3.1374 \(\int \frac{1}{(a+b x)^3 (c+d x)^8} \, dx\)

Optimal. Leaf size=276 \[ \frac{28 b^6 d^2}{(c+d x) (b c-a d)^9}+\frac{21 b^5 d^2}{2 (c+d x)^2 (b c-a d)^8}+\frac{5 b^4 d^2}{(c+d x)^3 (b c-a d)^7}+\frac{5 b^3 d^2}{2 (c+d x)^4 (b c-a d)^6}+\frac{6 b^2 d^2}{5 (c+d x)^5 (b c-a d)^5}+\frac{36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac{36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}+\frac{8 b^7 d}{(a+b x) (b c-a d)^9}-\frac{b^7}{2 (a+b x)^2 (b c-a d)^8}+\frac{b d^2}{2 (c+d x)^6 (b c-a d)^4}+\frac{d^2}{7 (c+d x)^7 (b c-a d)^3} \]

[Out]

-b^7/(2*(b*c - a*d)^8*(a + b*x)^2) + (8*b^7*d)/((b*c - a*d)^9*(a + b*x)) + d^2/(7*(b*c - a*d)^3*(c + d*x)^7) +
(b*d^2)/(2*(b*c - a*d)^4*(c + d*x)^6) + (6*b^2*d^2)/(5*(b*c - a*d)^5*(c + d*x)^5) + (5*b^3*d^2)/(2*(b*c - a*d
)^6*(c + d*x)^4) + (5*b^4*d^2)/((b*c - a*d)^7*(c + d*x)^3) + (21*b^5*d^2)/(2*(b*c - a*d)^8*(c + d*x)^2) + (28*
b^6*d^2)/((b*c - a*d)^9*(c + d*x)) + (36*b^7*d^2*Log[a + b*x])/(b*c - a*d)^10 - (36*b^7*d^2*Log[c + d*x])/(b*c
- a*d)^10

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Rubi [A]  time = 0.356038, antiderivative size = 276, normalized size of antiderivative = 1., 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 = {44} \[ \frac{28 b^6 d^2}{(c+d x) (b c-a d)^9}+\frac{21 b^5 d^2}{2 (c+d x)^2 (b c-a d)^8}+\frac{5 b^4 d^2}{(c+d x)^3 (b c-a d)^7}+\frac{5 b^3 d^2}{2 (c+d x)^4 (b c-a d)^6}+\frac{6 b^2 d^2}{5 (c+d x)^5 (b c-a d)^5}+\frac{36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac{36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}+\frac{8 b^7 d}{(a+b x) (b c-a d)^9}-\frac{b^7}{2 (a+b x)^2 (b c-a d)^8}+\frac{b d^2}{2 (c+d x)^6 (b c-a d)^4}+\frac{d^2}{7 (c+d x)^7 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^3*(c + d*x)^8),x]

[Out]

-b^7/(2*(b*c - a*d)^8*(a + b*x)^2) + (8*b^7*d)/((b*c - a*d)^9*(a + b*x)) + d^2/(7*(b*c - a*d)^3*(c + d*x)^7) +
(b*d^2)/(2*(b*c - a*d)^4*(c + d*x)^6) + (6*b^2*d^2)/(5*(b*c - a*d)^5*(c + d*x)^5) + (5*b^3*d^2)/(2*(b*c - a*d
)^6*(c + d*x)^4) + (5*b^4*d^2)/((b*c - a*d)^7*(c + d*x)^3) + (21*b^5*d^2)/(2*(b*c - a*d)^8*(c + d*x)^2) + (28*
b^6*d^2)/((b*c - a*d)^9*(c + d*x)) + (36*b^7*d^2*Log[a + b*x])/(b*c - a*d)^10 - (36*b^7*d^2*Log[c + d*x])/(b*c
- a*d)^10

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^3 (c+d x)^8} \, dx &=\int \left (\frac{b^8}{(b c-a d)^8 (a+b x)^3}-\frac{8 b^8 d}{(b c-a d)^9 (a+b x)^2}+\frac{36 b^8 d^2}{(b c-a d)^{10} (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)^8}-\frac{3 b d^3}{(b c-a d)^4 (c+d x)^7}-\frac{6 b^2 d^3}{(b c-a d)^5 (c+d x)^6}-\frac{10 b^3 d^3}{(b c-a d)^6 (c+d x)^5}-\frac{15 b^4 d^3}{(b c-a d)^7 (c+d x)^4}-\frac{21 b^5 d^3}{(b c-a d)^8 (c+d x)^3}-\frac{28 b^6 d^3}{(b c-a d)^9 (c+d x)^2}-\frac{36 b^7 d^3}{(b c-a d)^{10} (c+d x)}\right ) \, dx\\ &=-\frac{b^7}{2 (b c-a d)^8 (a+b x)^2}+\frac{8 b^7 d}{(b c-a d)^9 (a+b x)}+\frac{d^2}{7 (b c-a d)^3 (c+d x)^7}+\frac{b d^2}{2 (b c-a d)^4 (c+d x)^6}+\frac{6 b^2 d^2}{5 (b c-a d)^5 (c+d x)^5}+\frac{5 b^3 d^2}{2 (b c-a d)^6 (c+d x)^4}+\frac{5 b^4 d^2}{(b c-a d)^7 (c+d x)^3}+\frac{21 b^5 d^2}{2 (b c-a d)^8 (c+d x)^2}+\frac{28 b^6 d^2}{(b c-a d)^9 (c+d x)}+\frac{36 b^7 d^2 \log (a+b x)}{(b c-a d)^{10}}-\frac{36 b^7 d^2 \log (c+d x)}{(b c-a d)^{10}}\\ \end{align*}

