (b c-a d)^4 (c+d x)^6}+\frac {6 b^2 d^2}{5 (b c-a d)^5 (c+d x)^5}+\frac {5 b^3 d^2}{2 (b c-a d)^6 (c+d x)^4}+\frac {5 b^4 d^2}{(b c-a d)^7 (c+d x)^3}+\frac {21 b^5 d^2}{2 (b c-a d)^8 (c+d x)^2}+\frac {28 b^6 d^2}{(b c-a d)^9 (c+d x)}+\frac {36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac {36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 254, normalized size = 0.92 \begin {gather*} \frac {-\frac {35 b^7 (b c-a d)^2}{(a+b x)^2}+\frac {560 b^7 d (b c-a d)}{a+b x}+\frac {10 d^2 (b c-a d)^7}{(c+d x)^7}+\frac {35 b d^2 (b c-a d)^6}{(c+d x)^6}+\frac {84 b^2 d^2 (b c-a d)^5}{(c+d x)^5}+\frac {175 b^3 d^2 (b c-a d)^4}{(c+d x)^4}+\frac {350 b^4 d^2 (b c-a d)^3}{(c+d x)^3}+\frac {735 b^5 d^2 (b c-a d)^2}{(c+d x)^2}+\frac {1960 b^6 d^2 (b c-a d)}{c+d x}+2520 b^7 d^2 \log (a+b x)-2520 b^7 d^2 \log (c+d x)}{70 (b c-a d)^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)^3*(c + d*x)^8),x]
[Out]
((-35*b^7*(b*c - a*d)^2)/(a + b*x)^2 + (560*b^7*d*(b*c - a*d))/(a + b*x) + (10*d^2*(b*c - a*d)^7)/(c + d*x)^7
+ (35*b*d^2*(b*c - a*d)^6)/(c + d*x)^6 + (84*b^2*d^2*(b*c - a*d)^5)/(c + d*x)^5 + (175*b^3*d^2*(b*c - a*d)^4)/
(c + d*x)^4 + (350*b^4*d^2*(b*c - a*d)^3)/(c + d*x)^3 + (735*b^5*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (1960*b^6*d^
2*(b*c - a*d))/(c + d*x) + 2520*b^7*d^2*Log[a + b*x] - 2520*b^7*d^2*Log[c + d*x])/(70*(b*c - a*d)^10)
________________________________________________________________________________________
Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(2097\) vs. \(2(276)=552\).
time = 65.95, size = 2095, normalized size = 7.59
result too large to display
Antiderivative was successfully verified.
[In]
mathics('Integrate[(a + b*x)^(-3)/(c + d*x)^8,x]')
[Out]
-36 b ^ 7 d ^ 2 Log[c / d + x] / (a d - b c) ^ 10 + 36 b ^ 7 d ^ 2 Log[a / b + x] / (a d - b c) ^ 10 - (10 a ^
8 d ^ 8 - 95 a ^ 7 b c d ^ 7 + 409 a ^ 6 b ^ 2 c ^ 2 d ^ 6 - 1061 a ^ 5 b ^ 3 c ^ 3 d ^ 5 + 1879 a ^ 4 b ^ 4
c ^ 4 d ^ 4 - 2531 a ^ 3 b ^ 5 c ^ 5 d ^ 3 + 3349 a ^ 2 b ^ 6 c ^ 6 d ^ 2 + 595 a b ^ 7 c ^ 7 d - 35 b ^ 8 c ^
8 + 3 b d x (-5 a ^ 7 d ^ 7 + 51 a ^ 6 b c d ^ 6 - 243 a ^ 5 b ^ 2 c ^ 2 d ^ 5 + 737 a ^ 4 b ^ 3 c ^ 3 d ^ 4
- 1713 a ^ 3 b ^ 4 c ^ 4 d ^ 3 + 4167 a ^ 2 b ^ 5 c ^ 5 d ^ 2 + 3621 a b ^ 6 c ^ 6 d + 105 b ^ 7 c ^ 7) + 6 b
^ 2 d ^ 2 x ^ 2 (4 a ^ 6 d ^ 6 - 45 a ^ 5 b c d ^ 5 + 249 a ^ 4 b ^ 2 c ^ 2 d ^ 4 - 976 a ^ 3 b ^ 3 c ^ 3 d ^
3 + 3924 a ^ 2 b ^ 4 c ^ 4 d ^ 2 + 7515 a b ^ 5 c ^ 5 d + 1089 b ^ 6 c ^ 6) + 42 b ^ 3 d ^ 3 x ^ 3 (-a ^ 5 d ^
5 + 13 a ^ 4 b c d ^ 4 - 92 a ^ 3 b ^ 2 c ^ 2 d ^ 3 + 608 a ^ 2 b ^ 3 c ^ 3 d ^ 2 + 2163 a b ^ 4 c ^ 4 d + 66
