3.14.72 \(\int \frac {1}{(a+b x) (c+d x)^8} \, dx\) [1372]

Optimal. Leaf size=202 \[ \frac {1}{7 (b c-a d) (c+d x)^7}+\frac {b}{6 (b c-a d)^2 (c+d x)^6}+\frac {b^2}{5 (b c-a d)^3 (c+d x)^5}+\frac {b^3}{4 (b c-a d)^4 (c+d x)^4}+\frac {b^4}{3 (b c-a d)^5 (c+d x)^3}+\frac {b^5}{2 (b c-a d)^6 (c+d x)^2}+\frac {b^6}{(b c-a d)^7 (c+d x)}+\frac {b^7 \log (a+b x)}{(b c-a d)^8}-\frac {b^7 \log (c+d x)}{(b c-a d)^8} \]

[Out]

1/7/(-a*d+b*c)/(d*x+c)^7+1/6*b/(-a*d+b*c)^2/(d*x+c)^6+1/5*b^2/(-a*d+b*c)^3/(d*x+c)^5+1/4*b^3/(-a*d+b*c)^4/(d*x
+c)^4+1/3*b^4/(-a*d+b*c)^5/(d*x+c)^3+1/2*b^5/(-a*d+b*c)^6/(d*x+c)^2+b^6/(-a*d+b*c)^7/(d*x+c)+b^7*ln(b*x+a)/(-a
*d+b*c)^8-b^7*ln(d*x+c)/(-a*d+b*c)^8

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Rubi [A]
time = 0.12, antiderivative size = 202, 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 \log (a+b x)}{(b c-a d)^8}-\frac {b^7 \log (c+d x)}{(b c-a d)^8}+\frac {b^6}{(c+d x) (b c-a d)^7}+\frac {b^5}{2 (c+d x)^2 (b c-a d)^6}+\frac {b^4}{3 (c+d x)^3 (b c-a d)^5}+\frac {b^3}{4 (c+d x)^4 (b c-a d)^4}+\frac {b^2}{5 (c+d x)^5 (b c-a d)^3}+\frac {b}{6 (c+d x)^6 (b c-a d)^2}+\frac {1}{7 (c+d x)^7 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(c + d*x)^8),x]

[Out]

1/(7*(b*c - a*d)*(c + d*x)^7) + b/(6*(b*c - a*d)^2*(c + d*x)^6) + b^2/(5*(b*c - a*d)^3*(c + d*x)^5) + b^3/(4*(
b*c - a*d)^4*(c + d*x)^4) + b^4/(3*(b*c - a*d)^5*(c + d*x)^3) + b^5/(2*(b*c - a*d)^6*(c + d*x)^2) + b^6/((b*c
- a*d)^7*(c + d*x)) + (b^7*Log[a + b*x])/(b*c - a*d)^8 - (b^7*Log[c + d*x])/(b*c - a*d)^8

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) (c+d x)^8} \, dx &=\int \left (\frac {b^8}{(b c-a d)^8 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^8}-\frac {b d}{(b c-a d)^2 (c+d x)^7}-\frac {b^2 d}{(b c-a d)^3 (c+d x)^6}-\frac {b^3 d}{(b c-a d)^4 (c+d x)^5}-\frac {b^4 d}{(b c-a d)^5 (c+d x)^4}-\frac {b^5 d}{(b c-a d)^6 (c+d x)^3}-\frac {b^6 d}{(b c-a d)^7 (c+d x)^2}-\frac {b^7 d}{(b c-a d)^8 (c+d x)}\right ) \, dx\\ &=\frac {1}{7 (b c-a d) (c+d x)^7}+\frac {b}{6 (b c-a d)^2 (c+d x)^6}+\frac {b^2}{5 (b c-a d)^3 (c+d x)^5}+\frac {b^3}{4 (b c-a d)^4 (c+d x)^4}+\frac {b^4}{3 (b c-a d)^5 (c+d x)^3}+\frac {b^5}{2 (b c-a d)^6 (c+d x)^2}+\frac {b^6}{(b c-a d)^7 (c+d x)}+\frac {b^7 \log (a+b x)}{(b c-a d)^8}-\frac {b^7 \log (c+d x)}{(b c-a d)^8}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 196, normalized size = 0.97 \begin {gather*} \frac {60 (b c-a d)^7+70 b (b c-a d)^6 (c+d x)+84 b^2 (b c-a d)^5 (c+d x)^2+105 b^3 (b c-a d)^4 (c+d x)^3+140 b^4 (b c-a d)^3 (c+d x)^4+210 b^5 (b c-a d)^2 (c+d x)^5+420 b^6 (b c-a d) (c+d x)^6+420 b^7 (c+d x)^7 \log (a+b x)-420 b^7 (c+d x)^7 \log (c+d x)}{420 (b c-a d)^8 (c+d x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(c + d*x)^8),x]

[Out]

(60*(b*c - a*d)^7 + 70*b*(b*c - a*d)^6*(c + d*x) + 84*b^2*(b*c - a*d)^5*(c + d*x)^2 + 105*b^3*(b*c - a*d)^4*(c
+ d*x)^3 + 140*b^4*(b*c - a*d)^3*(c + d*x)^4 + 210*b^5*(b*c - a*d)^2*(c + d*x)^5 + 420*b^6*(b*c - a*d)*(c + d
*x)^6 + 420*b^7*(c + d*x)^7*Log[a + b*x] - 420*b^7*(c + d*x)^7*Log[c + d*x])/(420*(b*c - a*d)^8*(c + d*x)^7)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(1884\) vs. \(2(202)=404\).
time = 27.84, size = 1882, normalized size = 9.32

result too large to display

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^(-1)/(c + d*x)^8,x]')

[Out]

