3.14.73 \(\int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx\) [1373]
Optimal. Leaf size=231 \[ -\frac {b^7}{(b c-a d)^8 (a+b x)}-\frac {d}{7 (b c-a d)^2 (c+d x)^7}-\frac {b d}{3 (b c-a d)^3 (c+d x)^6}-\frac {3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac {b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac {5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac {3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac {7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9} \]
[Out]
-b^7/(-a*d+b*c)^8/(b*x+a)-1/7*d/(-a*d+b*c)^2/(d*x+c)^7-1/3*b*d/(-a*d+b*c)^3/(d*x+c)^6-3/5*b^2*d/(-a*d+b*c)^4/(
d*x+c)^5-b^3*d/(-a*d+b*c)^5/(d*x+c)^4-5/3*b^4*d/(-a*d+b*c)^6/(d*x+c)^3-3*b^5*d/(-a*d+b*c)^7/(d*x+c)^2-7*b^6*d/
(-a*d+b*c)^8/(d*x+c)-8*b^7*d*ln(b*x+a)/(-a*d+b*c)^9+8*b^7*d*ln(d*x+c)/(-a*d+b*c)^9
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Rubi [A]
time = 0.19, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46}
\begin {gather*} -\frac {b^7}{(a+b x) (b c-a d)^8}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9}-\frac {7 b^6 d}{(c+d x) (b c-a d)^8}-\frac {3 b^5 d}{(c+d x)^2 (b c-a d)^7}-\frac {5 b^4 d}{3 (c+d x)^3 (b c-a d)^6}-\frac {b^3 d}{(c+d x)^4 (b c-a d)^5}-\frac {3 b^2 d}{5 (c+d x)^5 (b c-a d)^4}-\frac {b d}{3 (c+d x)^6 (b c-a d)^3}-\frac {d}{7 (c+d x)^7 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((a + b*x)^2*(c + d*x)^8),x]
[Out]
-(b^7/((b*c - a*d)^8*(a + b*x))) - d/(7*(b*c - a*d)^2*(c + d*x)^7) - (b*d)/(3*(b*c - a*d)^3*(c + d*x)^6) - (3*
b^2*d)/(5*(b*c - a*d)^4*(c + d*x)^5) - (b^3*d)/((b*c - a*d)^5*(c + d*x)^4) - (5*b^4*d)/(3*(b*c - a*d)^6*(c + d
*x)^3) - (3*b^5*d)/((b*c - a*d)^7*(c + d*x)^2) - (7*b^6*d)/((b*c - a*d)^8*(c + d*x)) - (8*b^7*d*Log[a + b*x])/
(b*c - a*d)^9 + (8*b^7*d*Log[c + d*x])/(b*c - a*d)^9
Rule 46
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^8} \, dx &=\int \left (\frac {b^8}{(b c-a d)^8 (a+b x)^2}-\frac {8 b^8 d}{(b c-a d)^9 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^8}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)^7}+\frac {3 b^2 d^2}{(b c-a d)^4 (c+d x)^6}+\frac {4 b^3 d^2}{(b c-a d)^5 (c+d x)^5}+\frac {5 b^4 d^2}{(b c-a d)^6 (c+d x)^4}+\frac {6 b^5 d^2}{(b c-a d)^7 (c+d x)^3}+\frac {7 b^6 d^2}{(b c-a d)^8 (c+d x)^2}+\frac {8 b^7 d^2}{(b c-a d)^9 (c+d x)}\right ) \, dx\\ &=-\frac {b^7}{(b c-a d)^8 (a+b x)}-\frac {d}{7 (b c-a d)^2 (c+d x)^7}-\frac {b d}{3 (b c-a d)^3 (c+d x)^6}-\frac {3 b^2 d}{5 (b c-a d)^4 (c+d x)^5}-\frac {b^3 d}{(b c-a d)^5 (c+d x)^4}-\frac {5 b^4 d}{3 (b c-a d)^6 (c+d x)^3}-\frac {3 b^5 d}{(b c-a d)^7 (c+d x)^2}-\frac {7 b^6 d}{(b c-a d)^8 (c+d x)}-\frac {8 b^7 d \log (a+b x)}{(b c-a d)^9}+\frac {8 b^7 d \log (c+d x)}{(b c-a d)^9}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 213, normalized size = 0.92 \begin {gather*} -\frac {\frac {105 b^7 (b c-a d)}{a+b x}-\frac {15 d (-b c+a d)^7}{(c+d x)^7}+\frac {35 b d (b c-a d)^6}{(c+d x)^6}+\frac {63 b^2 d (b c-a d)^5}{(c+d x)^5}+\frac {105 b^3 d (b c-a d)^4}{(c+d x)^4}+\frac {175 b^4 d (b c-a d)^3}{(c+d x)^3}+\frac {315 b^5 d (b c-a d)^2}{(c+d x)^2}+\frac {735 b^6 d (b c-a d)}{c+d x}+840 b^7 d \log (a+b x)-840 b^7 d \log (c+d x)}{105 (b c-a d)^9} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b*x)^2*(c + d*x)^8),x]
[Out]
-1/105*((105*b^7*(b*c - a*d))/(a + b*x) - (15*d*(-(b*c) + a*d)^7)/(c + d*x)^7 + (35*b*d*(b*c - a*d)^6)/(c + d*
x)^6 + (63*b^2*d*(b*c - a*d)^5)/(c + d*x)^5 + (105*b^3*d*(b*c - a*d)^4)/(c + d*x)^4 + (175*b^4*d*(b*c - a*d)^3
)/(c + d*x)^3 + (315*b^5*d*(b*c - a*d)^2)/(c + d*x)^2 + (735*b^6*d*(b*c - a*d))/(c + d*x) + 840*b^7*d*Log[a +
b*x] - 840*b^7*d*Log[c + d*x])/(b*c - a*d)^9
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Maple [A]
time = 0.20, size = 223, normalized size = 0.97 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*x+a)^2/(d*x+c)^8,x,method=_RETURNVERBOSE)
[Out]
-1/7*d/(a*d-b*c)^2/(d*x+c)^7-8*d/(a*d-b*c)^9*b^7*ln(d*x+c)-7*d/(a*d-b*c)^8*b^6/(d*x+c)+3*d/(a*d-b*c)^7*b^5/(d*
x+c)^2-5/3*d/(a*d-b*c)^6*b^4/(d*x+c)^3+d/(a*d-b*c)^5*b^3/(d*x+c)^4-3/5*d/(a*d-b*c)^4*b^2/(d*x+c)^5+1/3*d/(a*d-
b*c)^3*b/(d*x+c)^6-b^7/(a*d-b*c)^8/(b*x+a)+8*d/(a*d-b*c)^9*b^7*ln(b*x+a)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1881 vs.