9 b ^ 5 c ^ 5) + 42 b ^ 4 d ^ 4 x ^ 4 (2 a ^ 4 d ^ 4 - 33 a ^ 3 b c d ^ 3 + 387 a ^ 2 b ^ 2 c ^ 2 d ^ 2 + 2467
a b ^ 3 c ^ 3 d + 1377 b ^ 4 c ^ 4) + 210 b ^ 5 d ^ 5 x ^ 5 (-a ^ 3 d ^ 3 + 27 a ^ 2 b c d ^ 2 + 327 a b ^ 2
c ^ 2 d + 319 b ^ 3 c ^ 3) + 420 b ^ 6 d ^ 6 x ^ 6 (2 a ^ 2 d ^ 2 + 59 a b c d + 107 b ^ 2 c ^ 2) + 1260 b ^ 7
d ^ 7 x ^ 7 (3 a d + 13 b c) + 2520 b ^ 8 d ^ 8 x ^ 8) / (70 (a ^ 11 c ^ 7 d ^ 9 - 9 a ^ 10 b c ^ 8 d ^ 8 + 3
6 a ^ 9 b ^ 2 c ^ 9 d ^ 7 - 84 a ^ 8 b ^ 3 c ^ 10 d ^ 6 + 126 a ^ 7 b ^ 4 c ^ 11 d ^ 5 - 126 a ^ 6 b ^ 5 c ^ 1
2 d ^ 4 + 84 a ^ 5 b ^ 6 c ^ 13 d ^ 3 - 36 a ^ 4 b ^ 7 c ^ 14 d ^ 2 + 9 a ^ 3 b ^ 8 c ^ 15 d - a ^ 2 b ^ 9 c ^
16 - a c ^ 6 x (-7 a ^ 10 d ^ 10 + 61 a ^ 9 b c d ^ 9 - 234 a ^ 8 b ^ 2 c ^ 2 d ^ 8 + 516 a ^ 7 b ^ 3 c ^ 3 d
^ 7 - 714 a ^ 6 b ^ 4 c ^ 4 d ^ 6 + 630 a ^ 5 b ^ 5 c ^ 5 d ^ 5 - 336 a ^ 4 b ^ 6 c ^ 6 d ^ 4 + 84 a ^ 3 b ^
7 c ^ 7 d ^ 3 + 9 a ^ 2 b ^ 8 c ^ 8 d ^ 2 - 11 a b ^ 9 c ^ 9 d + 2 b ^ 10 c ^ 10) + b d ^ 6 x ^ 8 (2 a ^ 10 d
^ 10 - 11 a ^ 9 b c d ^ 9 + 9 a ^ 8 b ^ 2 c ^ 2 d ^ 8 + 84 a ^ 7 b ^ 3 c ^ 3 d ^ 7 - 336 a ^ 6 b ^ 4 c ^ 4 d ^
6 + 630 a ^ 5 b ^ 5 c ^ 5 d ^ 5 - 714 a ^ 4 b ^ 6 c ^ 6 d ^ 4 + 516 a ^ 3 b ^ 7 c ^ 7 d ^ 3 - 234 a ^ 2 b ^ 8
c ^ 8 d ^ 2 + 61 a b ^ 9 c ^ 9 d - 7 b ^ 10 c ^ 10) + c ^ 5 x ^ 2 (21 a ^ 11 d ^ 11 - 175 a ^ 10 b c d ^ 10 +
631 a ^ 9 b ^ 2 c ^ 2 d ^ 9 - 1269 a ^ 8 b ^ 3 c ^ 3 d ^ 8 + 1506 a ^ 7 b ^ 4 c ^ 4 d ^ 7 - 966 a ^ 6 b ^ 5 c
^ 5 d ^ 6 + 126 a ^ 5 b ^ 6 c ^ 6 d ^ 5 + 294 a ^ 4 b ^ 7 c ^ 7 d ^ 4 - 231 a ^ 3 b ^ 8 c ^ 8 d ^ 3 + 69 a ^
2 b ^ 9 c ^ 9 d ^ 2 - 5 a b ^ 10 c ^ 10 d - b ^ 11 c ^ 11) + 7 c ^ 4 d x ^ 3 (5 a ^ 11 d ^ 11 - 39 a ^ 10 b c
d ^ 10 + 127 a ^ 9 b ^ 2 c ^ 2 d ^ 9 - 213 a ^ 8 b ^ 3 c ^ 3 d ^ 8 + 162 a ^ 7 b ^ 4 c ^ 4 d ^ 7 + 42 a ^ 6 b
^ 5 c ^ 5 d ^ 6 - 210 a ^ 5 b ^ 6 c ^ 6 d ^ 5 + 198 a ^ 4 b ^ 7 c ^ 7 d ^ 4 - 87 a ^ 3 b ^ 8 c ^ 8 d ^ 3 + 13
a ^ 2 b ^ 9 c ^ 9 d ^ 2 + 3 a b ^ 10 c ^ 10 d - b ^ 11 c ^ 11) - 7 c ^ 3 d ^ 2 x ^ 4 (-5 a ^ 11 d ^ 11 + 35 a
^ 10 b c d ^ 10 - 93 a ^ 9 b ^ 2 c ^ 2 d ^ 9 + 87 a ^ 8 b ^ 3 c ^ 3 d ^ 8 + 102 a ^ 7 b ^ 4 c ^ 4 d ^ 7 - 378
a ^ 6 b ^ 5 c ^ 5 d ^ 6 + 462 a ^ 5 b ^ 6 c ^ 6 d ^ 5 - 282 a ^ 4 b ^ 7 c ^ 7 d ^ 4 + 63 a ^ 3 b ^ 8 c ^ 8 d ^
3 + 23 a ^ 2 b ^ 9 c ^ 9 d ^ 2 - 17 a b ^ 10 c ^ 10 d + 3 b ^ 11 c ^ 11) + 7 c ^ 2 d ^ 3 x ^ 5 (3 a ^ 11 d ^
11 - 17 a ^ 10 b c d ^ 10 + 23 a ^ 9 b ^ 2 c ^ 2 d ^ 9 + 63 a ^ 8 b ^ 3 c ^ 3 d ^ 8 - 282 a ^ 7 b ^ 4 c ^ 4 d
^ 7 + 462 a ^ 6 b ^ 5 c ^ 5 d ^ 6 - 378 a ^ 5 b ^ 6 c ^ 6 d ^ 5 + 102 a ^ 4 b ^ 7 c ^ 7 d ^ 4 + 87 a ^ 3 b ^ 8