Mathematica [A]  time = 0.198945, size = 254, normalized size = 0.92 \[ \frac{\frac{1960 b^6 d^2 (b c-a d)}{c+d x}+\frac{735 b^5 d^2 (b c-a d)^2}{(c+d x)^2}+\frac{350 b^4 d^2 (b c-a d)^3}{(c+d x)^3}+\frac{175 b^3 d^2 (b c-a d)^4}{(c+d x)^4}+\frac{84 b^2 d^2 (b c-a d)^5}{(c+d x)^5}+\frac{560 b^7 d (b c-a d)}{a+b x}-\frac{35 b^7 (b c-a d)^2}{(a+b x)^2}+2520 b^7 d^2 \log (a+b x)+\frac{35 b d^2 (b c-a d)^6}{(c+d x)^6}+\frac{10 d^2 (b c-a d)^7}{(c+d x)^7}-2520 b^7 d^2 \log (c+d x)}{70 (b c-a d)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^3*(c + d*x)^8),x]

[Out]

((-35*b^7*(b*c - a*d)^2)/(a + b*x)^2 + (560*b^7*d*(b*c - a*d))/(a + b*x) + (10*d^2*(b*c - a*d)^7)/(c + d*x)^7
+ (35*b*d^2*(b*c - a*d)^6)/(c + d*x)^6 + (84*b^2*d^2*(b*c - a*d)^5)/(c + d*x)^5 + (175*b^3*d^2*(b*c - a*d)^4)/
(c + d*x)^4 + (350*b^4*d^2*(b*c - a*d)^3)/(c + d*x)^3 + (735*b^5*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (1960*b^6*d^
2*(b*c - a*d))/(c + d*x) + 2520*b^7*d^2*Log[a + b*x] - 2520*b^7*d^2*Log[c + d*x])/(70*(b*c - a*d)^10)

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Maple [A]  time = 0.02, size = 265, normalized size = 1. \begin{align*} -{\frac{{d}^{2}}{7\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{7}}}-36\,{\frac{{d}^{2}{b}^{7}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{10}}}-28\,{\frac{{d}^{2}{b}^{6}}{ \left ( ad-bc \right ) ^{9} \left ( dx+c \right ) }}+{\frac{21\,{d}^{2}{b}^{5}}{2\, \left ( ad-bc \right ) ^{8} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{d}^{2}{b}^{4}}{ \left ( ad-bc \right ) ^{7} \left ( dx+c \right ) ^{3}}}+{\frac{5\,{d}^{2}{b}^{3}}{2\, \left ( ad-bc \right ) ^{6} \left ( dx+c \right ) ^{4}}}-{\frac{6\,{d}^{2}{b}^{2}}{5\, \left ( ad-bc \right ) ^{5} \left ( dx+c \right ) ^{5}}}+{\frac{{d}^{2}b}{2\, \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) ^{6}}}-{\frac{{b}^{7}}{2\, \left ( ad-bc \right ) ^{8} \left ( bx+a \right ) ^{2}}}+36\,{\frac{{d}^{2}{b}^{7}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{10}}}-8\,{\frac{{b}^{7}d}{ \left ( ad-bc \right ) ^{9} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^3/(d*x+c)^8,x)

[Out]

-1/7*d^2/(a*d-b*c)^3/(d*x+c)^7-36*d^2/(a*d-b*c)^10*b^7*ln(d*x+c)-28*d^2/(a*d-b*c)^9*b^6/(d*x+c)+21/2*d^2/(a*d-
b*c)^8*b^5/(d*x+c)^2-5*d^2/(a*d-b*c)^7*b^4/(d*x+c)^3+5/2*d^2/(a*d-b*c)^6*b^3/(d*x+c)^4-6/5*d^2/(a*d-b*c)^5*b^2
/(d*x+c)^5+1/2*d^2/(a*d-b*c)^4*b/(d*x+c)^6-1/2*b^7/(a*d-b*c)^8/(b*x+a)^2+36*d^2/(a*d-b*c)^10*b^7*ln(b*x+a)-8*b
^7/(a*d-b*c)^9*d/(b*x+a)

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Maxima [B]  time = 2.18734, size = 3239, normalized size = 11.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3/(d*x+c)^8,x, algorithm="maxima")

[Out]

