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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Maple [A]
time = 0.20, size = 192, normalized size = 0.95 Too large to display

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 = 0.44, size = 1418, normalized size = 7.02 \begin {gather*} \frac {b^{7} \log \left (b x + a\right )}{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}} - \frac {b^{7} \log \left (d x + c\right )}{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}} + \frac {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 \, {\left (13 \, b^{6} c d^{5} - a b^{5} d^{6}\right )} x^{5} + 70 \, {\left (107 \, b^{6} c^{2} d^{4} - 19 \, a b^{5} c d^{5} + 2 \, a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x}{420 \, {\left (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} + {\left (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}\right )} x^{7} + 7 \, {\left (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}\right )} x^{6} + 21 \, {\left (b^{7} c^{9} d^{5} - 7 \, a b^{6} c^{8} d^{6} + 21 \, a^{2} b^{5} c^{7} d^{7} - 35 \, 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}\right )} x^{5} + 35 \, {\left (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}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x\right )}} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1589 vs. \(2 (190) = 380\).
time = 0.60, size = 1589, normalized size = 7.87 \begin {gather*} \frac {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 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 210 \, {\left (13 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (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}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x + 420 \, {\left (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}\right )} \log \left (b x + a\right ) - 420 \, {\left (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}\right )} \log \left (d x + c\right )}{420 \, {\left (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} + {\left (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}\right )} x^{7} + 7 \, {\left (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}\right )} x^{6} + 21 \, {\left (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}\right )} x^{5} + 35 \, {\left (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}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x\right )}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1776 vs. \(2 (170) = 340\).
time = 8.34, size = 1776, normalized size = 8.79 \begin {gather*} - \frac {b^{7} \log {\left (x + \frac {- \frac {a^{9} b^{7} d^{9}}{\left (a d - b c\right )^{8}} + \frac {9 a^{8} b^{8} c d^{8}}{\left (a d - b c\right )^{8}} - \frac {36 a^{7} b^{9} c^{2} d^{7}}{\left (a d - b c\right )^{8}} + \frac {84 a^{6} b^{10} c^{3} d^{6}}{\left (a d - b c\right )^{8}} - \frac {126 a^{5} b^{11} c^{4} d^{5}}{\left (a d - b c\right )^{8}} + \frac {126 a^{4} b^{12} c^{5} d^{4}}{\left (a d - b c\right )^{8}} - \frac {84 a^{3} b^{13} c^{6} d^{3}}{\left (a d - b c\right )^{8}} + \frac {36 a^{2} b^{14} c^{7} d^{2}}{\left (a d - b c\right )^{8}} - \frac {9 a b^{15} c^{8} d}{\left (a d - b c\right )^{8}} + a b^{7} d + \frac {b^{16} c^{9}}{\left (a d - b c\right )^{8}} + b^{8} c}{2 b^{8} d} \right )}}{\left (a d - b c\right )^{8}} + \frac {b^{7} \log {\left (x + \frac {\frac {a^{9} b^{7} d^{9}}{\left (a d - b c\right )^{8}} - \frac {9 a^{8} b^{8} c d^{8}}{\left (a d - b c\right )^{8}} + \frac {36 a^{7} b^{9} c^{2} d^{7}}{\left (a d - b c\right )^{8}} - \frac {84 a^{6} b^{10} c^{3} d^{6}}{\left (a d - b c\right )^{8}} + \frac {126 a^{5} b^{11} c^{4} d^{5}}{\left (a d - b c\right )^{8}} - \frac {126 a^{4} b^{12} c^{5} d^{4}}{\left (a d - b c\right )^{8}} + \frac {84 a^{3} b^{13} c^{6} d^{3}}{\left (a d - b c\right )^{8}} - \frac {36 a^{2} b^{14} c^{7} d^{2}}{\left (a d - b c\right )^{8}} + \frac {9 a b^{15} c^{8} d}{\left (a d - b c\right )^{8}} + a b^{7} d - \frac {b^{16} c^{9}}{\left (a d - b c\right )^{8}} + b^{8} c}{2 b^{8} d} \right )}}{\left (a d - b c\right )^{8}} + \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 b^{6} d^{6} x^{6} + x^{5} \cdot \left (210 a b^{5} d^{6} - 2730 b^{6} c d^{5}\right ) + x^{4} \left (- 140 a^{2} b^{4} d^{6} + 1330 a b^{5} c d^{5} - 7490 b^{6} c^{2} d^{4}\right ) + x^{3} \cdot \left (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}\right ) + x^{2} \left (- 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}\right ) + x \left (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\right )}{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} \cdot \left (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}\right ) + x^{6} \cdot \left (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}\right ) + x^{5} \cdot \left (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}\right ) + x^{4} \cdot \left (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}\right ) + x^{3} \cdot \left (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}\right ) + x^{2} \cdot \left (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}\right ) + x \left (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\right )} \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**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 = 1.83, size = 703, normalized size = 3.48 \begin {gather*} \frac {b^{8} \log \left ({\left | b x + a \right |}\right )}{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}} - \frac {b^{7} d \log \left ({\left | d x + c \right |}\right )}{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}} + \frac {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 \, {\left (b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 210 \, {\left (13 \, b^{7} c^{2} d^{5} - 14 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (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}\right )} x^{4} + 35 \, {\left (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}\right )} x^{3} + 21 \, {\left (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}\right )} x^{2} + 7 \, {\left (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}\right )} x}{420 \, {\left (b c - a d\right )}^{8} {\left (d x + 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, algorithm="giac")

[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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