c ^ 8 d ^ 3 - 93 a ^ 2 b ^ 9 c ^ 9 d ^ 2 + 35 a b ^ 10 c ^ 10 d - 5 b ^ 11 c ^ 11) + d ^ 5 x ^ 7 (a ^ 11 d ^
11 + 5 a ^ 10 b c d ^ 10 - 69 a ^ 9 b ^ 2 c ^ 2 d ^ 9 + 231 a ^ 8 b ^ 3 c ^ 3 d ^ 8 - 294 a ^ 7 b ^ 4 c ^ 4 d
^ 7 - 126 a ^ 6 b ^ 5 c ^ 5 d ^ 6 + 966 a ^ 5 b ^ 6 c ^ 6 d ^ 5 - 1506 a ^ 4 b ^ 7 c ^ 7 d ^ 4 + 1269 a ^ 3 b
^ 8 c ^ 8 d ^ 3 - 631 a ^ 2 b ^ 9 c ^ 9 d ^ 2 + 175 a b ^ 10 c ^ 10 d - 21 b ^ 11 c ^ 11) + b ^ 2 d ^ 7 x ^ 9
(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 c ^ 6 d ^ 3 - 36 a ^ 2 b ^ 7 c ^ 7 d ^ 2 + 9 a b ^ 8
c ^ 8 d - b ^ 9 c ^ 9) - 7 c d ^ 4 x ^ 6 (-a ^ 11 d ^ 11 + 3 a ^ 10 b c d ^ 10 + 13 a ^ 9 b ^ 2 c ^ 2 d ^ 9 -
87 a ^ 8 b ^ 3 c ^ 3 d ^ 8 + 198 a ^ 7 b ^ 4 c ^ 4 d ^ 7 - 210 a ^ 6 b ^ 5 c ^ 5 d ^ 6 + 42 a ^ 5 b ^ 6 c ^ 6
d ^ 5 + 162 a ^ 4 b ^ 7 c ^ 7 d ^ 4 - 213 a ^ 3 b ^ 8 c ^ 8 d ^ 3 + 127 a ^ 2 b ^ 9 c ^ 9 d ^ 2 - 39 a b ^ 10
c ^ 10 d + 5 b ^ 11 c ^ 11)))
________________________________________________________________________________________
Maple [A]
time = 0.24, size = 265, normalized size = 0.96
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {d^{2}}{7 \left (a d -b c \right )^{3} \left (d x +c \right )^{7}}-\frac {36 d^{2} b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{10}}-\frac {28 d^{2} b^{6}}{\left (a d -b c \right )^{9} \left (d x +c \right )}+\frac {21 d^{2} b^{5}}{2 \left (a d -b c \right )^{8} \left (d x +c \right )^{2}}-\frac {5 d^{2} b^{4}}{\left (a d -b c \right )^{7} \left (d x +c \right )^{3}}+\frac {5 d^{2} b^{3}}{2 \left (a d -b c \right )^{6} \left (d x +c \right )^{4}}-\frac {6 d^{2} b^{2}}{5 \left (a d -b c \right )^{5} \left (d x +c \right )^{5}}+\frac {d^{2} b}{2 \left (a d -b c \right )^{4} \left (d x +c \right )^{6}}-\frac {b^{7}}{2 \left (a d -b c \right )^{8} \left (b x +a \right )^{2}}+\frac {36 d^{2} b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{10}}-\frac {8 b^{7} d}{\left (a d -b c \right )^{9} \left (b x +a \right )}\) |
\(265\) |
| risch |
\(\text {Expression too large to display}\) |
\(1963\) |
| norman |
\(\text {Expression too large to display}\) |
\(2049\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)^3/(d*x+c)^8,x,method=_RETURNVERBOSE)
[Out]
-1/7*d^2/(a*d-b*c)^3/(d*x+c)^7-36*d^2/(a*d-b*c)^10*b^7*ln(d*x+c)-28*d^2/(a*d-b*c)^9*b^6/(d*x+c)+21/2*d^2/(a*d-
b*c)^8*b^5/(d*x+c)^2-5*d^2/(a*d-b*c)^7*b^4/(d*x+c)^3+5/2*d^2/(a*d-b*c)^6*b^3/(d*x+c)^4-6/5*d^2/(a*d-b*c)^5*b^2
/(d*x+c)^5+1/2*d^2/(a*d-b*c)^4*b/(d*x+c)^6-1/2*b^7/(a*d-b*c)^8/(b*x+a)^2+36*d^2/(a*d-b*c)^10*b^7*ln(b*x+a)-8*b
^7/(a*d-b*c)^9*d/(b*x+a)
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2399 vs.
\(2 (264) = 528\).