\(2 (223) = 446\).
time = 0.56, size = 1881, normalized size = 8.14 \begin {gather*} -\frac {8 \, b^{7} d \log \left (b x + a\right )}{b^{9} c^{9} - 9 \, a b^{8} c^{8} d + 36 \, a^{2} b^{7} c^{7} d^{2} - 84 \, a^{3} b^{6} c^{6} d^{3} + 126 \, a^{4} b^{5} c^{5} d^{4} - 126 \, a^{5} b^{4} c^{4} d^{5} + 84 \, a^{6} b^{3} c^{3} d^{6} - 36 \, a^{7} b^{2} c^{2} d^{7} + 9 \, a^{8} b c d^{8} - a^{9} d^{9}} + \frac {8 \, b^{7} d \log \left (d x + c\right )}{b^{9} c^{9} - 9 \, a b^{8} c^{8} d + 36 \, a^{2} b^{7} c^{7} d^{2} - 84 \, a^{3} b^{6} c^{6} d^{3} + 126 \, a^{4} b^{5} c^{5} d^{4} - 126 \, a^{5} b^{4} c^{4} d^{5} + 84 \, a^{6} b^{3} c^{3} d^{6} - 36 \, a^{7} b^{2} c^{2} d^{7} + 9 \, a^{8} b c d^{8} - a^{9} d^{9}} - \frac {840 \, b^{7} d^{7} x^{7} + 105 \, b^{7} c^{7} + 1443 \, a b^{6} c^{6} d - 1497 \, a^{2} b^{5} c^{5} d^{2} + 1443 \, a^{3} b^{4} c^{4} d^{3} - 1007 \, a^{4} b^{3} c^{3} d^{4} + 463 \, a^{5} b^{2} c^{2} d^{5} - 125 \, a^{6} b c d^{6} + 15 \, a^{7} d^{7} + 420 \, {\left (13 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 140 \, {\left (107 \, b^{7} c^{2} d^{5} + 20 \, a b^{6} c d^{6} - a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (319 \, b^{7} c^{3} d^{4} + 113 \, a b^{6} c^{2} d^{5} - 13 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 14 \, {\left (1377 \, b^{7} c^{4} d^{3} + 872 \, a b^{6} c^{3} d^{4} - 178 \, a^{2} b^{5} c^{2} d^{5} + 32 \, a^{3} b^{4} c d^{6} - 3 \, a^{4} b^{3} d^{7}\right )} x^{3} + 14 \, {\left (669 \, b^{7} c^{5} d^{2} + 786 \, a b^{6} c^{4} d^{3} - 264 \, a^{2} b^{5} c^{3} d^{4} + 86 \, a^{3} b^{4} c^{2} d^{5} - 19 \, a^{4} b^{3} c d^{6} + 2 \, a^{5} b^{2} d^{7}\right )} x^{2} + 2 \, {\left (1089 \, b^{7} c^{6} d + 2832 \, a b^{6} c^{5} d^{2} - 1578 \, a^{2} b^{5} c^{4} d^{3} + 872 \, a^{3} b^{4} c^{3} d^{4} - 353 \, a^{4} b^{3} c^{2} d^{5} + 88 \, a^{5} b^{2} c d^{6} - 10 \, a^{6} b d^{7}\right )} x}{105 \, {\left (a b^{8} c^{15} - 8 \, a^{2} b^{7} c^{14} d + 28 \, a^{3} b^{6} c^{13} d^{2} - 56 \, a^{4} b^{5} c^{12} d^{3} + 70 \, a^{5} b^{4} c^{11} d^{4} - 56 \, a^{6} b^{3} c^{10} d^{5} + 28 \, a^{7} b^{2} c^{9} d^{6} - 8 \, a^{8} b c^{8} d^{7} + a^{9} c^{7} d^{8} + {\left (b^{9} c^{8} d^{7} - 8 \, a b^{8} c^{7} d^{8} + 28 \, a^{2} b^{7} c^{6} d^{9} - 56 \, a^{3} b^{6} c^{5} d^{10} + 70 \, a^{4} b^{5} c^{4} d^{11} - 56 \, a^{5} b^{4} c^{3} d^{12} + 28 \, a^{6} b^{3} c^{2} d^{13} - 8 \, a^{7} b^{2} c d^{14} + a^{8} b d^{15}\right )} x^{8} + {\left (7 \, b^{9} c^{9} d^{6} - 55 \, a b^{8} c^{8} d^{7} + 188 \, a^{2} b^{7} c^{7} d^{8} - 364 \, a^{3} b^{6} c^{6} d^{9} + 434 \, a^{4} b^{5} c^{5} d^{10} - 322 \, a^{5} b^{4} c^{4} d^{11} + 140 \, a^{6} b^{3} c^{3} d^{12} - 28 \, a^{7} b^{2} c^{2} d^{13} - a^{8} b c d^{14} + a^{9} d^{15}\right )} x^{7} + 7 \, {\left (3 \, b^{9} c^{10} d^{5} - 23 \, a b^{8} c^{9} d^{6} + 76 \, a^{2} b^{7} c^{8} d^{7} - 140 \, a^{3} b^{6} c^{7} d^{8} + 154 \, a^{4} b^{5} c^{6} d^{9} - 98 \, a^{5} b^{4} c^{5} d^{10} + 28 \, a^{6} b^{3} c^{4} d^{11} + 4 \, a^{7} b^{2} c^{3} d^{12} - 5 \, a^{8} b c^{2} d^{13} + a^{9} c d^{14}\right )} x^{6} + 7 \, {\left (5 \, b^{9} c^{11} d^{4} - 37 \, a b^{8} c^{10} d^{5} + 116 \, a^{2} b^{7} c^{9} d^{6} - 