(-60 a ^ 7 d ^ 7 + 490 a ^ 6 b c d ^ 6 + 70 a ^ 6 b d ^ 7 x - 1764 a ^ 5 b ^ 2 c ^ 2 d ^ 5 - 588 a ^ 5 b ^ 2 c
d ^ 6 x - 84 a ^ 5 b ^ 2 d ^ 7 x ^ 2 + 3675 a ^ 4 b ^ 3 c ^ 3 d ^ 4 + 2205 a ^ 4 b ^ 3 c ^ 2 d ^ 5 x + 735 a
^ 4 b ^ 3 c d ^ 6 x ^ 2 + 105 a ^ 4 b ^ 3 d ^ 7 x ^ 3 - 4900 a ^ 3 b ^ 4 c ^ 4 d ^ 3 - 4900 a ^ 3 b ^ 4 c ^ 3
d ^ 4 x - 2940 a ^ 3 b ^ 4 c ^ 2 d ^ 5 x ^ 2 - 980 a ^ 3 b ^ 4 c d ^ 6 x ^ 3 - 140 a ^ 3 b ^ 4 d ^ 7 x ^ 4 + 4
410 a ^ 2 b ^ 5 c ^ 5 d ^ 2 + 7350 a ^ 2 b ^ 5 c ^ 4 d ^ 3 x + 7350 a ^ 2 b ^ 5 c ^ 3 d ^ 4 x ^ 2 + 4410 a ^ 2
b ^ 5 c ^ 2 d ^ 5 x ^ 3 + 1470 a ^ 2 b ^ 5 c d ^ 6 x ^ 4 + 210 a ^ 2 b ^ 5 d ^ 7 x ^ 5 - 2940 a b ^ 6 c ^ 6 d
- 8820 a b ^ 6 c ^ 5 d ^ 2 x - 14700 a b ^ 6 c ^ 4 d ^ 3 x ^ 2 - 14700 a b ^ 6 c ^ 3 d ^ 4 x ^ 3 - 8820 a b ^
6 c ^ 2 d ^ 5 x ^ 4 - 2940 a b ^ 6 c d ^ 6 x ^ 5 - 420 a b ^ 6 d ^ 7 x ^ 6 - 420 b ^ 7 c ^ 7 Log[c / d + x] +
420 b ^ 7 c ^ 7 Log[a / b + x] + 1089 b ^ 7 c ^ 7 - 2940 b ^ 7 c ^ 6 d x Log[c / d + x] + 2940 b ^ 7 c ^ 6 d
x Log[a / b + x] + 4683 b ^ 7 c ^ 6 d x - 8820 b ^ 7 c ^ 5 d ^ 2 x ^ 2 Log[c / d + x] + 8820 b ^ 7 c ^ 5 d ^ 2
x ^ 2 Log[a / b + x] + 9639 b ^ 7 c ^ 5 d ^ 2 x ^ 2 - 14700 b ^ 7 c ^ 4 d ^ 3 x ^ 3 Log[c / d + x] + 11165 b
^ 7 c ^ 4 d ^ 3 x ^ 3 + 14700 b ^ 7 c ^ 4 d ^ 3 x ^ 3 Log[a / b + x] - 14700 b ^ 7 c ^ 3 d ^ 4 x ^ 4 Log[c / d
+ x] + 7490 b ^ 7 c ^ 3 d ^ 4 x ^ 4 + 14700 b ^ 7 c ^ 3 d ^ 4 x ^ 4 Log[a / b + x] - 8820 b ^ 7 c ^ 2 d ^ 5 x
^ 5 Log[c / d + x] + 2730 b ^ 7 c ^ 2 d ^ 5 x ^ 5 + 8820 b ^ 7 c ^ 2 d ^ 5 x ^ 5 Log[a / b + x] - 2940 b ^ 7
c d ^ 6 x ^ 6 Log[c / d + x] + 420 b ^ 7 c d ^ 6 x ^ 6 + 2940 b ^ 7 c d ^ 6 x ^ 6 Log[a / b + x] - 420 b ^ 7 d
^ 7 x ^ 7 Log[c / d + x] + 420 b ^ 7 d ^ 7 x ^ 7 Log[a / b + x]) / (420 (a ^ 8 c ^ 7 d ^ 8 + 7 a ^ 8 c ^ 6 d
^ 9 x + 21 a ^ 8 c ^ 5 d ^ 10 x ^ 2 + 35 a ^ 8 c ^ 4 d ^ 11 x ^ 3 + 35 a ^ 8 c ^ 3 d ^ 12 x ^ 4 + 21 a ^ 8 c ^
2 d ^ 13 x ^ 5 + 7 a ^ 8 c d ^ 14 x ^ 6 + a ^ 8 d ^ 15 x ^ 7 - 8 a ^ 7 b c ^ 8 d ^ 7 - 56 a ^ 7 b c ^ 7 d ^ 8