time = 0.51, size = 2399, normalized size = 8.69
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^3/(d*x+c)^8,x, algorithm="maxima")
[Out]
36*b^7*d^2*log(b*x + a)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c
^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9
+ a^10*d^10) - 36*b^7*d^2*log(d*x + c)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3
+ 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8
- 10*a^9*b*c*d^9 + a^10*d^10) + 1/70*(2520*b^8*d^8*x^8 - 35*b^8*c^8 + 595*a*b^7*c^7*d + 3349*a^2*b^6*c^6*d^2 -
2531*a^3*b^5*c^5*d^3 + 1879*a^4*b^4*c^4*d^4 - 1061*a^5*b^3*c^3*d^5 + 409*a^6*b^2*c^2*d^6 - 95*a^7*b*c*d^7 + 1
0*a^8*d^8 + 1260*(13*b^8*c*d^7 + 3*a*b^7*d^8)*x^7 + 420*(107*b^8*c^2*d^6 + 59*a*b^7*c*d^7 + 2*a^2*b^6*d^8)*x^6
+ 210*(319*b^8*c^3*d^5 + 327*a*b^7*c^2*d^6 + 27*a^2*b^6*c*d^7 - a^3*b^5*d^8)*x^5 + 42*(1377*b^8*c^4*d^4 + 246
7*a*b^7*c^3*d^5 + 387*a^2*b^6*c^2*d^6 - 33*a^3*b^5*c*d^7 + 2*a^4*b^4*d^8)*x^4 + 42*(669*b^8*c^5*d^3 + 2163*a*b
^7*c^4*d^4 + 608*a^2*b^6*c^3*d^5 - 92*a^3*b^5*c^2*d^6 + 13*a^4*b^4*c*d^7 - a^5*b^3*d^8)*x^3 + 6*(1089*b^8*c^6*
d^2 + 7515*a*b^7*c^5*d^3 + 3924*a^2*b^6*c^4*d^4 - 976*a^3*b^5*c^3*d^5 + 249*a^4*b^4*c^2*d^6 - 45*a^5*b^3*c*d^7
+ 4*a^6*b^2*d^8)*x^2 + 3*(105*b^8*c^7*d + 3621*a*b^7*c^6*d^2 + 4167*a^2*b^6*c^5*d^3 - 1713*a^3*b^5*c^4*d^4 +
737*a^4*b^4*c^3*d^5 - 243*a^5*b^3*c^2*d^6 + 51*a^6*b^2*c*d^7 - 5*a^7*b*d^8)*x)/(a^2*b^9*c^16 - 9*a^3*b^8*c^15*
d + 36*a^4*b^7*c^14*d^2 - 84*a^5*b^6*c^13*d^3 + 126*a^6*b^5*c^12*d^4 - 126*a^7*b^4*c^11*d^5 + 84*a^8*b^3*c^10*
d^6 - 36*a^9*b^2*c^9*d^7 + 9*a^10*b*c^8*d^8 - a^11*c^7*d^9 + (b^11*c^9*d^7 - 9*a*b^10*c^8*d^8 + 36*a^2*b^9*c^7
*d^9 - 84*a^3*b^8*c^6*d^10 + 126*a^4*b^7*c^5*d^11 - 126*a^5*b^6*c^4*d^12 + 84*a^6*b^5*c^3*d^13 - 36*a^7*b^4*c^
2*d^14 + 9*a^8*b^3*c*d^15 - a^9*b^2*d^16)*x^9 + (7*b^11*c^10*d^6 - 61*a*b^10*c^9*d^7 + 234*a^2*b^9*c^8*d^8 - 5
16*a^3*b^8*c^7*d^9 + 714*a^4*b^7*c^6*d^10 - 630*a^5*b^6*c^5*d^11 + 336*a^6*b^5*c^4*d^12 - 84*a^7*b^4*c^3*d^13
- 9*a^8*b^3*c^2*d^14 + 11*a^9*b^2*c*d^15 - 2*a^10*b*d^16)*x^8 + (21*b^11*c^11*d^5 - 175*a*b^10*c^10*d^6 + 631*
a^2*b^9*c^9*d^7 - 1269*a^3*b^8*c^8*d^8 + 1506*a^4*b^7*c^7*d^9 - 966*a^5*b^6*c^6*d^10 + 126*a^6*b^5*c^5*d^11 +
294*a^7*b^4*c^4*d^12 - 231*a^8*b^3*c^3*d^13 + 69*a^9*b^2*c^2*d^14 - 5*a^10*b*c*d^15 - a^11*d^16)*x^7 + 7*(5*b^
11*c^12*d^4 - 39*a*b^10*c^11*d^5 + 127*a^2*b^9*c^10*d^6 - 213*a^3*b^8*c^9*d^7 + 162*a^4*b^7*c^8*d^8 + 42*a^5*b
^6*c^7*d^9 - 210*a^6*b^5*c^6*d^10 + 198*a^7*b^4*c^5*d^11 - 87*a^8*b^3*c^4*d^12 + 13*a^9*b^2*c^3*d^13 + 3*a^10*
b*c^2*d^14 - a^11*c*d^15)*x^6 + 7*(5*b^11*c^13*d^3 - 35*a*b^10*c^12*d^4 + 93*a^2*b^9*c^11*d^5 - 87*a^3*b^8*c^1
0*d^6 - 102*a^4*b^7*c^9*d^7 + 378*a^5*b^6*c^8*d^8 - 462*a^6*b^5*c^7*d^9 + 282*a^7*b^4*c^6*d^10 - 63*a^8*b^3*c^
5*d^11 - 23*a^9*b^2*c^4*d^12 + 17*a^10*b*c^3*d^13 - 3*a^11*c^2*d^14)*x^5 + 7*(3*b^11*c^14*d^2 - 17*a*b^10*c^13
*d^3 + 23*a^2*b^9*c^12*d^4 + 63*a^3*b^8*c^11*d^5 - 282*a^4*b^7*c^10*d^6 + 462*a^5*b^6*c^9*d^7 - 378*a^6*b^5*c^
8*d^8 + 102*a^7*b^4*c^7*d^9 + 87*a^8*b^3*c^6*d^10 - 93*a^9*b^2*c^5*d^11 + 35*a^10*b*c^4*d^12 - 5*a^11*c^3*d^13
)*x^4 + 7*(b^11*c^15*d - 3*a*b^10*c^14*d^2 - 13*a^2*b^9*c^13*d^3 + 87*a^3*b^8*c^12*d^4 - 198*a^4*b^7*c^11*d^5
+ 210*a^5*b^6*c^10*d^6 - 42*a^6*b^5*c^9*d^7 - 162*a^7*b^4*c^8*d^8 + 213*a^8*b^3*c^7*d^9 - 127*a^9*b^2*c^6*d^10
+ 39*a^10*b*c^5*d^11 - 5*a^11*c^4*d^12)*x^3 + (b^11*c^16 + 5*a*b^10*c^15*d - 69*a^2*b^9*c^14*d^2 + 231*a^3*b^
8*c^13*d^3 - 294*a^4*b^7*c^12*d^4 - 126*a^5*b^6*c^11*d^5 + 966*a^6*b^5*c^10*d^6 - 1506*a^7*b^4*c^9*d^7 + 1269*
a^8*b^3*c^8*d^8 - 631*a^9*b^2*c^7*d^9 + 175*a^10*b*c^6*d^10 - 21*a^11*c^5*d^11)*x^2 + (2*a*b^10*c^16 - 11*a^2*
b^9*c^15*d + 9*a^3*b^8*c^14*d^2 + 84*a^4*b^7*c^13*d^3 - 336*a^5*b^6*c^12*d^4 + 630*a^6*b^5*c^11*d^5 - 714*a^7*
b^4*c^10*d^6 + 516*a^8*b^3*c^9*d^7 - 234*a^9*b^2*c^8*d^8 + 61*a^10*b*c^7*d^9 - 7*a^11*c^6*d^10)*x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3016 vs.