36*b^7*d^2*log(b*x + a)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c
^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10*a^9*b*c*d^9
+ a^10*d^10) - 36*b^7*d^2*log(d*x + c)/(b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3
+ 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8
- 10*a^9*b*c*d^9 + a^10*d^10) + 1/70*(2520*b^8*d^8*x^8 - 35*b^8*c^8 + 595*a*b^7*c^7*d + 3349*a^2*b^6*c^6*d^2 -
2531*a^3*b^5*c^5*d^3 + 1879*a^4*b^4*c^4*d^4 - 1061*a^5*b^3*c^3*d^5 + 409*a^6*b^2*c^2*d^6 - 95*a^7*b*c*d^7 + 1
0*a^8*d^8 + 1260*(13*b^8*c*d^7 + 3*a*b^7*d^8)*x^7 + 420*(107*b^8*c^2*d^6 + 59*a*b^7*c*d^7 + 2*a^2*b^6*d^8)*x^6
+ 210*(319*b^8*c^3*d^5 + 327*a*b^7*c^2*d^6 + 27*a^2*b^6*c*d^7 - a^3*b^5*d^8)*x^5 + 42*(1377*b^8*c^4*d^4 + 246
7*a*b^7*c^3*d^5 + 387*a^2*b^6*c^2*d^6 - 33*a^3*b^5*c*d^7 + 2*a^4*b^4*d^8)*x^4 + 42*(669*b^8*c^5*d^3 + 2163*a*b
^7*c^4*d^4 + 608*a^2*b^6*c^3*d^5 - 92*a^3*b^5*c^2*d^6 + 13*a^4*b^4*c*d^7 - a^5*b^3*d^8)*x^3 + 6*(1089*b^8*c^6*
d^2 + 7515*a*b^7*c^5*d^3 + 3924*a^2*b^6*c^4*d^4 - 976*a^3*b^5*c^3*d^5 + 249*a^4*b^4*c^2*d^6 - 45*a^5*b^3*c*d^7
+ 4*a^6*b^2*d^8)*x^2 + 3*(105*b^8*c^7*d + 3621*a*b^7*c^6*d^2 + 4167*a^2*b^6*c^5*d^3 - 1713*a^3*b^5*c^4*d^4 +
737*a^4*b^4*c^3*d^5 - 243*a^5*b^3*c^2*d^6 + 51*a^6*b^2*c*d^7 - 5*a^7*b*d^8)*x)/(a^2*b^9*c^16 - 9*a^3*b^8*c^15*
d + 36*a^4*b^7*c^14*d^2 - 84*a^5*b^6*c^13*d^3 + 126*a^6*b^5*c^12*d^4 - 126*a^7*b^4*c^11*d^5 + 84*a^8*b^3*c^10*
d^6 - 36*a^9*b^2*c^9*d^7 + 9*a^10*b*c^8*d^8 - a^11*c^7*d^9 + (b^11*c^9*d^7 - 9*a*b^10*c^8*d^8 + 36*a^2*b^9*c^7
*d^9 - 84*a^3*b^8*c^6*d^10 + 126*a^4*b^7*c^5*d^11 - 126*a^5*b^6*c^4*d^12 + 84*a^6*b^5*c^3*d^13 - 36*a^7*b^4*c^
2*d^14 + 9*a^8*b^3*c*d^15 - a^9*b^2*d^16)*x^9 + (7*b^11*c^10*d^6 - 61*a*b^10*c^9*d^7 + 234*a^2*b^9*c^8*d^8 - 5
16*a^3*b^8*c^7*d^9 + 714*a^4*b^7*c^6*d^10 - 630*a^5*b^6*c^5*d^11 + 336*a^6*b^5*c^4*d^12 - 84*a^7*b^4*c^3*d^13
- 9*a^8*b^3*c^2*d^14 + 11*a^9*b^2*c*d^15 - 2*a^10*b*d^16)*x^8 + (21*b^11*c^11*d^5 - 175*a*b^10*c^10*d^6 + 631*
a^2*b^9*c^9*d^7 - 1269*a^3*b^8*c^8*d^8 + 1506*a^4*b^7*c^7*d^9 - 966*a^5*b^6*c^6*d^10 + 126*a^6*b^5*c^5*d^11 +
294*a^7*b^4*c^4*d^12 - 231*a^8*b^3*c^3*d^13 + 69*a^9*b^2*c^2*d^14 - 5*a^10*b*c*d^15 - a^11*d^16)*x^7 + 7*(5*b^
11*c^12*d^4 - 39*a*b^10*c^11*d^5 + 127*a^2*b^9*c^10*d^6 - 213*a^3*b^8*c^9*d^7 + 162*a^4*b^7*c^8*d^8 + 42*a^5*b
^6*c^7*d^9 - 210*a^6*b^5*c^6*d^10 + 198*a^7*b^4*c^5*d^11 - 87*a^8*b^3*c^4*d^12 + 13*a^9*b^2*c^3*d^13 + 3*a^10*
b*c^2*d^14 - a^11*c*d^15)*x^6 + 7*(5*b^11*c^13*d^3 - 35*a*b^10*c^12*d^4 + 93*a^2*b^9*c^11*d^5 - 87*a^3*b^8*c^1
0*d^6 - 102*a^4*b^7*c^9*d^7 + 378*a^5*b^6*c^8*d^8 - 462*a^6*b^5*c^7*d^9 + 282*a^7*b^4*c^6*d^10 - 63*a^8*b^3*c^
5*d^11 - 23*a^9*b^2*c^4*d^12 + 17*a^10*b*c^3*d^13 - 3*a^11*c^2*d^14)*x^5 + 7*(3*b^11*c^14*d^2 - 17*a*b^10*c^13
*d^3 + 23*a^2*b^9*c^12*d^4 + 63*a^3*b^8*c^11*d^5 - 282*a^4*b^7*c^10*d^6 + 462*a^5*b^6*c^9*d^7 - 378*a^6*b^5*c^
8*d^8 + 102*a^7*b^4*c^7*d^9 + 87*a^8*b^3*c^6*d^10 - 93*a^9*b^2*c^5*d^11 + 35*a^10*b*c^4*d^12 - 5*a^11*c^3*d^13
)*x^4 + 7*(b^11*c^15*d - 3*a*b^10*c^14*d^2 - 13*a^2*b^9*c^13*d^3 + 87*a^3*b^8*c^12*d^4 - 198*a^4*b^7*c^11*d^5
+ 210*a^5*b^6*c^10*d^6 - 42*a^6*b^5*c^9*d^7 - 162*a^7*b^4*c^8*d^8 + 213*a^8*b^3*c^7*d^9 - 127*a^9*b^2*c^6*d^10
+ 39*a^10*b*c^5*d^11 - 5*a^11*c^4*d^12)*x^3 + (b^11*c^16 + 5*a*b^10*c^15*d - 69*a^2*b^9*c^14*d^2 + 231*a^3*b^
8*c^13*d^3 - 294*a^4*b^7*c^12*d^4 - 126*a^5*b^6*c^11*d^5 + 966*a^6*b^5*c^10*d^6 - 1506*a^7*b^4*c^9*d^7 + 1269*
a^8*b^3*c^8*d^8 - 631*a^9*b^2*c^7*d^9 + 175*a^10*b*c^6*d^10 - 21*a^11*c^5*d^11)*x^2 + (2*a*b^10*c^16 - 11*a^2*
b^9*c^15*d + 9*a^3*b^8*c^14*d^2 + 84*a^4*b^7*c^13*d^3 - 336*a^5*b^6*c^12*d^4 + 630*a^6*b^5*c^11*d^5 - 714*a^7*
b^4*c^10*d^6 + 516*a^8*b^3*c^9*d^7 - 234*a^9*b^2*c^8*d^8 + 61*a^10*b*c^7*d^9 - 7*a^11*c^6*d^10)*x)