x - 168 a ^ 7 b c ^ 6 d ^ 9 x ^ 2 - 280 a ^ 7 b c ^ 5 d ^ 10 x ^ 3 - 280 a ^ 7 b c ^ 4 d ^ 11 x ^ 4 - 168 a ^
7 b c ^ 3 d ^ 12 x ^ 5 - 56 a ^ 7 b c ^ 2 d ^ 13 x ^ 6 - 8 a ^ 7 b c d ^ 14 x ^ 7 + 28 a ^ 6 b ^ 2 c ^ 9 d ^
6 + 196 a ^ 6 b ^ 2 c ^ 8 d ^ 7 x + 588 a ^ 6 b ^ 2 c ^ 7 d ^ 8 x ^ 2 + 980 a ^ 6 b ^ 2 c ^ 6 d ^ 9 x ^ 3 + 98
0 a ^ 6 b ^ 2 c ^ 5 d ^ 10 x ^ 4 + 588 a ^ 6 b ^ 2 c ^ 4 d ^ 11 x ^ 5 + 196 a ^ 6 b ^ 2 c ^ 3 d ^ 12 x ^ 6 + 2
8 a ^ 6 b ^ 2 c ^ 2 d ^ 13 x ^ 7 - 56 a ^ 5 b ^ 3 c ^ 10 d ^ 5 - 392 a ^ 5 b ^ 3 c ^ 9 d ^ 6 x - 1176 a ^ 5 b
^ 3 c ^ 8 d ^ 7 x ^ 2 - 1960 a ^ 5 b ^ 3 c ^ 7 d ^ 8 x ^ 3 - 1960 a ^ 5 b ^ 3 c ^ 6 d ^ 9 x ^ 4 - 1176 a ^ 5 b
^ 3 c ^ 5 d ^ 10 x ^ 5 - 392 a ^ 5 b ^ 3 c ^ 4 d ^ 11 x ^ 6 - 56 a ^ 5 b ^ 3 c ^ 3 d ^ 12 x ^ 7 + 70 a ^ 4 b
^ 4 c ^ 11 d ^ 4 + 490 a ^ 4 b ^ 4 c ^ 10 d ^ 5 x + 1470 a ^ 4 b ^ 4 c ^ 9 d ^ 6 x ^ 2 + 2450 a ^ 4 b ^ 4 c ^
8 d ^ 7 x ^ 3 + 2450 a ^ 4 b ^ 4 c ^ 7 d ^ 8 x ^ 4 + 1470 a ^ 4 b ^ 4 c ^ 6 d ^ 9 x ^ 5 + 490 a ^ 4 b ^ 4 c ^
5 d ^ 10 x ^ 6 + 70 a ^ 4 b ^ 4 c ^ 4 d ^ 11 x ^ 7 - 56 a ^ 3 b ^ 5 c ^ 12 d ^ 3 - 392 a ^ 3 b ^ 5 c ^ 11 d ^
4 x - 1176 a ^ 3 b ^ 5 c ^ 10 d ^ 5 x ^ 2 - 1960 a ^ 3 b ^ 5 c ^ 9 d ^ 6 x ^ 3 - 1960 a ^ 3 b ^ 5 c ^ 8 d ^ 7
x ^ 4 - 1176 a ^ 3 b ^ 5 c ^ 7 d ^ 8 x ^ 5 - 392 a ^ 3 b ^ 5 c ^ 6 d ^ 9 x ^ 6 - 56 a ^ 3 b ^ 5 c ^ 5 d ^ 10 x
^ 7 + 28 a ^ 2 b ^ 6 c ^ 13 d ^ 2 + 196 a ^ 2 b ^ 6 c ^ 12 d ^ 3 x + 588 a ^ 2 b ^ 6 c ^ 11 d ^ 4 x ^ 2 + 980
a ^ 2 b ^ 6 c ^ 10 d ^ 5 x ^ 3 + 980 a ^ 2 b ^ 6 c ^ 9 d ^ 6 x ^ 4 + 588 a ^ 2 b ^ 6 c ^ 8 d ^ 7 x ^ 5 + 196
a ^ 2 b ^ 6 c ^ 7 d ^ 8 x ^ 6 + 28 a ^ 2 b ^ 6 c ^ 6 d ^ 9 x ^ 7 - 8 a b ^ 7 c ^ 14 d - 56 a b ^ 7 c ^ 13 d ^
2 x - 168 a b ^ 7 c ^ 12 d ^ 3 x ^ 2 - 280 a b ^ 7 c ^ 11 d ^ 4 x ^ 3 - 280 a b ^ 7 c ^ 10 d ^ 5 x ^ 4 - 168 a
b ^ 7 c ^ 9 d ^ 6 x ^ 5 - 56 a b ^ 7 c ^ 8 d ^ 7 x ^ 6 - 8 a b ^ 7 c ^ 7 d ^ 8 x ^ 7 + b ^ 8 c ^ 15 + 7 b ^ 8
c ^ 14 d x + 21 b ^ 8 c ^ 13 d ^ 2 x ^ 2 + 35 b ^ 8 c ^ 12 d ^ 3 x ^ 3 + 35 b ^ 8 c ^ 11 d ^ 4 x ^ 4 + 21 b ^
8 c ^ 10 d ^ 5 x ^ 5 + 7 b ^ 8 c ^ 9 d ^ 6 x ^ 6 + b ^ 8 c ^ 8 d ^ 7 x ^ 7))