196 \, a^{3} b^{6} c^{8} d^{7} + 182 \, a^{4} b^{5} c^{7} d^{8} - 70 \, a^{5} b^{4} c^{6} d^{9} - 28 \, a^{6} b^{3} c^{5} d^{10} + 44 \, a^{7} b^{2} c^{4} d^{11} - 19 \, a^{8} b c^{3} d^{12} + 3 \, a^{9} c^{2} d^{13}\right )} x^{5} + 35 \, {\left (b^{9} c^{12} d^{3} - 7 \, a b^{8} c^{11} d^{4} + 20 \, a^{2} b^{7} c^{10} d^{5} - 28 \, a^{3} b^{6} c^{9} d^{6} + 14 \, a^{4} b^{5} c^{8} d^{7} + 14 \, a^{5} b^{4} c^{7} d^{8} - 28 \, a^{6} b^{3} c^{6} d^{9} + 20 \, a^{7} b^{2} c^{5} d^{10} - 7 \, a^{8} b c^{4} d^{11} + a^{9} c^{3} d^{12}\right )} x^{4} + 7 \, {\left (3 \, b^{9} c^{13} d^{2} - 19 \, a b^{8} c^{12} d^{3} + 44 \, a^{2} b^{7} c^{11} d^{4} - 28 \, a^{3} b^{6} c^{10} d^{5} - 70 \, a^{4} b^{5} c^{9} d^{6} + 182 \, a^{5} b^{4} c^{8} d^{7} - 196 \, a^{6} b^{3} c^{7} d^{8} + 116 \, a^{7} b^{2} c^{6} d^{9} - 37 \, a^{8} b c^{5} d^{10} + 5 \, a^{9} c^{4} d^{11}\right )} x^{3} + 7 \, {\left (b^{9} c^{14} d - 5 \, a b^{8} c^{13} d^{2} + 4 \, a^{2} b^{7} c^{12} d^{3} + 28 \, a^{3} b^{6} c^{11} d^{4} - 98 \, a^{4} b^{5} c^{10} d^{5} + 154 \, a^{5} b^{4} c^{9} d^{6} - 140 \, a^{6} b^{3} c^{8} d^{7} + 76 \, a^{7} b^{2} c^{7} d^{8} - 23 \, a^{8} b c^{6} d^{9} + 3 \, a^{9} c^{5} d^{10}\right )} x^{2} + {\left (b^{9} c^{15} - a b^{8} c^{14} d - 28 \, a^{2} b^{7} c^{13} d^{2} + 140 \, a^{3} b^{6} c^{12} d^{3} - 322 \, a^{4} b^{5} c^{11} d^{4} + 434 \, a^{5} b^{4} c^{10} d^{5} - 364 \, a^{6} b^{3} c^{9} d^{6} + 188 \, a^{7} b^{2} c^{8} d^{7} - 55 \, a^{8} b c^{7} d^{8} + 7 \, a^{9} c^{6} d^{9}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="maxima")
[Out]
-8*b^7*d*log(b*x + a)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4
- 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) + 8*b^7*d*log(d*x
+ c)/(b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^
4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8*b*c*d^8 - a^9*d^9) - 1/105*(840*b^7*d^7*x^7 + 105*b^7*
c^7 + 1443*a*b^6*c^6*d - 1497*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*
d^5 - 125*a^6*b*c*d^6 + 15*a^7*d^7 + 420*(13*b^7*c*d^6 + a*b^6*d^7)*x^6 + 140*(107*b^7*c^2*d^5 + 20*a*b^6*c*d^
6 - a^2*b^5*d^7)*x^5 + 70*(319*b^7*c^3*d^4 + 113*a*b^6*c^2*d^5 - 13*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 14*(137
7*b^7*c^4*d^3 + 872*a*b^6*c^3*d^4 - 178*a^2*b^5*c^2*d^5 + 32*a^3*b^4*c*d^6 - 3*a^4*b^3*d^7)*x^3 + 14*(669*b^7*
c^5*d^2 + 786*a*b^6*c^4*d^3 - 264*a^2*b^5*c^3*d^4 + 86*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 2*a^5*b^2*d^7)*x^2
+ 2*(1089*b^7*c^6*d + 2832*a*b^6*c^5*d^2 - 1578*a^2*b^5*c^4*d^3 + 872*a^3*b^4*c^3*d^4 - 353*a^4*b^3*c^2*d^5 +
88*a^5*b^2*c*d^6 - 10*a^6*b*d^7)*x)/(a*b^8*c^15 - 8*a^2*b^7*c^14*d + 28*a^3*b^6*c^13*d^2 - 56*a^4*b^5*c^12*d^
3 + 70*a^5*b^4*c^11*d^4 - 56*a^6*b^3*c^10*d^5 + 28*a^7*b^2*c^9*d^6 - 8*a^8*b*c^8*d^7 + a^9*c^7*d^8 + (b^9*c^8*
d^7 - 8*a*b^8*c^7*d^8 + 28*a^2*b^7*c^6*d^9 - 56*a^3*b^6*c^5*d^10 + 70*a^4*b^5*c^4*d^11 - 56*a^5*b^4*c^3*d^12 +
28*a^6*b^3*c^2*d^13 - 8*a^7*b^2*c*d^14 + a^8*b*d^15)*x^8 + (7*b^9*c^9*d^6 - 55*a*b^8*c^8*d^7 + 188*a^2*b^7*c^
7*d^8 - 364*a^3*b^6*c^6*d^9 + 434*a^4*b^5*c^5*d^10 - 322*a^5*b^4*c^4*d^11 + 140*a^6*b^3*c^3*d^12 - 28*a^7*b^2*