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Fricas [B]  time = 3.05477, size = 6507, normalized size = 23.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3/(d*x+c)^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^9*c^9 - 630*a*b^8*c^8*d - 2754*a^2*b^7*c^7*d^2 + 5880*a^3*b^6*c^6*d^3 - 4410*a^4*b^5*c^5*d^4 + 294
0*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d^6 + 504*a^7*b^2*c^2*d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^
8 - a*b^8*d^9)*x^8 - 1260*(13*b^9*c^2*d^7 - 10*a*b^8*c*d^8 - 3*a^2*b^7*d^9)*x^7 - 420*(107*b^9*c^3*d^6 - 48*a*
b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b^6*d^9)*x^6 - 210*(319*b^9*c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2
*d^7 - 28*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^5 - 42*(1377*b^9*c^5*d^4 + 1090*a*b^8*c^4*d^5 - 2080*a^2*b^7*c^3*d^6
- 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 - 42*(669*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 155
5*a^2*b^7*c^4*d^5 - 700*a^3*b^6*c^3*d^6 + 105*a^4*b^5*c^2*d^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(1089*
b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2*b^7*c^5*d^4 - 4900*a^3*b^6*c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^
5*b^4*c^2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(105*b^9*c^8*d + 3516*a*b^8*c^7*d^2 + 546*a^2*b^7*c^
6*d^3 - 5880*a^3*b^6*c^5*d^4 + 2450*a^4*b^5*c^4*d^5 - 980*a^5*b^4*c^3*d^6 + 294*a^6*b^3*c^2*d^7 - 56*a^7*b^2*c
*d^8 + 5*a^8*b*d^9)*x - 2520*(b^9*d^9*x^9 + a^2*b^7*c^7*d^2 + (7*b^9*c*d^8 + 2*a*b^8*d^9)*x^8 + (21*b^9*c^2*d^
7 + 14*a*b^8*c*d^8 + a^2*b^7*d^9)*x^7 + 7*(5*b^9*c^3*d^6 + 6*a*b^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 7*(5*b^9*c^4
*d^5 + 10*a*b^8*c^3*d^6 + 3*a^2*b^7*c^2*d^7)*x^5 + 7*(3*b^9*c^5*d^4 + 10*a*b^8*c^4*d^5 + 5*a^2*b^7*c^3*d^6)*x^
4 + 7*(b^9*c^6*d^3 + 6*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5)*x^3 + (b^9*c^7*d^2 + 14*a*b^8*c^6*d^3 + 21*a^2*b^7*c
^5*d^4)*x^2 + (2*a*b^8*c^7*d^2 + 7*a^2*b^7*c^6*d^3)*x)*log(b*x + a) + 2520*(b^9*d^9*x^9 + a^2*b^7*c^7*d^2 + (7
*b^9*c*d^8 + 2*a*b^8*d^9)*x^8 + (21*b^9*c^2*d^7 + 14*a*b^8*c*d^8 + a^2*b^7*d^9)*x^7 + 7*(5*b^9*c^3*d^6 + 6*a*b
^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 7*(5*b^9*c^4*d^5 + 10*a*b^8*c^3*d^6 + 3*a^2*b^7*c^2*d^7)*x^5 + 7*(3*b^9*c^5*
d^4 + 10*a*b^8*c^4*d^5 + 5*a^2*b^7*c^3*d^6)*x^4 + 7*(b^9*c^6*d^3 + 6*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5)*x^3 +
(b^9*c^7*d^2 + 14*a*b^8*c^6*d^3 + 21*a^2*b^7*c^5*d^4)*x^2 + (2*a*b^8*c^7*d^2 + 7*a^2*b^7*c^6*d^3)*x)*log(d*x +
c))/(a^2*b^10*c^17 - 10*a^3*b^9*c^16*d + 45*a^4*b^8*c^15*d^2 - 120*a^5*b^7*c^14*d^3 + 210*a^6*b^6*c^13*d^4 -