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Maple [A]
time = 0.20, size = 192, normalized size = 0.95

method	result	size
default	\(-\frac {1}{7 \left (a d -b c \right ) \left (d x +c \right )^{7}}-\frac {b^{2}}{5 \left (a d -b c \right )^{3} \left (d x +c \right )^{5}}-\frac {b^{4}}{3 \left (a d -b c \right )^{5} \left (d x +c \right )^{3}}-\frac {b^{6}}{\left (a d -b c \right )^{7} \left (d x +c \right )}+\frac {b^{3}}{4 \left (a d -b c \right )^{4} \left (d x +c \right )^{4}}+\frac {b^{5}}{2 \left (a d -b c \right )^{6} \left (d x +c \right )^{2}}-\frac {b^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{8}}+\frac {b}{6 \left (a d -b c \right )^{2} \left (d x +c \right )^{6}}+\frac {b^{7} \ln \left (b x +a \right )}{\left (a d -b c \right )^{8}}\)	\(192\)
risch	\(\frac {-\frac {b^{6} d^{6} x^{6}}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}+\frac {\left (a d -13 b c \right ) b^{5} d^{5} x^{5}}{2 a^{7} d^{7}-14 a^{6} b c \,d^{6}+42 a^{5} b^{2} c^{2} d^{5}-70 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d -2 b^{7} c^{7}}-\frac {d^{4} b^{4} \left (2 a^{2} d^{2}-19 a b c d +107 b^{2} c^{2}\right ) x^{4}}{6 \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {d^{3} b^{3} \left (3 a^{3} d^{3}-25 a^{2} b c \,d^{2}+101 a \,b^{2} c^{2} d -319 b^{3} c^{3}\right ) x^{3}}{12 a^{7} d^{7}-84 a^{6} b c \,d^{6}+252 a^{5} b^{2} c^{2} d^{5}-420 a^{4} b^{3} c^{3} d^{4}+420 a^{3} b^{4} c^{4} d^{3}-252 a^{2} b^{5} c^{5} d^{2}+84 a \,b^{6} c^{6} d -12 b^{7} c^{7}}-\frac {d^{2} b^{2} \left (4 a^{4} d^{4}-31 a^{3} b c \,d^{3}+109 a^{2} b^{2} c^{2} d^{2}-241 a \,b^{3} c^{3} d +459 b^{4} c^{4}\right ) x^{2}}{20 \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (10 a^{5} d^{5}-74 a^{4} b c \,d^{4}+241 a^{3} b^{2} c^{2} d^{3}-459 a^{2} b^{3} c^{3} d^{2}+591 a \,b^{4} c^{4} d -669 b^{5} c^{5}\right ) b d x}{60 a^{7} d^{7}-420 a^{6} b c \,d^{6}+1260 a^{5} b^{2} c^{2} d^{5}-2100 a^{4} b^{3} c^{3} d^{4}+2100 a^{3} b^{4} c^{4} d^{3}-1260 a^{2} b^{5} c^{5} d^{2}+420 a \,b^{6} c^{6} d -60 b^{7} c^{7}}-\frac {60 a^{6} d^{6}-430 a^{5} b c \,d^{5}+1334 a^{4} b^{2} c^{2} d^{4}-2341 a^{3} b^{3} c^{3} d^{3}+2559 a^{2} b^{4} c^{4} d^{2}-1851 a \,b^{5} c^{5} d +1089 b^{6} c^{6}}{420 \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (-b x -a \right )}{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 {b^{7} \ln \left (d x +c \right )}{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}}\)	\(1232\)
norman	\(\frac {-\frac {b^{6} d^{6} x^{6}}{a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}}+\frac {-60 a^{6} d^{13}+430 a^{5} b c \,d^{12}-1334 a^{4} b^{2} c^{2} d^{11}+2341 a^{3} b^{3} c^{3} d^{10}-2559 a^{2} b^{4} c^{4} d^{9}+1851 a \,b^{5} c^{5} d^{8}-1089 b^{6} c^{6} d^{7}}{420 d^{7} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (a \,b^{5} d^{8}-13 b^{6} c \,d^{7}\right ) x^{5}}{2 d^{2} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (-2 a^{2} b^{4} d^{9}+19 a \,b^{5} c \,d^{8}-107 b^{6} c^{2} d^{7}\right ) x^{4}}{6 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (3 a^{3} b^{3} d^{10}-25 a^{2} b^{4} c \,d^{9}+101 a \,b^{5} c^{2} d^{8}-319 b^{6} c^{3} d^{7}\right ) x^{3}}{12 d^{4} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (-4 a^{4} b^{2} d^{11}+31 a^{3} b^{3} c \,d^{10}-109 a^{2} b^{4} c^{2} d^{9}+241 a \,b^{5} c^{3} d^{8}-459 b^{6} c^{4} d^{7}\right ) x^{2}}{20 d^{5} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}+\frac {\left (10 a^{5} b \,d^{12}-74 a^{4} b^{2} c \,d^{11}+241 a^{3} b^{3} c^{2} d^{10}-459 a^{2} b^{4} c^{3} d^{9}+591 a \,b^{5} c^{4} d^{8}-669 b^{6} c^{5} d^{7}\right ) x}{60 d^{6} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}}{\left (d x +c \right )^{7}}+\frac {b^{7} \ln \left (b x +a \right )}{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 {b^{7} \ln \left (d x +c \right )}{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}}\)	\(1274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(d*x+c)^8,x,method=_RETURNVERBOSE)

[Out]

-1/7/(a*d-b*c)/(d*x+c)^7-1/5*b^2/(a*d-b*c)^3/(d*x+c)^5-1/3*b^4/(a*d-b*c)^5/(d*x+c)^3-b^6/(a*d-b*c)^7/(d*x+c)+1
/4*b^3/(a*d-b*c)^4/(d*x+c)^4+1/2*b^5/(a*d-b*c)^6/(d*x+c)^2-b^7/(a*d-b*c)^8*ln(d*x+c)+1/6*b/(a*d-b*c)^2/(d*x+c)
^6+b^7/(a*d-b*c)^8*ln(b*x+a)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1418 vs. \(2 (190) = 380\).
time = 0.36, size = 1418, normalized size = 7.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)^8,x, algorithm="maxima")

[Out]