c^2*d^13 - a^8*b*c*d^14 + a^9*d^15)*x^7 + 7*(3*b^9*c^10*d^5 - 23*a*b^8*c^9*d^6 + 76*a^2*b^7*c^8*d^7 - 140*a^3*
b^6*c^7*d^8 + 154*a^4*b^5*c^6*d^9 - 98*a^5*b^4*c^5*d^10 + 28*a^6*b^3*c^4*d^11 + 4*a^7*b^2*c^3*d^12 - 5*a^8*b*c
^2*d^13 + a^9*c*d^14)*x^6 + 7*(5*b^9*c^11*d^4 - 37*a*b^8*c^10*d^5 + 116*a^2*b^7*c^9*d^6 - 196*a^3*b^6*c^8*d^7
+ 182*a^4*b^5*c^7*d^8 - 70*a^5*b^4*c^6*d^9 - 28*a^6*b^3*c^5*d^10 + 44*a^7*b^2*c^4*d^11 - 19*a^8*b*c^3*d^12 + 3
*a^9*c^2*d^13)*x^5 + 35*(b^9*c^12*d^3 - 7*a*b^8*c^11*d^4 + 20*a^2*b^7*c^10*d^5 - 28*a^3*b^6*c^9*d^6 + 14*a^4*b
^5*c^8*d^7 + 14*a^5*b^4*c^7*d^8 - 28*a^6*b^3*c^6*d^9 + 20*a^7*b^2*c^5*d^10 - 7*a^8*b*c^4*d^11 + a^9*c^3*d^12)*
x^4 + 7*(3*b^9*c^13*d^2 - 19*a*b^8*c^12*d^3 + 44*a^2*b^7*c^11*d^4 - 28*a^3*b^6*c^10*d^5 - 70*a^4*b^5*c^9*d^6 +
182*a^5*b^4*c^8*d^7 - 196*a^6*b^3*c^7*d^8 + 116*a^7*b^2*c^6*d^9 - 37*a^8*b*c^5*d^10 + 5*a^9*c^4*d^11)*x^3 + 7
*(b^9*c^14*d - 5*a*b^8*c^13*d^2 + 4*a^2*b^7*c^12*d^3 + 28*a^3*b^6*c^11*d^4 - 98*a^4*b^5*c^10*d^5 + 154*a^5*b^4
*c^9*d^6 - 140*a^6*b^3*c^8*d^7 + 76*a^7*b^2*c^7*d^8 - 23*a^8*b*c^6*d^9 + 3*a^9*c^5*d^10)*x^2 + (b^9*c^15 - a*b
^8*c^14*d - 28*a^2*b^7*c^13*d^2 + 140*a^3*b^6*c^12*d^3 - 322*a^4*b^5*c^11*d^4 + 434*a^5*b^4*c^10*d^5 - 364*a^6
*b^3*c^9*d^6 + 188*a^7*b^2*c^8*d^7 - 55*a^8*b*c^7*d^8 + 7*a^9*c^6*d^9)*x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2264 vs.
\(2 (223) = 446\).
time = 0.60, size = 2264, normalized size = 9.80 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="fricas")
[Out]
-1/105*(105*b^8*c^8 + 1338*a*b^7*c^7*d - 2940*a^2*b^6*c^6*d^2 + 2940*a^3*b^5*c^5*d^3 - 2450*a^4*b^4*c^4*d^4 +
1470*a^5*b^3*c^3*d^5 - 588*a^6*b^2*c^2*d^6 + 140*a^7*b*c*d^7 - 15*a^8*d^8 + 840*(b^8*c*d^7 - a*b^7*d^8)*x^7 +
420*(13*b^8*c^2*d^6 - 12*a*b^7*c*d^7 - a^2*b^6*d^8)*x^6 + 140*(107*b^8*c^3*d^5 - 87*a*b^7*c^2*d^6 - 21*a^2*b^6
*c*d^7 + a^3*b^5*d^8)*x^5 + 70*(319*b^8*c^4*d^4 - 206*a*b^7*c^3*d^5 - 126*a^2*b^6*c^2*d^6 + 14*a^3*b^5*c*d^7 -
a^4*b^4*d^8)*x^4 + 14*(1377*b^8*c^5*d^3 - 505*a*b^7*c^4*d^4 - 1050*a^2*b^6*c^3*d^5 + 210*a^3*b^5*c^2*d^6 - 35
*a^4*b^4*c*d^7 + 3*a^5*b^3*d^8)*x^3 + 14*(669*b^8*c^6*d^2 + 117*a*b^7*c^5*d^3 - 1050*a^2*b^6*c^4*d^4 + 350*a^3
*b^5*c^3*d^5 - 105*a^4*b^4*c^2*d^6 + 21*a^5*b^3*c*d^7 - 2*a^6*b^2*d^8)*x^2 + 2*(1089*b^8*c^7*d + 1743*a*b^7*c^
6*d^2 - 4410*a^2*b^6*c^5*d^3 + 2450*a^3*b^5*c^4*d^4 - 1225*a^4*b^4*c^3*d^5 + 441*a^5*b^3*c^2*d^6 - 98*a^6*b^2*
c*d^7 + 10*a^7*b*d^8)*x + 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 +
a*b^7*c*d^7)*x^6 + 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c
^5*d^3 + 5*a*b^7*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(b
*x + a) - 840*(b^8*d^8*x^8 + a*b^7*c^7*d + (7*b^8*c*d^7 + a*b^7*d^8)*x^7 + 7*(3*b^8*c^2*d^6 + a*b^7*c*d^7)*x^6
+ 7*(5*b^8*c^3*d^5 + 3*a*b^7*c^2*d^6)*x^5 + 35*(b^8*c^4*d^4 + a*b^7*c^3*d^5)*x^4 + 7*(3*b^8*c^5*d^3 + 5*a*b^7
*c^4*d^4)*x^3 + 7*(b^8*c^6*d^2 + 3*a*b^7*c^5*d^3)*x^2 + (b^8*c^7*d + 7*a*b^7*c^6*d^2)*x)*log(d*x + c))/(a*b^9*
c^16 - 9*a^2*b^8*c^15*d + 36*a^3*b^7*c^14*d^2 - 84*a^4*b^6*c^13*d^3 + 126*a^5*b^5*c^12*d^4 - 126*a^6*b^4*c^11*
d^5 + 84*a^7*b^3*c^10*d^6 - 36*a^8*b^2*c^9*d^7 + 9*a^9*b*c^8*d^8 - a^10*c^7*d^9 + (b^10*c^9*d^7 - 9*a*b^9*c^8*