252*a^7*b^5*c^12*d^5 + 210*a^8*b^4*c^11*d^6 - 120*a^9*b^3*c^10*d^7 + 45*a^10*b^2*c^9*d^8 - 10*a^11*b*c^8*d^9 +
a^12*c^7*d^10 + (b^12*c^10*d^7 - 10*a*b^11*c^9*d^8 + 45*a^2*b^10*c^8*d^9 - 120*a^3*b^9*c^7*d^10 + 210*a^4*b^8
*c^6*d^11 - 252*a^5*b^7*c^5*d^12 + 210*a^6*b^6*c^4*d^13 - 120*a^7*b^5*c^3*d^14 + 45*a^8*b^4*c^2*d^15 - 10*a^9*
b^3*c*d^16 + a^10*b^2*d^17)*x^9 + (7*b^12*c^11*d^6 - 68*a*b^11*c^10*d^7 + 295*a^2*b^10*c^9*d^8 - 750*a^3*b^9*c
^8*d^9 + 1230*a^4*b^8*c^7*d^10 - 1344*a^5*b^7*c^6*d^11 + 966*a^6*b^6*c^5*d^12 - 420*a^7*b^5*c^4*d^13 + 75*a^8*
b^4*c^3*d^14 + 20*a^9*b^3*c^2*d^15 - 13*a^10*b^2*c*d^16 + 2*a^11*b*d^17)*x^8 + (21*b^12*c^12*d^5 - 196*a*b^11*
c^11*d^6 + 806*a^2*b^10*c^10*d^7 - 1900*a^3*b^9*c^9*d^8 + 2775*a^4*b^8*c^8*d^9 - 2472*a^5*b^7*c^7*d^10 + 1092*
a^6*b^6*c^6*d^11 + 168*a^7*b^5*c^5*d^12 - 525*a^8*b^4*c^4*d^13 + 300*a^9*b^3*c^3*d^14 - 74*a^10*b^2*c^2*d^15 +
4*a^11*b*c*d^16 + a^12*d^17)*x^7 + 7*(5*b^12*c^13*d^4 - 44*a*b^11*c^12*d^5 + 166*a^2*b^10*c^11*d^6 - 340*a^3*
b^9*c^10*d^7 + 375*a^4*b^8*c^9*d^8 - 120*a^5*b^7*c^8*d^9 - 252*a^6*b^6*c^7*d^10 + 408*a^7*b^5*c^6*d^11 - 285*a
^8*b^4*c^5*d^12 + 100*a^9*b^3*c^4*d^13 - 10*a^10*b^2*c^3*d^14 - 4*a^11*b*c^2*d^15 + a^12*c*d^16)*x^6 + 7*(5*b^
12*c^14*d^3 - 40*a*b^11*c^13*d^4 + 128*a^2*b^10*c^12*d^5 - 180*a^3*b^9*c^11*d^6 - 15*a^4*b^8*c^10*d^7 + 480*a^
5*b^7*c^9*d^8 - 840*a^6*b^6*c^8*d^9 + 744*a^7*b^5*c^7*d^10 - 345*a^8*b^4*c^6*d^11 + 40*a^9*b^3*c^5*d^12 + 40*a
^10*b^2*c^4*d^13 - 20*a^11*b*c^3*d^14 + 3*a^12*c^2*d^15)*x^5 + 7*(3*b^12*c^15*d^2 - 20*a*b^11*c^14*d^3 + 40*a^
2*b^10*c^13*d^4 + 40*a^3*b^9*c^12*d^5 - 345*a^4*b^8*c^11*d^6 + 744*a^5*b^7*c^10*d^7 - 840*a^6*b^6*c^9*d^8 + 48
0*a^7*b^5*c^8*d^9 - 15*a^8*b^4*c^7*d^10 - 180*a^9*b^3*c^6*d^11 + 128*a^10*b^2*c^5*d^12 - 40*a^11*b*c^4*d^13 +
5*a^12*c^3*d^14)*x^4 + 7*(b^12*c^16*d - 4*a*b^11*c^15*d^2 - 10*a^2*b^10*c^14*d^3 + 100*a^3*b^9*c^13*d^4 - 285*
a^4*b^8*c^12*d^5 + 408*a^5*b^7*c^11*d^6 - 252*a^6*b^6*c^10*d^7 - 120*a^7*b^5*c^9*d^8 + 375*a^8*b^4*c^8*d^9 - 3
40*a^9*b^3*c^7*d^10 + 166*a^10*b^2*c^6*d^11 - 44*a^11*b*c^5*d^12 + 5*a^12*c^4*d^13)*x^3 + (b^12*c^17 + 4*a*b^1
1*c^16*d - 74*a^2*b^10*c^15*d^2 + 300*a^3*b^9*c^14*d^3 - 525*a^4*b^8*c^13*d^4 + 168*a^5*b^7*c^12*d^5 + 1092*a^
6*b^6*c^11*d^6 - 2472*a^7*b^5*c^10*d^7 + 2775*a^8*b^4*c^9*d^8 - 1900*a^9*b^3*c^8*d^9 + 806*a^10*b^2*c^7*d^10 -
196*a^11*b*c^6*d^11 + 21*a^12*c^5*d^12)*x^2 + (2*a*b^11*c^17 - 13*a^2*b^10*c^16*d + 20*a^3*b^9*c^15*d^2 + 75*
a^4*b^8*c^14*d^3 - 420*a^5*b^7*c^13*d^4 + 966*a^6*b^6*c^12*d^5 - 1344*a^7*b^5*c^11*d^6 + 1230*a^8*b^4*c^10*d^7
- 750*a^9*b^3*c^9*d^8 + 295*a^10*b^2*c^8*d^9 - 68*a^11*b*c^7*d^10 + 7*a^12*c^6*d^11)*x)