b^7*log(b*x + a)/(b^8*c^8 - 8*a*b^7*c^7*d + 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^7*b*c*d^7 + a^8*d^8) - b^7*log(d*x + c)/(b^8*c^8 - 8*a*b^7*c^7*d +
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^7
*b*c*d^7 + a^8*d^8) + 1/420*(420*b^6*d^6*x^6 + 1089*b^6*c^6 - 1851*a*b^5*c^5*d + 2559*a^2*b^4*c^4*d^2 - 2341*a
^3*b^3*c^3*d^3 + 1334*a^4*b^2*c^2*d^4 - 430*a^5*b*c*d^5 + 60*a^6*d^6 + 210*(13*b^6*c*d^5 - a*b^5*d^6)*x^5 + 70
*(107*b^6*c^2*d^4 - 19*a*b^5*c*d^5 + 2*a^2*b^4*d^6)*x^4 + 35*(319*b^6*c^3*d^3 - 101*a*b^5*c^2*d^4 + 25*a^2*b^4
*c*d^5 - 3*a^3*b^3*d^6)*x^3 + 21*(459*b^6*c^4*d^2 - 241*a*b^5*c^3*d^3 + 109*a^2*b^4*c^2*d^4 - 31*a^3*b^3*c*d^5
+ 4*a^4*b^2*d^6)*x^2 + 7*(669*b^6*c^5*d - 591*a*b^5*c^4*d^2 + 459*a^2*b^4*c^3*d^3 - 241*a^3*b^3*c^2*d^4 + 74*
a^4*b^2*c*d^5 - 10*a^5*b*d^6)*x)/(b^7*c^14 - 7*a*b^6*c^13*d + 21*a^2*b^5*c^12*d^2 - 35*a^3*b^4*c^11*d^3 + 35*a
^4*b^3*c^10*d^4 - 21*a^5*b^2*c^9*d^5 + 7*a^6*b*c^8*d^6 - a^7*c^7*d^7 + (b^7*c^7*d^7 - 7*a*b^6*c^6*d^8 + 21*a^2
*b^5*c^5*d^9 - 35*a^3*b^4*c^4*d^10 + 35*a^4*b^3*c^3*d^11 - 21*a^5*b^2*c^2*d^12 + 7*a^6*b*c*d^13 - a^7*d^14)*x^
7 + 7*(b^7*c^8*d^6 - 7*a*b^6*c^7*d^7 + 21*a^2*b^5*c^6*d^8 - 35*a^3*b^4*c^5*d^9 + 35*a^4*b^3*c^4*d^10 - 21*a^5*
b^2*c^3*d^11 + 7*a^6*b*c^2*d^12 - a^7*c*d^13)*x^6 + 21*(b^7*c^9*d^5 - 7*a*b^6*c^8*d^6 + 21*a^2*b^5*c^7*d^7 - 3
5*a^3*b^4*c^6*d^8 + 35*a^4*b^3*c^5*d^9 - 21*a^5*b^2*c^4*d^10 + 7*a^6*b*c^3*d^11 - a^7*c^2*d^12)*x^5 + 35*(b^7*
c^10*d^4 - 7*a*b^6*c^9*d^5 + 21*a^2*b^5*c^8*d^6 - 35*a^3*b^4*c^7*d^7 + 35*a^4*b^3*c^6*d^8 - 21*a^5*b^2*c^5*d^9
+ 7*a^6*b*c^4*d^10 - a^7*c^3*d^11)*x^4 + 35*(b^7*c^11*d^3 - 7*a*b^6*c^10*d^4 + 21*a^2*b^5*c^9*d^5 - 35*a^3*b^
4*c^8*d^6 + 35*a^4*b^3*c^7*d^7 - 21*a^5*b^2*c^6*d^8 + 7*a^6*b*c^5*d^9 - a^7*c^4*d^10)*x^3 + 21*(b^7*c^12*d^2 -
7*a*b^6*c^11*d^3 + 21*a^2*b^5*c^10*d^4 - 35*a^3*b^4*c^9*d^5 + 35*a^4*b^3*c^8*d^6 - 21*a^5*b^2*c^7*d^7 + 7*a^6
*b*c^6*d^8 - a^7*c^5*d^9)*x^2 + 7*(b^7*c^13*d - 7*a*b^6*c^12*d^2 + 21*a^2*b^5*c^11*d^3 - 35*a^3*b^4*c^10*d^4 +
35*a^4*b^3*c^9*d^5 - 21*a^5*b^2*c^8*d^6 + 7*a^6*b*c^7*d^7 - a^7*c^6*d^8)*x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1589 vs. \(2 (190) = 380\).
time = 0.33, size = 1589, normalized size = 7.87

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)^8,x, algorithm="fricas")

[Out]

1/420*(1089*b^7*c^7 - 2940*a*b^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900*a^3*b^4*c^4*d^3 + 3675*a^4*b^3*c^3*d^4 -
1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d^6 - 60*a^7*d^7 + 420*(b^7*c*d^6 - a*b^6*d^7)*x^6 + 210*(13*b^7*c^2*d^5 -
14*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 70*(107*b^7*c^3*d^4 - 126*a*b^6*c^2*d^5 + 21*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7
)*x^4 + 35*(319*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 126*a^2*b^5*c^2*d^5 - 28*a^3*b^4*c*d^6 + 3*a^4*b^3*d^7)*x^3
+ 21*(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^3 + 350*a^2*b^5*c^3*d^4 - 140*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a
^5*b^2*d^7)*x^2 + 7*(669*b^7*c^6*d - 1260*a*b^6*c^5*d^2 + 1050*a^2*b^5*c^4*d^3 - 700*a^3*b^4*c^3*d^4 + 315*a^4
*b^3*c^2*d^5 - 84*a^5*b^2*c*d^6 + 10*a^6*b*d^7)*x + 420*(b^7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 +
35*b^7*c^3*d^4*x^4 + 35*b^7*c^4*d^3*x^3 + 21*b^7*c^5*d^2*x^2 + 7*b^7*c^6*d*x + b^7*c^7)*log(b*x + a) - 420*(b^
7*d^7*x^7 + 7*b^7*c*d^6*x^6 + 21*b^7*c^2*d^5*x^5 + 35*b^7*c^3*d^4*x^4 + 35*b^7*c^4*d^3*x^3 + 21*b^7*c^5*d^2*x^
2 + 7*b^7*c^6*d*x + b^7*c^7)*log(d*x + c))/(b^8*c^15 - 8*a*b^7*c^14*d + 28*a^2*b^6*c^13*d^2 - 56*a^3*b^5*c^12*
d^3 + 70*a^4*b^4*c^11*d^4 - 56*a^5*b^3*c^10*d^5 + 28*a^6*b^2*c^9*d^6 - 8*a^7*b*c^8*d^7 + a^8*c^7*d^8 + (b^8*c^
8*d^7 - 8*a*b^7*c^7*d^8 + 28*a^2*b^6*c^6*d^9 - 56*a^3*b^5*c^5*d^10 + 70*a^4*b^4*c^4*d^11 - 56*a^5*b^3*c^3*d^12
+ 28*a^6*b^2*c^2*d^13 - 8*a^7*b*c*d^14 + a^8*d^15)*x^7 + 7*(b^8*c^9*d^6 - 8*a*b^7*c^8*d^7 + 28*a^2*b^6*c^7*d^
8 - 56*a^3*b^5*c^6*d^9 + 70*a^4*b^4*c^5*d^10 - 56*a^5*b^3*c^4*d^11 + 28*a^6*b^2*c^3*d^12 - 8*a^7*b*c^2*d^13 +
a^8*c*d^14)*x^6 + 21*(b^8*c^10*d^5 - 8*a*b^7*c^9*d^6 + 28*a^2*b^6*c^8*d^7 - 56*a^3*b^5*c^7*d^8 + 70*a^4*b^4*c^
6*d^9 - 56*a^5*b^3*c^5*d^10 + 28*a^6*b^2*c^4*d^11 - 8*a^7*b*c^3*d^12 + a^8*c^2*d^13)*x^5 + 35*(b^8*c^11*d^4 -
8*a*b^7*c^10*d^5 + 28*a^2*b^6*c^9*d^6 - 56*a^3*b^5*c^8*d^7 + 70*a^4*b^4*c^7*d^8 - 56*a^5*b^3*c^6*d^9 + 28*a^6*
b^2*c^5*d^10 - 8*a^7*b*c^4*d^11 + a^8*c^3*d^12)*x^4 + 35*(b^8*c^12*d^3 - 8*a*b^7*c^11*d^4 + 28*a^2*b^6*c^10*d^
5 - 56*a^3*b^5*c^9*d^6 + 70*a^4*b^4*c^8*d^7 - 56*a^5*b^3*c^7*d^8 + 28*a^6*b^2*c^6*d^9 - 8*a^7*b*c^5*d^10 + a^8
*c^4*d^11)*x^3 + 21*(b^8*c^13*d^2 - 8*a*b^7*c^12*d^3 + 28*a^2*b^6*c^11*d^4 - 56*a^3*b^5*c^10*d^5 + 70*a^4*b^4*
c^9*d^6 - 56*a^5*b^3*c^8*d^7 + 28*a^6*b^2*c^7*d^8 - 8*a^7*b*c^6*d^9 + a^8*c^5*d^10)*x^2 + 7*(b^8*c^14*d - 8*a*
b^7*c^13*d^2 + 28*a^2*b^6*c^12*d^3 - 56*a^3*b^5*c^11*d^4 + 70*a^4*b^4*c^10*d^5 - 56*a^5*b^3*c^9*d^6 + 28*a^6*b
^2*c^8*d^7 - 8*a^7*b*c^7*d^8 + a^8*c^6*d^9)*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1776 vs. \(2 (170) = 340\).
time = 7.99, size = 1776, normalized size = 8.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)**8,x)