d^8 + 36*a^2*b^8*c^7*d^9 - 84*a^3*b^7*c^6*d^10 + 126*a^4*b^6*c^5*d^11 - 126*a^5*b^5*c^4*d^12 + 84*a^6*b^4*c^3*
d^13 - 36*a^7*b^3*c^2*d^14 + 9*a^8*b^2*c*d^15 - a^9*b*d^16)*x^8 + (7*b^10*c^10*d^6 - 62*a*b^9*c^9*d^7 + 243*a^
2*b^8*c^8*d^8 - 552*a^3*b^7*c^7*d^9 + 798*a^4*b^6*c^6*d^10 - 756*a^5*b^5*c^5*d^11 + 462*a^6*b^4*c^4*d^12 - 168
*a^7*b^3*c^3*d^13 + 27*a^8*b^2*c^2*d^14 + 2*a^9*b*c*d^15 - a^10*d^16)*x^7 + 7*(3*b^10*c^11*d^5 - 26*a*b^9*c^10
*d^6 + 99*a^2*b^8*c^9*d^7 - 216*a^3*b^7*c^8*d^8 + 294*a^4*b^6*c^7*d^9 - 252*a^5*b^5*c^6*d^10 + 126*a^6*b^4*c^5
*d^11 - 24*a^7*b^3*c^4*d^12 - 9*a^8*b^2*c^3*d^13 + 6*a^9*b*c^2*d^14 - a^10*c*d^15)*x^6 + 7*(5*b^10*c^12*d^4 -
42*a*b^9*c^11*d^5 + 153*a^2*b^8*c^10*d^6 - 312*a^3*b^7*c^9*d^7 + 378*a^4*b^6*c^8*d^8 - 252*a^5*b^5*c^7*d^9 + 4
2*a^6*b^4*c^6*d^10 + 72*a^7*b^3*c^5*d^11 - 63*a^8*b^2*c^4*d^12 + 22*a^9*b*c^3*d^13 - 3*a^10*c^2*d^14)*x^5 + 35
*(b^10*c^13*d^3 - 8*a*b^9*c^12*d^4 + 27*a^2*b^8*c^11*d^5 - 48*a^3*b^7*c^10*d^6 + 42*a^4*b^6*c^9*d^7 - 42*a^6*b
^4*c^7*d^9 + 48*a^7*b^3*c^6*d^10 - 27*a^8*b^2*c^5*d^11 + 8*a^9*b*c^4*d^12 - a^10*c^3*d^13)*x^4 + 7*(3*b^10*c^1
4*d^2 - 22*a*b^9*c^13*d^3 + 63*a^2*b^8*c^12*d^4 - 72*a^3*b^7*c^11*d^5 - 42*a^4*b^6*c^10*d^6 + 252*a^5*b^5*c^9*
d^7 - 378*a^6*b^4*c^8*d^8 + 312*a^7*b^3*c^7*d^9 - 153*a^8*b^2*c^6*d^10 + 42*a^9*b*c^5*d^11 - 5*a^10*c^4*d^12)*
x^3 + 7*(b^10*c^15*d - 6*a*b^9*c^14*d^2 + 9*a^2*b^8*c^13*d^3 + 24*a^3*b^7*c^12*d^4 - 126*a^4*b^6*c^11*d^5 + 25
2*a^5*b^5*c^10*d^6 - 294*a^6*b^4*c^9*d^7 + 216*a^7*b^3*c^8*d^8 - 99*a^8*b^2*c^7*d^9 + 26*a^9*b*c^6*d^10 - 3*a^
10*c^5*d^11)*x^2 + (b^10*c^16 - 2*a*b^9*c^15*d - 27*a^2*b^8*c^14*d^2 + 168*a^3*b^7*c^13*d^3 - 462*a^4*b^6*c^12
*d^4 + 756*a^5*b^5*c^11*d^5 - 798*a^6*b^4*c^10*d^6 + 552*a^7*b^3*c^9*d^7 - 243*a^8*b^2*c^8*d^8 + 62*a^9*b*c^7*
d^9 - 7*a^10*c^6*d^10)*x)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2336 vs.
\(2 (209) = 418\).
time = 26.93, size = 2336, normalized size = 10.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)**2/(d*x+c)**8,x)
[Out]
-8*b**7*d*log(x + (-8*a**10*b**7*d**11/(a*d - b*c)**9 + 80*a**9*b**8*c*d**10/(a*d - b*c)**9 - 360*a**8*b**9*c*
*2*d**9/(a*d - b*c)**9 + 960*a**7*b**10*c**3*d**8/(a*d - b*c)**9 - 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 +
2016*a**5*b**12*c**5*d**6/(a*d - b*c)**9 - 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 + 960*a**3*b**14*c**7*d**4
/(a*d - b*c)**9 - 360*a**2*b**15*c**8*d**3/(a*d - b*c)**9 + 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**
2 - 8*b**17*c**10*d/(a*d - b*c)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 + 8*b**7*d*log(x + (8*a**10*b*
*7*d**11/(a*d - b*c)**9 - 80*a**9*b**8*c*d**10/(a*d - b*c)**9 + 360*a**8*b**9*c**2*d**9/(a*d - b*c)**9 - 960*a
**7*b**10*c**3*d**8/(a*d - b*c)**9 + 1680*a**6*b**11*c**4*d**7/(a*d - b*c)**9 - 2016*a**5*b**12*c**5*d**6/(a*d
- b*c)**9 + 1680*a**4*b**13*c**6*d**5/(a*d - b*c)**9 - 960*a**3*b**14*c**7*d**4/(a*d - b*c)**9 + 360*a**2*b**
15*c**8*d**3/(a*d - b*c)**9 - 80*a*b**16*c**9*d**2/(a*d - b*c)**9 + 8*a*b**7*d**2 + 8*b**17*c**10*d/(a*d - b*c
)**9 + 8*b**8*c*d)/(16*b**8*d**2))/(a*d - b*c)**9 + (-15*a**7*d**7 + 125*a**6*b*c*d**6 - 463*a**5*b**2*c**2*d*
*5 + 1007*a**4*b**3*c**3*d**4 - 1443*a**3*b**4*c**4*d**3 + 1497*a**2*b**5*c**5*d**2 - 1443*a*b**6*c**6*d - 105