\(2 (264) = 528\).
time = 0.37, size = 3016, normalized size = 10.93
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^3/(d*x+c)^8,x, algorithm="fricas")
[Out]
-1/70*(35*b^9*c^9 - 630*a*b^8*c^8*d - 2754*a^2*b^7*c^7*d^2 + 5880*a^3*b^6*c^6*d^3 - 4410*a^4*b^5*c^5*d^4 + 294
0*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d^6 + 504*a^7*b^2*c^2*d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^
8 - a*b^8*d^9)*x^8 - 1260*(13*b^9*c^2*d^7 - 10*a*b^8*c*d^8 - 3*a^2*b^7*d^9)*x^7 - 420*(107*b^9*c^3*d^6 - 48*a*
b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b^6*d^9)*x^6 - 210*(319*b^9*c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2
*d^7 - 28*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^5 - 42*(1377*b^9*c^5*d^4 + 1090*a*b^8*c^4*d^5 - 2080*a^2*b^7*c^3*d^6
- 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 - 42*(669*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 155
5*a^2*b^7*c^4*d^5 - 700*a^3*b^6*c^3*d^6 + 105*a^4*b^5*c^2*d^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(1089*
b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2*b^7*c^5*d^4 - 4900*a^3*b^6*c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^
5*b^4*c^2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(105*b^9*c^8*d + 3516*a*b^8*c^7*d^2 + 546*a^2*b^7*c^
6*d^3 - 5880*a^3*b^6*c^5*d^4 + 2450*a^4*b^5*c^4*d^5 - 980*a^5*b^4*c^3*d^6 + 294*a^6*b^3*c^2*d^7 - 56*a^7*b^2*c
*d^8 + 5*a^8*b*d^9)*x - 2520*(b^9*d^9*x^9 + a^2*b^7*c^7*d^2 + (7*b^9*c*d^8 + 2*a*b^8*d^9)*x^8 + (21*b^9*c^2*d^
7 + 14*a*b^8*c*d^8 + a^2*b^7*d^9)*x^7 + 7*(5*b^9*c^3*d^6 + 6*a*b^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 7*(5*b^9*c^4
*d^5 + 10*a*b^8*c^3*d^6 + 3*a^2*b^7*c^2*d^7)*x^5 + 7*(3*b^9*c^5*d^4 + 10*a*b^8*c^4*d^5 + 5*a^2*b^7*c^3*d^6)*x^
4 + 7*(b^9*c^6*d^3 + 6*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5)*x^3 + (b^9*c^7*d^2 + 14*a*b^8*c^6*d^3 + 21*a^2*b^7*c
^5*d^4)*x^2 + (2*a*b^8*c^7*d^2 + 7*a^2*b^7*c^6*d^3)*x)*log(b*x + a) + 2520*(b^9*d^9*x^9 + a^2*b^7*c^7*d^2 + (7
*b^9*c*d^8 + 2*a*b^8*d^9)*x^8 + (21*b^9*c^2*d^7 + 14*a*b^8*c*d^8 + a^2*b^7*d^9)*x^7 + 7*(5*b^9*c^3*d^6 + 6*a*b
^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 7*(5*b^9*c^4*d^5 + 10*a*b^8*c^3*d^6 + 3*a^2*b^7*c^2*d^7)*x^5 + 7*(3*b^9*c^5*
d^4 + 10*a*b^8*c^4*d^5 + 5*a^2*b^7*c^3*d^6)*x^4 + 7*(b^9*c^6*d^3 + 6*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5)*x^3 +
(b^9*c^7*d^2 + 14*a*b^8*c^6*d^3 + 21*a^2*b^7*c^5*d^4)*x^2 + (2*a*b^8*c^7*d^2 + 7*a^2*b^7*c^6*d^3)*x)*log(d*x +
c))/(a^2*b^10*c^17 - 10*a^3*b^9*c^16*d + 45*a^4*b^8*c^15*d^2 - 120*a^5*b^7*c^14*d^3 + 210*a^6*b^6*c^13*d^4 -
252*a^7*b^5*c^12*d^5 + 210*a^8*b^4*c^11*d^6 - 120*a^9*b^3*c^10*d^7 + 45*a^10*b^2*c^9*d^8 - 10*a^11*b*c^8*d^9 +