________________________________________________________________________________________

Sympy [B]  time = 43.5253, size = 2914, normalized size = 10.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**3/(d*x+c)**8,x)

[Out]

-36*b**7*d**2*log(x + (-36*a**11*b**7*d**13/(a*d - b*c)**10 + 396*a**10*b**8*c*d**12/(a*d - b*c)**10 - 1980*a*
*9*b**9*c**2*d**11/(a*d - b*c)**10 + 5940*a**8*b**10*c**3*d**10/(a*d - b*c)**10 - 11880*a**7*b**11*c**4*d**9/(
a*d - b*c)**10 + 16632*a**6*b**12*c**5*d**8/(a*d - b*c)**10 - 16632*a**5*b**13*c**6*d**7/(a*d - b*c)**10 + 118
80*a**4*b**14*c**7*d**6/(a*d - b*c)**10 - 5940*a**3*b**15*c**8*d**5/(a*d - b*c)**10 + 1980*a**2*b**16*c**9*d**
4/(a*d - b*c)**10 - 396*a*b**17*c**10*d**3/(a*d - b*c)**10 + 36*a*b**7*d**3 + 36*b**18*c**11*d**2/(a*d - b*c)*
*10 + 36*b**8*c*d**2)/(72*b**8*d**3))/(a*d - b*c)**10 + 36*b**7*d**2*log(x + (36*a**11*b**7*d**13/(a*d - b*c)*
*10 - 396*a**10*b**8*c*d**12/(a*d - b*c)**10 + 1980*a**9*b**9*c**2*d**11/(a*d - b*c)**10 - 5940*a**8*b**10*c**
3*d**10/(a*d - b*c)**10 + 11880*a**7*b**11*c**4*d**9/(a*d - b*c)**10 - 16632*a**6*b**12*c**5*d**8/(a*d - b*c)*
*10 + 16632*a**5*b**13*c**6*d**7/(a*d - b*c)**10 - 11880*a**4*b**14*c**7*d**6/(a*d - b*c)**10 + 5940*a**3*b**1
5*c**8*d**5/(a*d - b*c)**10 - 1980*a**2*b**16*c**9*d**4/(a*d - b*c)**10 + 396*a*b**17*c**10*d**3/(a*d - b*c)**
10 + 36*a*b**7*d**3 - 36*b**18*c**11*d**2/(a*d - b*c)**10 + 36*b**8*c*d**2)/(72*b**8*d**3))/(a*d - b*c)**10 -
(10*a**8*d**8 - 95*a**7*b*c*d**7 + 409*a**6*b**2*c**2*d**6 - 1061*a**5*b**3*c**3*d**5 + 1879*a**4*b**4*c**4*d*
*4 - 2531*a**3*b**5*c**5*d**3 + 3349*a**2*b**6*c**6*d**2 + 595*a*b**7*c**7*d - 35*b**8*c**8 + 2520*b**8*d**8*x
**8 + x**7*(3780*a*b**7*d**8 + 16380*b**8*c*d**7) + x**6*(840*a**2*b**6*d**8 + 24780*a*b**7*c*d**7 + 44940*b**
8*c**2*d**6) + x**5*(-210*a**3*b**5*d**8 + 5670*a**2*b**6*c*d**7 + 68670*a*b**7*c**2*d**6 + 66990*b**8*c**3*d*
*5) + x**4*(84*a**4*b**4*d**8 - 1386*a**3*b**5*c*d**7 + 16254*a**2*b**6*c**2*d**6 + 103614*a*b**7*c**3*d**5 +
57834*b**8*c**4*d**4) + x**3*(-42*a**5*b**3*d**8 + 546*a**4*b**4*c*d**7 - 3864*a**3*b**5*c**2*d**6 + 25536*a**
2*b**6*c**3*d**5 + 90846*a*b**7*c**4*d**4 + 28098*b**8*c**5*d**3) + x**2*(24*a**6*b**2*d**8 - 270*a**5*b**3*c*
d**7 + 1494*a**4*b**4*c**2*d**6 - 5856*a**3*b**5*c**3*d**5 + 23544*a**2*b**6*c**4*d**4 + 45090*a*b**7*c**5*d**
3 + 6534*b**8*c**6*d**2) + x*(-15*a**7*b*d**8 + 153*a**6*b**2*c*d**7 - 729*a**5*b**3*c**2*d**6 + 2211*a**4*b**
4*c**3*d**5 - 5139*a**3*b**5*c**4*d**4 + 12501*a**2*b**6*c**5*d**3 + 10863*a*b**7*c**6*d**2 + 315*b**8*c**7*d)
)/(70*a**11*c**7*d**9 - 630*a**10*b*c**8*d**8 + 2520*a**9*b**2*c**9*d**7 - 5880*a**8*b**3*c**10*d**6 + 8820*a*
*7*b**4*c**11*d**5 - 8820*a**6*b**5*c**12*d**4 + 5880*a**5*b**6*c**13*d**3 - 2520*a**4*b**7*c**14*d**2 + 630*a