*b**7*c**7 - 840*b**7*d**7*x**7 + x**6*(-420*a*b**6*d**7 - 5460*b**7*c*d**6) + x**5*(140*a**2*b**5*d**7 - 2800
*a*b**6*c*d**6 - 14980*b**7*c**2*d**5) + x**4*(-70*a**3*b**4*d**7 + 910*a**2*b**5*c*d**6 - 7910*a*b**6*c**2*d*
*5 - 22330*b**7*c**3*d**4) + x**3*(42*a**4*b**3*d**7 - 448*a**3*b**4*c*d**6 + 2492*a**2*b**5*c**2*d**5 - 12208
*a*b**6*c**3*d**4 - 19278*b**7*c**4*d**3) + x**2*(-28*a**5*b**2*d**7 + 266*a**4*b**3*c*d**6 - 1204*a**3*b**4*c
**2*d**5 + 3696*a**2*b**5*c**3*d**4 - 11004*a*b**6*c**4*d**3 - 9366*b**7*c**5*d**2) + x*(20*a**6*b*d**7 - 176*
a**5*b**2*c*d**6 + 706*a**4*b**3*c**2*d**5 - 1744*a**3*b**4*c**3*d**4 + 3156*a**2*b**5*c**4*d**3 - 5664*a*b**6
*c**5*d**2 - 2178*b**7*c**6*d))/(105*a**9*c**7*d**8 - 840*a**8*b*c**8*d**7 + 2940*a**7*b**2*c**9*d**6 - 5880*a
**6*b**3*c**10*d**5 + 7350*a**5*b**4*c**11*d**4 - 5880*a**4*b**5*c**12*d**3 + 2940*a**3*b**6*c**13*d**2 - 840*
a**2*b**7*c**14*d + 105*a*b**8*c**15 + x**8*(105*a**8*b*d**15 - 840*a**7*b**2*c*d**14 + 2940*a**6*b**3*c**2*d*
*13 - 5880*a**5*b**4*c**3*d**12 + 7350*a**4*b**5*c**4*d**11 - 5880*a**3*b**6*c**5*d**10 + 2940*a**2*b**7*c**6*
d**9 - 840*a*b**8*c**7*d**8 + 105*b**9*c**8*d**7) + x**7*(105*a**9*d**15 - 105*a**8*b*c*d**14 - 2940*a**7*b**2
*c**2*d**13 + 14700*a**6*b**3*c**3*d**12 - 33810*a**5*b**4*c**4*d**11 + 45570*a**4*b**5*c**5*d**10 - 38220*a**
3*b**6*c**6*d**9 + 19740*a**2*b**7*c**7*d**8 - 5775*a*b**8*c**8*d**7 + 735*b**9*c**9*d**6) + x**6*(735*a**9*c*
d**14 - 3675*a**8*b*c**2*d**13 + 2940*a**7*b**2*c**3*d**12 + 20580*a**6*b**3*c**4*d**11 - 72030*a**5*b**4*c**5
*d**10 + 113190*a**4*b**5*c**6*d**9 - 102900*a**3*b**6*c**7*d**8 + 55860*a**2*b**7*c**8*d**7 - 16905*a*b**8*c*
*9*d**6 + 2205*b**9*c**10*d**5) + x**5*(2205*a**9*c**2*d**13 - 13965*a**8*b*c**3*d**12 + 32340*a**7*b**2*c**4*
d**11 - 20580*a**6*b**3*c**5*d**10 - 51450*a**5*b**4*c**6*d**9 + 133770*a**4*b**5*c**7*d**8 - 144060*a**3*b**6
*c**8*d**7 + 85260*a**2*b**7*c**9*d**6 - 27195*a*b**8*c**10*d**5 + 3675*b**9*c**11*d**4) + x**4*(3675*a**9*c**
3*d**12 - 25725*a**8*b*c**4*d**11 + 73500*a**7*b**2*c**5*d**10 - 102900*a**6*b**3*c**6*d**9 + 51450*a**5*b**4*
c**7*d**8 + 51450*a**4*b**5*c**8*d**7 - 102900*a**3*b**6*c**9*d**6 + 73500*a**2*b**7*c**10*d**5 - 25725*a*b**8
*c**11*d**4 + 3675*b**9*c**12*d**3) + x**3*(3675*a**9*c**4*d**11 - 27195*a**8*b*c**5*d**10 + 85260*a**7*b**2*c
**6*d**9 - 144060*a**6*b**3*c**7*d**8 + 133770*a**5*b**4*c**8*d**7 - 51450*a**4*b**5*c**9*d**6 - 20580*a**3*b*
*6*c**10*d**5 + 32340*a**2*b**7*c**11*d**4 - 13965*a*b**8*c**12*d**3 + 2205*b**9*c**13*d**2) + x**2*(2205*a**9
*c**5*d**10 - 16905*a**8*b*c**6*d**9 + 55860*a**7*b**2*c**7*d**8 - 102900*a**6*b**3*c**8*d**7 + 113190*a**5*b*
*4*c**9*d**6 - 72030*a**4*b**5*c**10*d**5 + 20580*a**3*b**6*c**11*d**4 + 2940*a**2*b**7*c**12*d**3 - 3675*a*b*
*8*c**13*d**2 + 735*b**9*c**14*d) + x*(735*a**9*c**6*d**9 - 5775*a**8*b*c**7*d**8 + 19740*a**7*b**2*c**8*d**7
- 38220*a**6*b**3*c**9*d**6 + 45570*a**5*b**4*c**10*d**5 - 33810*a**4*b**5*c**11*d**4 + 14700*a**3*b**6*c**12*
d**3 - 2940*a**2*b**7*c**13*d**2 - 105*a*b**8*c**14*d + 105*b**9*c**15))
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 714 vs.