a^12*c^7*d^10 + (b^12*c^10*d^7 - 10*a*b^11*c^9*d^8 + 45*a^2*b^10*c^8*d^9 - 120*a^3*b^9*c^7*d^10 + 210*a^4*b^8
*c^6*d^11 - 252*a^5*b^7*c^5*d^12 + 210*a^6*b^6*c^4*d^13 - 120*a^7*b^5*c^3*d^14 + 45*a^8*b^4*c^2*d^15 - 10*a^9*
b^3*c*d^16 + a^10*b^2*d^17)*x^9 + (7*b^12*c^11*d^6 - 68*a*b^11*c^10*d^7 + 295*a^2*b^10*c^9*d^8 - 750*a^3*b^9*c
^8*d^9 + 1230*a^4*b^8*c^7*d^10 - 1344*a^5*b^7*c^6*d^11 + 966*a^6*b^6*c^5*d^12 - 420*a^7*b^5*c^4*d^13 + 75*a^8*
b^4*c^3*d^14 + 20*a^9*b^3*c^2*d^15 - 13*a^10*b^2*c*d^16 + 2*a^11*b*d^17)*x^8 + (21*b^12*c^12*d^5 - 196*a*b^11*
c^11*d^6 + 806*a^2*b^10*c^10*d^7 - 1900*a^3*b^9*c^9*d^8 + 2775*a^4*b^8*c^8*d^9 - 2472*a^5*b^7*c^7*d^10 + 1092*
a^6*b^6*c^6*d^11 + 168*a^7*b^5*c^5*d^12 - 525*a^8*b^4*c^4*d^13 + 300*a^9*b^3*c^3*d^14 - 74*a^10*b^2*c^2*d^15 +
4*a^11*b*c*d^16 + a^12*d^17)*x^7 + 7*(5*b^12*c^13*d^4 - 44*a*b^11*c^12*d^5 + 166*a^2*b^10*c^11*d^6 - 340*a^3*
b^9*c^10*d^7 + 375*a^4*b^8*c^9*d^8 - 120*a^5*b^7*c^8*d^9 - 252*a^6*b^6*c^7*d^10 + 408*a^7*b^5*c^6*d^11 - 285*a
^8*b^4*c^5*d^12 + 100*a^9*b^3*c^4*d^13 - 10*a^10*b^2*c^3*d^14 - 4*a^11*b*c^2*d^15 + a^12*c*d^16)*x^6 + 7*(5*b^
12*c^14*d^3 - 40*a*b^11*c^13*d^4 + 128*a^2*b^10*c^12*d^5 - 180*a^3*b^9*c^11*d^6 - 15*a^4*b^8*c^10*d^7 + 480*a^
5*b^7*c^9*d^8 - 840*a^6*b^6*c^8*d^9 + 744*a^7*b^5*c^7*d^10 - 345*a^8*b^4*c^6*d^11 + 40*a^9*b^3*c^5*d^12 + 40*a
^10*b^2*c^4*d^13 - 20*a^11*b*c^3*d^14 + 3*a^12*c^2*d^15)*x^5 + 7*(3*b^12*c^15*d^2 - 20*a*b^11*c^14*d^3 + 40*a^
2*b^10*c^13*d^4 + 40*a^3*b^9*c^12*d^5 - 345*a^4*b^8*c^11*d^6 + 744*a^5*b^7*c^10*d^7 - 840*a^6*b^6*c^9*d^8 + 48
0*a^7*b^5*c^8*d^9 - 15*a^8*b^4*c^7*d^10 - 180*a^9*b^3*c^6*d^11 + 128*a^10*b^2*c^5*d^12 - 40*a^11*b*c^4*d^13 +
5*a^12*c^3*d^14)*x^4 + 7*(b^12*c^16*d - 4*a*b^11*c^15*d^2 - 10*a^2*b^10*c^14*d^3 + 100*a^3*b^9*c^13*d^4 - 285*
a^4*b^8*c^12*d^5 + 408*a^5*b^7*c^11*d^6 - 252*a^6*b^6*c^10*d^7 - 120*a^7*b^5*c^9*d^8 + 375*a^8*b^4*c^8*d^9 - 3
40*a^9*b^3*c^7*d^10 + 166*a^10*b^2*c^6*d^11 - 44*a^11*b*c^5*d^12 + 5*a^12*c^4*d^13)*x^3 + (b^12*c^17 + 4*a*b^1
1*c^16*d - 74*a^2*b^10*c^15*d^2 + 300*a^3*b^9*c^14*d^3 - 525*a^4*b^8*c^13*d^4 + 168*a^5*b^7*c^12*d^5 + 1092*a^
6*b^6*c^11*d^6 - 2472*a^7*b^5*c^10*d^7 + 2775*a^8*b^4*c^9*d^8 - 1900*a^9*b^3*c^8*d^9 + 806*a^10*b^2*c^7*d^10 -
196*a^11*b*c^6*d^11 + 21*a^12*c^5*d^12)*x^2 + (2*a*b^11*c^17 - 13*a^2*b^10*c^16*d + 20*a^3*b^9*c^15*d^2 + 75*
a^4*b^8*c^14*d^3 - 420*a^5*b^7*c^13*d^4 + 966*a^6*b^6*c^12*d^5 - 1344*a^7*b^5*c^11*d^6 + 1230*a^8*b^4*c^10*d^7
- 750*a^9*b^3*c^9*d^8 + 295*a^10*b^2*c^8*d^9 - 68*a^11*b*c^7*d^10 + 7*a^12*c^6*d^11)*x)
________________________________________________________________________________________
Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)**3/(d*x+c)**8,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1029 vs.