[Out]

-b**7*log(x + (-a**9*b**7*d**9/(a*d - b*c)**8 + 9*a**8*b**8*c*d**8/(a*d - b*c)**8 - 36*a**7*b**9*c**2*d**7/(a*
d - b*c)**8 + 84*a**6*b**10*c**3*d**6/(a*d - b*c)**8 - 126*a**5*b**11*c**4*d**5/(a*d - b*c)**8 + 126*a**4*b**1
2*c**5*d**4/(a*d - b*c)**8 - 84*a**3*b**13*c**6*d**3/(a*d - b*c)**8 + 36*a**2*b**14*c**7*d**2/(a*d - b*c)**8 -
9*a*b**15*c**8*d/(a*d - b*c)**8 + a*b**7*d + b**16*c**9/(a*d - b*c)**8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 +
b**7*log(x + (a**9*b**7*d**9/(a*d - b*c)**8 - 9*a**8*b**8*c*d**8/(a*d - b*c)**8 + 36*a**7*b**9*c**2*d**7/(a*d
- b*c)**8 - 84*a**6*b**10*c**3*d**6/(a*d - b*c)**8 + 126*a**5*b**11*c**4*d**5/(a*d - b*c)**8 - 126*a**4*b**12
*c**5*d**4/(a*d - b*c)**8 + 84*a**3*b**13*c**6*d**3/(a*d - b*c)**8 - 36*a**2*b**14*c**7*d**2/(a*d - b*c)**8 +
9*a*b**15*c**8*d/(a*d - b*c)**8 + a*b**7*d - b**16*c**9/(a*d - b*c)**8 + b**8*c)/(2*b**8*d))/(a*d - b*c)**8 +
(-60*a**6*d**6 + 430*a**5*b*c*d**5 - 1334*a**4*b**2*c**2*d**4 + 2341*a**3*b**3*c**3*d**3 - 2559*a**2*b**4*c**4
*d**2 + 1851*a*b**5*c**5*d - 1089*b**6*c**6 - 420*b**6*d**6*x**6 + x**5*(210*a*b**5*d**6 - 2730*b**6*c*d**5) +
x**4*(-140*a**2*b**4*d**6 + 1330*a*b**5*c*d**5 - 7490*b**6*c**2*d**4) + x**3*(105*a**3*b**3*d**6 - 875*a**2*b
**4*c*d**5 + 3535*a*b**5*c**2*d**4 - 11165*b**6*c**3*d**3) + x**2*(-84*a**4*b**2*d**6 + 651*a**3*b**3*c*d**5 -
2289*a**2*b**4*c**2*d**4 + 5061*a*b**5*c**3*d**3 - 9639*b**6*c**4*d**2) + x*(70*a**5*b*d**6 - 518*a**4*b**2*c
*d**5 + 1687*a**3*b**3*c**2*d**4 - 3213*a**2*b**4*c**3*d**3 + 4137*a*b**5*c**4*d**2 - 4683*b**6*c**5*d))/(420*
a**7*c**7*d**7 - 2940*a**6*b*c**8*d**6 + 8820*a**5*b**2*c**9*d**5 - 14700*a**4*b**3*c**10*d**4 + 14700*a**3*b*
*4*c**11*d**3 - 8820*a**2*b**5*c**12*d**2 + 2940*a*b**6*c**13*d - 420*b**7*c**14 + x**7*(420*a**7*d**14 - 2940
*a**6*b*c*d**13 + 8820*a**5*b**2*c**2*d**12 - 14700*a**4*b**3*c**3*d**11 + 14700*a**3*b**4*c**4*d**10 - 8820*a
**2*b**5*c**5*d**9 + 2940*a*b**6*c**6*d**8 - 420*b**7*c**7*d**7) + x**6*(2940*a**7*c*d**13 - 20580*a**6*b*c**2
*d**12 + 61740*a**5*b**2*c**3*d**11 - 102900*a**4*b**3*c**4*d**10 + 102900*a**3*b**4*c**5*d**9 - 61740*a**2*b*
*5*c**6*d**8 + 20580*a*b**6*c**7*d**7 - 2940*b**7*c**8*d**6) + x**5*(8820*a**7*c**2*d**12 - 61740*a**6*b*c**3*
d**11 + 185220*a**5*b**2*c**4*d**10 - 308700*a**4*b**3*c**5*d**9 + 308700*a**3*b**4*c**6*d**8 - 185220*a**2*b*
*5*c**7*d**7 + 61740*a*b**6*c**8*d**6 - 8820*b**7*c**9*d**5) + x**4*(14700*a**7*c**3*d**11 - 102900*a**6*b*c**
4*d**10 + 308700*a**5*b**2*c**5*d**9 - 514500*a**4*b**3*c**6*d**8 + 514500*a**3*b**4*c**7*d**7 - 308700*a**2*b
**5*c**8*d**6 + 102900*a*b**6*c**9*d**5 - 14700*b**7*c**10*d**4) + x**3*(14700*a**7*c**4*d**10 - 102900*a**6*b
*c**5*d**9 + 308700*a**5*b**2*c**6*d**8 - 514500*a**4*b**3*c**7*d**7 + 514500*a**3*b**4*c**8*d**6 - 308700*a**
2*b**5*c**9*d**5 + 102900*a*b**6*c**10*d**4 - 14700*b**7*c**11*d**3) + x**2*(8820*a**7*c**5*d**9 - 61740*a**6*
b*c**6*d**8 + 185220*a**5*b**2*c**7*d**7 - 308700*a**4*b**3*c**8*d**6 + 308700*a**3*b**4*c**9*d**5 - 185220*a*
*2*b**5*c**10*d**4 + 61740*a*b**6*c**11*d**3 - 8820*b**7*c**12*d**2) + x*(2940*a**7*c**6*d**8 - 20580*a**6*b*c