\(2 (223) = 446\).
time = 1.19, size = 714, normalized size = 3.09 \begin {gather*} -\frac {b^{15}}{{\left (b^{16} c^{8} - 8 \, a b^{15} c^{7} d + 28 \, a^{2} b^{14} c^{6} d^{2} - 56 \, a^{3} b^{13} c^{5} d^{3} + 70 \, a^{4} b^{12} c^{4} d^{4} - 56 \, a^{5} b^{11} c^{3} d^{5} + 28 \, a^{6} b^{10} c^{2} d^{6} - 8 \, a^{7} b^{9} c d^{7} + a^{8} b^{8} d^{8}\right )} {\left (b x + a\right )}} + \frac {8 \, b^{8} d \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{10} c^{9} - 9 \, a b^{9} c^{8} d + 36 \, a^{2} b^{8} c^{7} d^{2} - 84 \, a^{3} b^{7} c^{6} d^{3} + 126 \, a^{4} b^{6} c^{5} d^{4} - 126 \, a^{5} b^{5} c^{4} d^{5} + 84 \, a^{6} b^{4} c^{3} d^{6} - 36 \, a^{7} b^{3} c^{2} d^{7} + 9 \, a^{8} b^{2} c d^{8} - a^{9} b d^{9}} + \frac {1443 \, b^{7} d^{8} + \frac {9366 \, {\left (b^{9} c d^{7} - a b^{8} d^{8}\right )}}{{\left (b x + a\right )} b} + \frac {25578 \, {\left (b^{11} c^{2} d^{6} - 2 \, a b^{10} c d^{7} + a^{2} b^{9} d^{8}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {37730 \, {\left (b^{13} c^{3} d^{5} - 3 \, a b^{12} c^{2} d^{6} + 3 \, a^{2} b^{11} c d^{7} - a^{3} b^{10} d^{8}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {31850 \, {\left (b^{15} c^{4} d^{4} - 4 \, a b^{14} c^{3} d^{5} + 6 \, a^{2} b^{13} c^{2} d^{6} - 4 \, a^{3} b^{12} c d^{7} + a^{4} b^{11} d^{8}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {14700 \, {\left (b^{17} c^{5} d^{3} - 5 \, a b^{16} c^{4} d^{4} + 10 \, a^{2} b^{15} c^{3} d^{5} - 10 \, a^{3} b^{14} c^{2} d^{6} + 5 \, a^{4} b^{13} c d^{7} - a^{5} b^{12} d^{8}\right )}}{{\left (b x + a\right )}^{5} b^{5}} + \frac {2940 \, {\left (b^{19} c^{6} d^{2} - 6 \, a b^{18} c^{5} d^{3} + 15 \, a^{2} b^{17} c^{4} d^{4} - 20 \, a^{3} b^{16} c^{3} d^{5} + 15 \, a^{4} b^{15} c^{2} d^{6} - 6 \, a^{5} b^{14} c d^{7} + a^{6} b^{13} d^{8}\right )}}{{\left (b x + a\right )}^{6} b^{6}}}{105 \, {\left (b c - a d\right )}^{9} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*x+a)^2/(d*x+c)^8,x, algorithm="giac")
[Out]
-b^15/((b^16*c^8 - 8*a*b^15*c^7*d + 28*a^2*b^14*c^6*d^2 - 56*a^3*b^13*c^5*d^3 + 70*a^4*b^12*c^4*d^4 - 56*a^5*b
^11*c^3*d^5 + 28*a^6*b^10*c^2*d^6 - 8*a^7*b^9*c*d^7 + a^8*b^8*d^8)*(b*x + a)) + 8*b^8*d*log(abs(b*c/(b*x + a)
- a*d/(b*x + a) + d))/(b^10*c^9 - 9*a*b^9*c^8*d + 36*a^2*b^8*c^7*d^2 - 84*a^3*b^7*c^6*d^3 + 126*a^4*b^6*c^5*d^
4 - 126*a^5*b^5*c^4*d^5 + 84*a^6*b^4*c^3*d^6 - 36*a^7*b^3*c^2*d^7 + 9*a^8*b^2*c*d^8 - a^9*b*d^9) + 1/105*(1443
*b^7*d^8 + 9366*(b^9*c*d^7 - a*b^8*d^8)/((b*x + a)*b) + 25578*(b^11*c^2*d^6 - 2*a*b^10*c*d^7 + a^2*b^9*d^8)/((
b*x + a)^2*b^2) + 37730*(b^13*c^3*d^5 - 3*a*b^12*c^2*d^6 + 3*a^2*b^11*c*d^7 - a^3*b^10*d^8)/((b*x + a)^3*b^3)
+ 31850*(b^15*c^4*d^4 - 4*a*b^14*c^3*d^5 + 6*a^2*b^13*c^2*d^6 - 4*a^3*b^12*c*d^7 + a^4*b^11*d^8)/((b*x + a)^4*
b^4) + 14700*(b^17*c^5*d^3 - 5*a*b^16*c^4*d^4 + 10*a^2*b^15*c^3*d^5 - 10*a^3*b^14*c^2*d^6 + 5*a^4*b^13*c*d^7 -
a^5*b^12*d^8)/((b*x + a)^5*b^5) + 2940*(b^19*c^6*d^2 - 6*a*b^18*c^5*d^3 + 15*a^2*b^17*c^4*d^4 - 20*a^3*b^16*c
^3*d^5 + 15*a^4*b^15*c^2*d^6 - 6*a^5*b^14*c*d^7 + a^6*b^13*d^8)/((b*x + a)^6*b^6))/((b*c - a*d)^9*(b*c/(b*x +
a) - a*d/(b*x + a) + d)^7)
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Mupad [B]
time = 1.39, size = 1738, normalized size = 7.52 \begin {gather*} \frac {16\,b^7\,d\,\mathrm {atanh}\left (\frac {a^9\,d^9-7\,a^8\,b\,c\,d^8+20\,a^7\,b^2\,c^2\,d^7-28\,a^6\,b^3\,c^3\,d^6+14\,a^5\,b^4\,c^4\,d^5+14\,a^4\,b^5\,c^5\,d^4-28\,a^3\,b^6\,c^6\,d^3+20\,a^2\,b^7\,c^7\,d^2-7\,a\,b^8\,c^8\,d+b^9\,c^9}{{\left (a\,d-b\,c\right )}^9}+\frac {2\,b\,d\,x\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}{{\left (a\,d-b\,c\right )}^9}\right )}{{\left (a\,d-b\,c\right )}^9}-\frac {\frac {15\,a^7\,d^7-125\,a^6\,b\,c\,d^6+463\,a^5\,b^2\,c^2\,d^5-1007\,a^4\,b^3\,c^3\,d^4+1443\,a^3\,b^4\,c^4\,d^3-1497\,a^2\,b^5\,c^5\,d^2+1443\,a\,b^6\,c^6\,d+105\,b^7\,c^7}{105\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}+\frac {4\,b^5\,x^5\,\left (-a^2\,d^7+20\,a\,b\,c\,d^6+107\,b^2\,c^2\,d^5\right )}{3\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}+\frac {2\,b^2\,x^2\,\left (2\,a^5\,d^7-19\,a^4\,b\,c\,d^6+86\,a^3\,b^2\,c^2\,d^5-264\,a^2\,b^3\,c^3\,d^4+786\,a\,b^4\,c^4\,d^3+669\,b^5\,c^5\,d^2\right )}{15\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}+\frac {2\,b^4\,x^4\,\left (a^3\,d^7-13\,a^2\,b\,c\,d^6+113\,a\,b^2\,c^2\,d^5+319\,b^3\,c^3\,d^4\right )}{3\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}+\frac {2\,b\,x\,\left (-10\,a^6\,d^7+88\,a^5\,b\,c\,d^6-353\,a^4\,b^2\,c^2\,d^5+872\,a^3\,b^3\,c^3\,d^4-1578\,a^2\,b^4\,c^4\,d^3+2832\,a\,b^5\,c^5\,d^2+1089\,b^6\,c^6\,d\right )}{105\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}+\frac {8\,b^7\,d^7\,x^7}{a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8}+\frac {4\,b^6\,x^6\,\left (a\,d^7+13\,b\,c\,d^6\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 {2\,b^3\,x^3\,\left (-3\,a^4\,d^7+32\,a^3\,b\,c\,d^6-178\,a^2\,b^2\,c^2\,d^5+872\,a\,b^3\,c^3\,d^4+1377\,b^4\,c^4\,d^3\right )}{15\,\left (a^8\,d^8-8\,a^7\,b\,c\,d^7+28\,a^6\,b^2\,c^2\,d^6-56\,a^5\,b^3\,c^3\,d^5+70\,a^4\,b^4\,c^4\,d^4-56\,a^3\,b^5\,c^5\,d^3+28\,a^2\,b^6\,c^6\,d^2-8\,a\,b^7\,c^7\,d+b^8\,c^8\right )}}{x^7\,\left (a\,d^7+7\,b\,c\,d^6\right )+x^3\,\left (21\,b\,c^5\,d^2+35\,a\,c^4\,d^3\right )+x^5\,\left (35\,b\,c^3\,d^4+21\,a\,c^2\,d^5\right )+x^4\,\left (35\,b\,c^4\,d^3+35\,a\,c^3\,d^4\right )+a\,c^7+x\,\left (b\,c^7+7\,a\,d\,c^6\right )+x^2\,\left (7\,b\,c^6\,d+21\,a\,c^5\,d^2\right )+x^6\,\left (21\,b\,c^2\,d^5+7\,a\,c\,d^6\right )+b\,d^7\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b*x)^2*(c + d*x)^8),x)
[Out]
(16*b^7*d*atanh((a^9*d^9 + b^9*c^9 + 20*a^2*b^7*c^7*d^2 - 28*a^3*b^6*c^6*d^3 + 14*a^4*b^5*c^5*d^4 + 14*a^5*b^4
*c^4*d^5 - 28*a^6*b^3*c^3*d^6 + 20*a^7*b^2*c^2*d^7 - 7*a*b^8*c^8*d - 7*a^8*b*c*d^8)/(a*d - b*c)^9 + (2*b*d*x*(
a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6
*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7))/(a*d - b*c)^9))/(a*d - b*c)^9 - ((15*a^7*d^7 + 105*b^7*c^7 - 14
97*a^2*b^5*c^5*d^2 + 1443*a^3*b^4*c^4*d^3 - 1007*a^4*b^3*c^3*d^4 + 463*a^5*b^2*c^2*d^5 + 1443*a*b^6*c^6*d - 12
5*a^6*b*c*d^6)/(105*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5
*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (4*b^5*x^5*(107*b^2*c^2*d^5 - a^2*d^7 +
20*a*b*c*d^6))/(3*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b
^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b^2*x^2*(2*a^5*d^7 + 669*b^5*c^5*d^2 +
786*a*b^4*c^4*d^3 - 264*a^2*b^3*c^3*d^4 + 86*a^3*b^2*c^2*d^5 - 19*a^4*b*c*d^6))/(15*(a^8*d^8 + b^8*c^8 + 28*a^
2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^
7*d - 8*a^7*b*c*d^7)) + (2*b^4*x^4*(a^3*d^7 + 319*b^3*c^3*d^4 + 113*a*b^2*c^2*d^5 - 13*a^2*b*c*d^6))/(3*(a^8*d
^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*
c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)) + (2*b*x*(1089*b^6*c^6*d - 10*a^6*d^7 + 2832*a*b^5*c^5*d^2 - 1578*a^
2*b^4*c^4*d^3 + 872*a^3*b^3*c^3*d^4 - 353*a^4*b^2*c^2*d^5 + 88*a^5*b*c*d^6))/(105*(a^8*d^8 + b^8*c^8 + 28*a^2*
b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*
d - 8*a^7*b*c*d^7)) + (8*b^7*d^7*x^7)/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^
4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (4*b^6*x^6*(a*d^7 + 13*
b*c*d^6))/(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d
^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7) + (2*b^3*x^3*(1377*b^4*c^4*d^3 - 3*a^4*d^7 + 872*a*b^
3*c^3*d^4 - 178*a^2*b^2*c^2*d^5 + 32*a^3*b*c*d^6))/(15*(a^8*d^8 + b^8*c^8 + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^
5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a*b^7*c^7*d - 8*a^7*b*c*d^7)))/(x^7*(
a*d^7 + 7*b*c*d^6) + x^3*(35*a*c^4*d^3 + 21*b*c^5*d^2) + x^5*(21*a*c^2*d^5 + 35*b*c^3*d^4) + x^4*(35*a*c^3*d^4
+ 35*b*c^4*d^3) + a*c^7 + x*(b*c^7 + 7*a*c^6*d) + x^2*(21*a*c^5*d^2 + 7*b*c^6*d) + x^6*(21*b*c^2*d^5 + 7*a*c*
d^6) + b*d^7*x^8)
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