\(2 (264) = 528\).
time = 0.01, size = 1064, normalized size = 3.86 \begin {gather*} \frac {36 b^{8} d^{2} \ln \left |x b+a\right |}{b^{11} c^{10}-10 b^{10} a d c^{9}+45 b^{9} a^{2} d^{2} c^{8}-120 b^{8} a^{3} d^{3} c^{7}+210 b^{7} a^{4} d^{4} c^{6}-252 b^{6} a^{5} d^{5} c^{5}+210 b^{5} a^{6} d^{6} c^{4}-120 b^{4} a^{7} d^{7} c^{3}+45 b^{3} a^{8} d^{8} c^{2}-10 b^{2} a^{9} d^{9} c+b a^{10} d^{10}}-\frac {36 b^{7} d^{3} \ln \left |x d+c\right |}{b^{10} d c^{10}-10 b^{9} a d^{2} c^{9}+45 b^{8} a^{2} d^{3} c^{8}-120 b^{7} a^{3} d^{4} c^{7}+210 b^{6} a^{4} d^{5} c^{6}-252 b^{5} a^{5} d^{6} c^{5}+210 b^{4} a^{6} d^{7} c^{4}-120 b^{3} a^{7} d^{8} c^{3}+45 b^{2} a^{8} d^{9} c^{2}-10 b a^{9} d^{10} c+a^{10} d^{11}}+\frac {\frac {1}{70} \left (\left (2520 b^{9} d^{8} c-2520 b^{8} d^{9} a\right ) x^{8}+\left (16380 b^{9} d^{7} c^{2}-12600 b^{8} d^{8} c a-3780 b^{7} d^{9} a^{2}\right ) x^{7}+\left (44940 b^{9} d^{6} c^{3}-20160 b^{8} d^{7} c^{2} a-23940 b^{7} d^{8} c a^{2}-840 b^{6} d^{9} a^{3}\right ) x^{6}+\left (66990 b^{9} d^{5} c^{4}+1680 b^{8} d^{6} c^{3} a-63000 b^{7} d^{7} c^{2} a^{2}-5880 b^{6} d^{8} c a^{3}+210 b^{5} d^{9} a^{4}\right ) x^{5}+\left (57834 b^{9} d^{4} c^{5}+45780 b^{8} d^{5} c^{4} a-87360 b^{7} d^{6} c^{3} a^{2}-17640 b^{6} d^{7} c^{2} a^{3}+1470 b^{5} d^{8} c a^{4}-84 b^{4} d^{9} a^{5}\right ) x^{4}+\left (28098 b^{9} d^{3} c^{6}+62748 b^{8} d^{4} c^{5} a-65310 b^{7} d^{5} c^{4} a^{2}-29400 b^{6} d^{6} c^{3} a^{3}+4410 b^{5} d^{7} c^{2} a^{4}-588 b^{4} d^{8} c a^{5}+42 b^{3} d^{9} a^{6}\right ) x^{3}+\left (6534 b^{9} d^{2} c^{7}+38556 b^{8} d^{3} c^{6} a-21546 b^{7} d^{4} c^{5} a^{2}-29400 b^{6} d^{5} c^{4} a^{3}+7350 b^{5} d^{6} c^{3} a^{4}-1764 b^{4} d^{7} c^{2} a^{5}+294 b^{3} d^{8} c a^{6}-24 b^{2} d^{9} a^{7}\right ) x^{2}+\left (315 b^{9} d c^{8}+10548 b^{8} d^{2} c^{7} a+1638 b^{7} d^{3} c^{6} a^{2}-17640 b^{6} d^{4} c^{5} a^{3}+7350 b^{5} d^{5} c^{4} a^{4}-2940 b^{4} d^{6} c^{3} a^{5}+882 b^{3} d^{7} c^{2} a^{6}-168 b^{2} d^{8} c a^{7}+15 b d^{9} a^{8}\right ) x-35 b^{9} c^{9}+630 b^{8} d c^{8} a+2754 b^{7} d^{2} c^{7} a^{2}-5880 b^{6} d^{3} c^{6} a^{3}+4410 b^{5} d^{4} c^{5} a^{4}-2940 b^{4} d^{5} c^{4} a^{5}+1470 b^{3} d^{6} c^{3} a^{6}-504 b^{2} d^{7} c^{2} a^{7}+105 b d^{8} c a^{8}-10 d^{9} a^{9}\right )}{\left (b c-d a\right )^{10} \left (x d+c\right )^{7} \left (x b+a\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^3/(d*x+c)^8,x)
[Out]
36*b^8*d^2*log(abs(b*x + a))/(b^11*c^10 - 10*a*b^10*c^9*d + 45*a^2*b^9*c^8*d^2 - 120*a^3*b^8*c^7*d^3 + 210*a^4
*b^7*c^6*d^4 - 252*a^5*b^6*c^5*d^5 + 210*a^6*b^5*c^4*d^6 - 120*a^7*b^4*c^3*d^7 + 45*a^8*b^3*c^2*d^8 - 10*a^9*b
^2*c*d^9 + a^10*b*d^10) - 36*b^7*d^3*log(abs(d*x + c))/(b^10*c^10*d - 10*a*b^9*c^9*d^2 + 45*a^2*b^8*c^8*d^3 -
120*a^3*b^7*c^7*d^4 + 210*a^4*b^6*c^6*d^5 - 252*a^5*b^5*c^5*d^6 + 210*a^6*b^4*c^4*d^7 - 120*a^7*b^3*c^3*d^8 +
45*a^8*b^2*c^2*d^9 - 10*a^9*b*c*d^10 + a^10*d^11) - 1/70*(35*b^9*c^9 - 630*a*b^8*c^8*d - 2754*a^2*b^7*c^7*d^2
+ 5880*a^3*b^6*c^6*d^3 - 4410*a^4*b^5*c^5*d^4 + 2940*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d^6 + 504*a^7*b^2*c^2*
d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^8 - a*b^8*d^9)*x^8 - 1260*(13*b^9*c^2*d^7 - 10*a*b^8*c*d^8
- 3*a^2*b^7*d^9)*x^7 - 420*(107*b^9*c^3*d^6 - 48*a*b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b^6*d^9)*x^6 - 210*(
319*b^9*c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2*d^7 - 28*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^5 - 42*(1377*b^9*c