**3*b**8*c**15*d - 70*a**2*b**9*c**16 + x**9*(70*a**9*b**2*d**16 - 630*a**8*b**3*c*d**15 + 2520*a**7*b**4*c**2
*d**14 - 5880*a**6*b**5*c**3*d**13 + 8820*a**5*b**6*c**4*d**12 - 8820*a**4*b**7*c**5*d**11 + 5880*a**3*b**8*c*
*6*d**10 - 2520*a**2*b**9*c**7*d**9 + 630*a*b**10*c**8*d**8 - 70*b**11*c**9*d**7) + x**8*(140*a**10*b*d**16 -
770*a**9*b**2*c*d**15 + 630*a**8*b**3*c**2*d**14 + 5880*a**7*b**4*c**3*d**13 - 23520*a**6*b**5*c**4*d**12 + 44
100*a**5*b**6*c**5*d**11 - 49980*a**4*b**7*c**6*d**10 + 36120*a**3*b**8*c**7*d**9 - 16380*a**2*b**9*c**8*d**8
+ 4270*a*b**10*c**9*d**7 - 490*b**11*c**10*d**6) + x**7*(70*a**11*d**16 + 350*a**10*b*c*d**15 - 4830*a**9*b**2
*c**2*d**14 + 16170*a**8*b**3*c**3*d**13 - 20580*a**7*b**4*c**4*d**12 - 8820*a**6*b**5*c**5*d**11 + 67620*a**5
*b**6*c**6*d**10 - 105420*a**4*b**7*c**7*d**9 + 88830*a**3*b**8*c**8*d**8 - 44170*a**2*b**9*c**9*d**7 + 12250*
a*b**10*c**10*d**6 - 1470*b**11*c**11*d**5) + x**6*(490*a**11*c*d**15 - 1470*a**10*b*c**2*d**14 - 6370*a**9*b*
*2*c**3*d**13 + 42630*a**8*b**3*c**4*d**12 - 97020*a**7*b**4*c**5*d**11 + 102900*a**6*b**5*c**6*d**10 - 20580*
a**5*b**6*c**7*d**9 - 79380*a**4*b**7*c**8*d**8 + 104370*a**3*b**8*c**9*d**7 - 62230*a**2*b**9*c**10*d**6 + 19
110*a*b**10*c**11*d**5 - 2450*b**11*c**12*d**4) + x**5*(1470*a**11*c**2*d**14 - 8330*a**10*b*c**3*d**13 + 1127
0*a**9*b**2*c**4*d**12 + 30870*a**8*b**3*c**5*d**11 - 138180*a**7*b**4*c**6*d**10 + 226380*a**6*b**5*c**7*d**9
- 185220*a**5*b**6*c**8*d**8 + 49980*a**4*b**7*c**9*d**7 + 42630*a**3*b**8*c**10*d**6 - 45570*a**2*b**9*c**11
*d**5 + 17150*a*b**10*c**12*d**4 - 2450*b**11*c**13*d**3) + x**4*(2450*a**11*c**3*d**13 - 17150*a**10*b*c**4*d
**12 + 45570*a**9*b**2*c**5*d**11 - 42630*a**8*b**3*c**6*d**10 - 49980*a**7*b**4*c**7*d**9 + 185220*a**6*b**5*
c**8*d**8 - 226380*a**5*b**6*c**9*d**7 + 138180*a**4*b**7*c**10*d**6 - 30870*a**3*b**8*c**11*d**5 - 11270*a**2
*b**9*c**12*d**4 + 8330*a*b**10*c**13*d**3 - 1470*b**11*c**14*d**2) + x**3*(2450*a**11*c**4*d**12 - 19110*a**1
0*b*c**5*d**11 + 62230*a**9*b**2*c**6*d**10 - 104370*a**8*b**3*c**7*d**9 + 79380*a**7*b**4*c**8*d**8 + 20580*a
**6*b**5*c**9*d**7 - 102900*a**5*b**6*c**10*d**6 + 97020*a**4*b**7*c**11*d**5 - 42630*a**3*b**8*c**12*d**4 + 6
370*a**2*b**9*c**13*d**3 + 1470*a*b**10*c**14*d**2 - 490*b**11*c**15*d) + x**2*(1470*a**11*c**5*d**11 - 12250*
a**10*b*c**6*d**10 + 44170*a**9*b**2*c**7*d**9 - 88830*a**8*b**3*c**8*d**8 + 105420*a**7*b**4*c**9*d**7 - 6762
0*a**6*b**5*c**10*d**6 + 8820*a**5*b**6*c**11*d**5 + 20580*a**4*b**7*c**12*d**4 - 16170*a**3*b**8*c**13*d**3 +
4830*a**2*b**9*c**14*d**2 - 350*a*b**10*c**15*d - 70*b**11*c**16) + x*(490*a**11*c**6*d**10 - 4270*a**10*b*c*
*7*d**9 + 16380*a**9*b**2*c**8*d**8 - 36120*a**8*b**3*c**9*d**7 + 49980*a**7*b**4*c**10*d**6 - 44100*a**6*b**5
*c**11*d**5 + 23520*a**5*b**6*c**12*d**4 - 5880*a**4*b**7*c**13*d**3 - 630*a**3*b**8*c**14*d**2 + 770*a**2*b**
9*c**15*d - 140*a*b**10*c**16))