**7*d**7 + 61740*a**5*b**2*c**8*d**6 - 102900*a**4*b**3*c**9*d**5 + 102900*a**3*b**4*c**10*d**4 - 61740*a**2*b
**5*c**11*d**3 + 20580*a*b**6*c**12*d**2 - 2940*b**7*c**13*d))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (190) = 380\).
time = 0.01, size = 725, normalized size = 3.59 \begin {gather*} \frac {b^{8} \ln \left |x b+a\right |}{b^{9} c^{8}-8 b^{8} a d c^{7}+28 b^{7} a^{2} d^{2} c^{6}-56 b^{6} a^{3} d^{3} c^{5}+70 b^{5} a^{4} d^{4} c^{4}-56 b^{4} a^{5} d^{5} c^{3}+28 b^{3} a^{6} d^{6} c^{2}-8 b^{2} a^{7} d^{7} c+b a^{8} d^{8}}-\frac {b^{7} d \ln \left |x d+c\right |}{b^{8} d c^{8}-8 b^{7} a d^{2} c^{7}+28 b^{6} a^{2} d^{3} c^{6}-56 b^{5} a^{3} d^{4} c^{5}+70 b^{4} a^{4} d^{5} c^{4}-56 b^{3} a^{5} d^{6} c^{3}+28 b^{2} a^{6} d^{7} c^{2}-8 b a^{7} d^{8} c+a^{8} d^{9}}+\frac {\frac {1}{420} \left (\left (420 b^{7} d^{6} c-420 b^{6} d^{7} a\right ) x^{6}+\left (2730 b^{7} d^{5} c^{2}-2940 b^{6} d^{6} c a+210 b^{5} d^{7} a^{2}\right ) x^{5}+\left (7490 b^{7} d^{4} c^{3}-8820 b^{6} d^{5} c^{2} a+1470 b^{5} d^{6} c a^{2}-140 b^{4} d^{7} a^{3}\right ) x^{4}+\left (11165 b^{7} d^{3} c^{4}-14700 b^{6} d^{4} c^{3} a+4410 b^{5} d^{5} c^{2} a^{2}-980 b^{4} d^{6} c a^{3}+105 b^{3} d^{7} a^{4}\right ) x^{3}+\left (9639 b^{7} d^{2} c^{5}-14700 b^{6} d^{3} c^{4} a+7350 b^{5} d^{4} c^{3} a^{2}-2940 b^{4} d^{5} c^{2} a^{3}+735 b^{3} d^{6} c a^{4}-84 b^{2} d^{7} a^{5}\right ) x^{2}+\left (4683 b^{7} d c^{6}-8820 b^{6} d^{2} c^{5} a+7350 b^{5} d^{3} c^{4} a^{2}-4900 b^{4} d^{4} c^{3} a^{3}+2205 b^{3} d^{5} c^{2} a^{4}-588 b^{2} d^{6} c a^{5}+70 b d^{7} a^{6}\right ) x+1089 b^{7} c^{7}-2940 b^{6} d c^{6} a+4410 b^{5} d^{2} c^{5} a^{2}-4900 b^{4} d^{3} c^{4} a^{3}+3675 b^{3} d^{4} c^{3} a^{4}-1764 b^{2} d^{5} c^{2} a^{5}+490 b d^{6} c a^{6}-60 d^{7} a^{7}\right )}{\left (b c-d a\right )^{8} \left (x d+c\right )^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(d*x+c)^8,x)

[Out]

b^8*log(abs(b*x + a))/(b^9*c^8 - 8*a*b^8*c^7*d + 28*a^2*b^7*c^6*d^2 - 56*a^3*b^6*c^5*d^3 + 70*a^4*b^5*c^4*d^4
- 56*a^5*b^4*c^3*d^5 + 28*a^6*b^3*c^2*d^6 - 8*a^7*b^2*c*d^7 + a^8*b*d^8) - b^7*d*log(abs(d*x + c))/(b^8*c^8*d
- 8*a*b^7*c^7*d^2 + 28*a^2*b^6*c^6*d^3 - 56*a^3*b^5*c^5*d^4 + 70*a^4*b^4*c^4*d^5 - 56*a^5*b^3*c^3*d^6 + 28*a^6
*b^2*c^2*d^7 - 8*a^7*b*c*d^8 + a^8*d^9) + 1/420*(1089*b^7*c^7 - 2940*a*b^6*c^6*d + 4410*a^2*b^5*c^5*d^2 - 4900
*a^3*b^4*c^4*d^3 + 3675*a^4*b^3*c^3*d^4 - 1764*a^5*b^2*c^2*d^5 + 490*a^6*b*c*d^6 - 60*a^7*d^7 + 420*(b^7*c*d^6
- a*b^6*d^7)*x^6 + 210*(13*b^7*c^2*d^5 - 14*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 70*(107*b^7*c^3*d^4 - 126*a*b^6*
c^2*d^5 + 21*a^2*b^5*c*d^6 - 2*a^3*b^4*d^7)*x^4 + 35*(319*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 126*a^2*b^5*c^2*d^
5 - 28*a^3*b^4*c*d^6 + 3*a^4*b^3*d^7)*x^3 + 21*(459*b^7*c^5*d^2 - 700*a*b^6*c^4*d^3 + 350*a^2*b^5*c^3*d^4 - 14
0*a^3*b^4*c^2*d^5 + 35*a^4*b^3*c*d^6 - 4*a^5*b^2*d^7)*x^2 + 7*(669*b^7*c^6*d - 1260*a*b^6*c^5*d^2 + 1050*a^2*b
^5*c^4*d^3 - 700*a^3*b^4*c^3*d^4 + 315*a^4*b^3*c^2*d^5 - 84*a^5*b^2*c*d^6 + 10*a^6*b*d^7)*x)/((b*c - a*d)^8*(d
*x + c)^7)