^5*d^4 + 1090*a*b^8*c^4*d^5 - 2080*a^2*b^7*c^3*d^6 - 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x
^4 - 42*(669*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 1555*a^2*b^7*c^4*d^5 - 700*a^3*b^6*c^3*d^6 + 105*a^4*b^5*c^2*d
^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(1089*b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2*b^7*c^5*d^4 - 4
900*a^3*b^6*c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^5*b^4*c^2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(
105*b^9*c^8*d + 3516*a*b^8*c^7*d^2 + 546*a^2*b^7*c^6*d^3 - 5880*a^3*b^6*c^5*d^4 + 2450*a^4*b^5*c^4*d^5 - 980*a
^5*b^4*c^3*d^6 + 294*a^6*b^3*c^2*d^7 - 56*a^7*b^2*c*d^8 + 5*a^8*b*d^9)*x)/((b*c - a*d)^10*(b*x + a)^2*(d*x + c
)^7)
________________________________________________________________________________________
Mupad [B]
time = 1.91, size = 2224, normalized size = 8.06
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*x)^3*(c + d*x)^8),x)
[Out]
(72*b^7*d^2*atanh((a^10*d^10 - b^10*c^10 - 27*a^2*b^8*c^8*d^2 + 48*a^3*b^7*c^7*d^3 - 42*a^4*b^6*c^6*d^4 + 42*a
^6*b^4*c^4*d^6 - 48*a^7*b^3*c^3*d^7 + 27*a^8*b^2*c^2*d^8 + 8*a*b^9*c^9*d - 8*a^9*b*c*d^9)/(a*d - b*c)^10 + (2*
b*d*x*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8))/(a*d - b*c)^10))/(a*d - b*c)^10 -
((10*a^8*d^8 - 35*b^8*c^8 + 3349*a^2*b^6*c^6*d^2 - 2531*a^3*b^5*c^5*d^3 + 1879*a^4*b^4*c^4*d^4 - 1061*a^5*b^3
*c^3*d^5 + 409*a^6*b^2*c^2*d^6 + 595*a*b^7*c^7*d - 95*a^7*b*c*d^7)/(70*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8)) + (3*b^2*x^2*(4*a^6*d^8 + 1089*b^6*c^6*d^2 + 7515*a*b^5*c^5*d^3 + 3924*a^2*b^4
*c^4*d^4 - 976*a^3*b^3*c^3*d^5 + 249*a^4*b^2*c^2*d^6 - 45*a^5*b*c*d^7))/(35*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8)) + (3*b^4*x^4*(2*a^4*d^8 + 1377*b^4*c^4*d^4 + 2467*a*b^3*c^3*d^5 + 387*a^2
*b^2*c^2*d^6 - 33*a^3*b*c*d^7))/(5*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8)) + (3
*b*x*(105*b^7*c^7*d - 5*a^7*d^8 + 3621*a*b^6*c^6*d^2 + 4167*a^2*b^5*c^5*d^3 - 1713*a^3*b^4*c^4*d^4 + 737*a^4*b
^3*c^3*d^5 - 243*a^5*b^2*c^2*d^6 + 51*a^6*b*c*d^7))/(70*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d -
9*a^8*b*c*d^8)) + (6*b^6*x^6*(2*a^2*d^8 + 107*b^2*c^2*d^6 + 59*a*b*c*d^7))/(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8) + (3*b^3*x^3*(669*b^5*c^5*d^3 - a^5*d^8 + 2163*a*b^4*c^4*d^4 + 608*a^2*b^3
*c^3*d^5 - 92*a^3*b^2*c^2*d^6 + 13*a^4*b*c*d^7))/(5*(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a
^8*b*c*d^8)) + (3*b^5*x^5*(319*b^3*c^3*d^5 - a^3*d^8 + 327*a*b^2*c^2*d^6 + 27*a^2*b*c*d^7))/(a^9*d^9 - b^9*c^9
- 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*b^8*c^8*d - 9*a^8*b*c*d^8) + (36*b^8*d^8*x^8)/(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8) + (18*b^6*d*x^7*(13*b^2*c*d^6 + 3*a*b*d^7))/(a^9*d^9 - b^9*c^9 - 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*b^8*c^8*d - 9*a^8*b*c*d^8))/(x^3*(7*b^2*c^6*d + 35*a^2*c^4*d^3 + 42*a*b*c^5*d^2) + x^6*(7*a^2*c*d^6 +
35*b^2*c^3*d^4 + 42*a*b*c^2*d^5) + x*(7*a^2*c^6*d + 2*a*b*c^7) + x^2*(b^2*c^7 + 21*a^2*c^5*d^2 + 14*a*b*c^6*d)
+ x^7*(a^2*d^7 + 21*b^2*c^2*d^5 + 14*a*b*c*d^6) + x^4*(35*a^2*c^3*d^4 + 21*b^2*c^5*d^2 + 70*a*b*c^4*d^3) + x^
5*(21*a^2*c^2*d^5 + 35*b^2*c^4*d^3 + 70*a*b*c^3*d^4) + x^8*(7*b^2*c*d^6 + 2*a*b*d^7) + a^2*c^7 + b^2*d^7*x^9)
________________________________________________________________________________________