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Giac [B]  time = 1.08716, size = 1389, normalized size = 5.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^3/(d*x+c)^8,x, algorithm="giac")

[Out]

36*b^8*d^2*log(abs(b*x + a))/(b^11*c^10 - 10*a*b^10*c^9*d + 45*a^2*b^9*c^8*d^2 - 120*a^3*b^8*c^7*d^3 + 210*a^4
*b^7*c^6*d^4 - 252*a^5*b^6*c^5*d^5 + 210*a^6*b^5*c^4*d^6 - 120*a^7*b^4*c^3*d^7 + 45*a^8*b^3*c^2*d^8 - 10*a^9*b
^2*c*d^9 + a^10*b*d^10) - 36*b^7*d^3*log(abs(d*x + c))/(b^10*c^10*d - 10*a*b^9*c^9*d^2 + 45*a^2*b^8*c^8*d^3 -
120*a^3*b^7*c^7*d^4 + 210*a^4*b^6*c^6*d^5 - 252*a^5*b^5*c^5*d^6 + 210*a^6*b^4*c^4*d^7 - 120*a^7*b^3*c^3*d^8 +
45*a^8*b^2*c^2*d^9 - 10*a^9*b*c*d^10 + a^10*d^11) - 1/70*(35*b^9*c^9 - 630*a*b^8*c^8*d - 2754*a^2*b^7*c^7*d^2
+ 5880*a^3*b^6*c^6*d^3 - 4410*a^4*b^5*c^5*d^4 + 2940*a^5*b^4*c^4*d^5 - 1470*a^6*b^3*c^3*d^6 + 504*a^7*b^2*c^2*
d^7 - 105*a^8*b*c*d^8 + 10*a^9*d^9 - 2520*(b^9*c*d^8 - a*b^8*d^9)*x^8 - 1260*(13*b^9*c^2*d^7 - 10*a*b^8*c*d^8
- 3*a^2*b^7*d^9)*x^7 - 420*(107*b^9*c^3*d^6 - 48*a*b^8*c^2*d^7 - 57*a^2*b^7*c*d^8 - 2*a^3*b^6*d^9)*x^6 - 210*(
319*b^9*c^4*d^5 + 8*a*b^8*c^3*d^6 - 300*a^2*b^7*c^2*d^7 - 28*a^3*b^6*c*d^8 + a^4*b^5*d^9)*x^5 - 42*(1377*b^9*c
^5*d^4 + 1090*a*b^8*c^4*d^5 - 2080*a^2*b^7*c^3*d^6 - 420*a^3*b^6*c^2*d^7 + 35*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x
^4 - 42*(669*b^9*c^6*d^3 + 1494*a*b^8*c^5*d^4 - 1555*a^2*b^7*c^4*d^5 - 700*a^3*b^6*c^3*d^6 + 105*a^4*b^5*c^2*d
^7 - 14*a^5*b^4*c*d^8 + a^6*b^3*d^9)*x^3 - 6*(1089*b^9*c^7*d^2 + 6426*a*b^8*c^6*d^3 - 3591*a^2*b^7*c^5*d^4 - 4
900*a^3*b^6*c^4*d^5 + 1225*a^4*b^5*c^3*d^6 - 294*a^5*b^4*c^2*d^7 + 49*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 - 3*(
105*b^9*c^8*d + 3516*a*b^8*c^7*d^2 + 546*a^2*b^7*c^6*d^3 - 5880*a^3*b^6*c^5*d^4 + 2450*a^4*b^5*c^4*d^5 - 980*a
^5*b^4*c^3*d^6 + 294*a^6*b^3*c^2*d^7 - 56*a^7*b^2*c*d^8 + 5*a^8*b*d^9)*x)/((b*c - a*d)^10*(b*x + a)^2*(d*x + c
)^7)