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Mupad [B]
time = 0.87, size = 1299, normalized size = 6.43 \begin {gather*} \frac {2\,b^7\,\mathrm {atanh}\left (\frac {a^8\,d^8-6\,a^7\,b\,c\,d^7+14\,a^6\,b^2\,c^2\,d^6-14\,a^5\,b^3\,c^3\,d^5+14\,a^3\,b^5\,c^5\,d^3-14\,a^2\,b^6\,c^6\,d^2+6\,a\,b^7\,c^7\,d-b^8\,c^8}{{\left (a\,d-b\,c\right )}^8}+\frac {2\,b\,d\,x\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}{{\left (a\,d-b\,c\right )}^8}\right )}{{\left (a\,d-b\,c\right )}^8}-\frac {\frac {60\,a^6\,d^6-430\,a^5\,b\,c\,d^5+1334\,a^4\,b^2\,c^2\,d^4-2341\,a^3\,b^3\,c^3\,d^3+2559\,a^2\,b^4\,c^4\,d^2-1851\,a\,b^5\,c^5\,d+1089\,b^6\,c^6}{420\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}-\frac {b^3\,x^3\,\left (3\,a^3\,d^6-25\,a^2\,b\,c\,d^5+101\,a\,b^2\,c^2\,d^4-319\,b^3\,c^3\,d^3\right )}{12\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}+\frac {b^6\,d^6\,x^6}{a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7}-\frac {b^5\,x^5\,\left (a\,d^6-13\,b\,c\,d^5\right )}{2\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}+\frac {b^2\,x^2\,\left (4\,a^4\,d^6-31\,a^3\,b\,c\,d^5+109\,a^2\,b^2\,c^2\,d^4-241\,a\,b^3\,c^3\,d^3+459\,b^4\,c^4\,d^2\right )}{20\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}+\frac {b^4\,x^4\,\left (2\,a^2\,d^6-19\,a\,b\,c\,d^5+107\,b^2\,c^2\,d^4\right )}{6\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}-\frac {b\,x\,\left (10\,a^5\,d^6-74\,a^4\,b\,c\,d^5+241\,a^3\,b^2\,c^2\,d^4-459\,a^2\,b^3\,c^3\,d^3+591\,a\,b^4\,c^4\,d^2-669\,b^5\,c^5\,d\right )}{60\,\left (a^7\,d^7-7\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5-35\,a^4\,b^3\,c^3\,d^4+35\,a^3\,b^4\,c^4\,d^3-21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d-b^7\,c^7\right )}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)*(c + d*x)^8),x)

[Out]

(2*b^7*atanh((a^8*d^8 - b^8*c^8 - 14*a^2*b^6*c^6*d^2 + 14*a^3*b^5*c^5*d^3 - 14*a^5*b^3*c^3*d^5 + 14*a^6*b^2*c^
2*d^6 + 6*a*b^7*c^7*d - 6*a^7*b*c*d^7)/(a*d - b*c)^8 + (2*b*d*x*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a
^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6))/(a*d - b*c)^8))/(a*
d - b*c)^8 - ((60*a^6*d^6 + 1089*b^6*c^6 + 2559*a^2*b^4*c^4*d^2 - 2341*a^3*b^3*c^3*d^3 + 1334*a^4*b^2*c^2*d^4
- 1851*a*b^5*c^5*d - 430*a^5*b*c*d^5)/(420*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a
^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) - (b^3*x^3*(3*a^3*d^6 - 319*b^3*c^3*d^3
+ 101*a*b^2*c^2*d^4 - 25*a^2*b*c*d^5))/(12*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a
^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) + (b^6*d^6*x^6)/(a^7*d^7 - b^7*c^7 - 21*
a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6
) - (b^5*x^5*(a*d^6 - 13*b*c*d^5))/(2*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^
3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) + (b^2*x^2*(4*a^4*d^6 + 459*b^4*c^4*d^2 - 241
*a*b^3*c^3*d^3 + 109*a^2*b^2*c^2*d^4 - 31*a^3*b*c*d^5))/(20*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b
^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) + (b^4*x^4*(2*a^2*d^6 +
107*b^2*c^2*d^4 - 19*a*b*c*d^5))/(6*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3
*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)) - (b*x*(10*a^5*d^6 - 669*b^5*c^5*d + 591*a*b^4
*c^4*d^2 - 459*a^2*b^3*c^3*d^3 + 241*a^3*b^2*c^2*d^4 - 74*a^4*b*c*d^5))/(60*(a^7*d^7 - b^7*c^7 - 21*a^2*b^5*c^
5*d^2 + 35*a^3*b^4*c^4*d^3 - 35*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 7*a^6*b*c*d^6)))/(c^7 +
d^7*x^7 + 7*c*d^6*x^6 + 21*c^5*d^2*x^2 + 35*c^4*d^3*x^3 + 35*c^3*d^4*x^4 + 21*c^2*d^5*x^5 + 7*c^6*d*x)

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