3.1362 \(\int \frac{(a+b x)^9}{(c+d x)^8} \, dx\)

Optimal. Leaf size=232 \[ -\frac{b^8 x (8 b c-9 a d)}{d^9}+\frac{84 b^6 (b c-a d)^3}{d^{10} (c+d x)}-\frac{63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac{42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac{21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac{36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}+\frac{36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}-\frac{3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac{(b c-a d)^9}{7 d^{10} (c+d x)^7}+\frac{b^9 x^2}{2 d^8} \]

[Out]

-((b^8*(8*b*c - 9*a*d)*x)/d^9) + (b^9*x^2)/(2*d^8) + (b*c - a*d)^9/(7*d^10*(c + d*x)^7) - (3*b*(b*c - a*d)^8)/
(2*d^10*(c + d*x)^6) + (36*b^2*(b*c - a*d)^7)/(5*d^10*(c + d*x)^5) - (21*b^3*(b*c - a*d)^6)/(d^10*(c + d*x)^4)
+ (42*b^4*(b*c - a*d)^5)/(d^10*(c + d*x)^3) - (63*b^5*(b*c - a*d)^4)/(d^10*(c + d*x)^2) + (84*b^6*(b*c - a*d)
^3)/(d^10*(c + d*x)) + (36*b^7*(b*c - a*d)^2*Log[c + d*x])/d^10

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Rubi [A]  time = 0.356882, antiderivative size = 232, 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 = {43} \[ -\frac{b^8 x (8 b c-9 a d)}{d^9}+\frac{84 b^6 (b c-a d)^3}{d^{10} (c+d x)}-\frac{63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac{42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac{21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac{36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}+\frac{36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}-\frac{3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac{(b c-a d)^9}{7 d^{10} (c+d x)^7}+\frac{b^9 x^2}{2 d^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^9/(c + d*x)^8,x]

[Out]

-((b^8*(8*b*c - 9*a*d)*x)/d^9) + (b^9*x^2)/(2*d^8) + (b*c - a*d)^9/(7*d^10*(c + d*x)^7) - (3*b*(b*c - a*d)^8)/
(2*d^10*(c + d*x)^6) + (36*b^2*(b*c - a*d)^7)/(5*d^10*(c + d*x)^5) - (21*b^3*(b*c - a*d)^6)/(d^10*(c + d*x)^4)
+ (42*b^4*(b*c - a*d)^5)/(d^10*(c + d*x)^3) - (63*b^5*(b*c - a*d)^4)/(d^10*(c + d*x)^2) + (84*b^6*(b*c - a*d)
^3)/(d^10*(c + d*x)) + (36*b^7*(b*c - a*d)^2*Log[c + d*x])/d^10

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^9}{(c+d x)^8} \, dx &=\int \left (-\frac{b^8 (8 b c-9 a d)}{d^9}+\frac{b^9 x}{d^8}+\frac{(-b c+a d)^9}{d^9 (c+d x)^8}+\frac{9 b (b c-a d)^8}{d^9 (c+d x)^7}-\frac{36 b^2 (b c-a d)^7}{d^9 (c+d x)^6}+\frac{84 b^3 (b c-a d)^6}{d^9 (c+d x)^5}-\frac{126 b^4 (b c-a d)^5}{d^9 (c+d x)^4}+\frac{126 b^5 (b c-a d)^4}{d^9 (c+d x)^3}-\frac{84 b^6 (b c-a d)^3}{d^9 (c+d x)^2}+\frac{36 b^7 (b c-a d)^2}{d^9 (c+d x)}\right ) \, dx\\ &=-\frac{b^8 (8 b c-9 a d) x}{d^9}+\frac{b^9 x^2}{2 d^8}+\frac{(b c-a d)^9}{7 d^{10} (c+d x)^7}-\frac{3 b (b c-a d)^8}{2 d^{10} (c+d x)^6}+\frac{36 b^2 (b c-a d)^7}{5 d^{10} (c+d x)^5}-\frac{21 b^3 (b c-a d)^6}{d^{10} (c+d x)^4}+\frac{42 b^4 (b c-a d)^5}{d^{10} (c+d x)^3}-\frac{63 b^5 (b c-a d)^4}{d^{10} (c+d x)^2}+\frac{84 b^6 (b c-a d)^3}{d^{10} (c+d x)}+\frac{36 b^7 (b c-a d)^2 \log (c+d x)}{d^{10}}\\ \end{align*}

Mathematica [B]  time = 0.25977, size = 584, normalized size = 2.52 \[ -\frac{-6 a^2 b^7 c d^2 \left (20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+7203 c^5 d x+1089 c^6+13230 c d^5 x^5+2940 d^6 x^6\right )+840 a^3 b^6 d^3 \left (21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+7 c^5 d x+c^6+21 c d^5 x^5+7 d^6 x^6\right )+210 a^4 b^5 d^4 \left (21 c^3 d^2 x^2+35 c^2 d^3 x^3+7 c^4 d x+c^5+35 c d^4 x^4+21 d^5 x^5\right )+84 a^5 b^4 d^5 \left (21 c^2 d^2 x^2+7 c^3 d x+c^4+35 c d^3 x^3+35 d^4 x^4\right )+42 a^6 b^3 d^6 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )+24 a^7 b^2 d^7 \left (c^2+7 c d x+21 d^2 x^2\right )+15 a^8 b d^8 (c+7 d x)+10 a^9 d^9+6 a b^8 d \left (24843 c^6 d^2 x^2+35525 c^5 d^3 x^3+28175 c^4 d^4 x^4+11025 c^3 d^5 x^5+735 c^2 d^6 x^6+9261 c^7 d x+1443 c^8-735 c d^7 x^7-105 d^8 x^8\right )-2520 b^7 (c+d x)^7 (b c-a d)^2 \log (c+d x)+b^9 \left (-\left (53949 c^7 d^2 x^2+72275 c^6 d^3 x^3+50225 c^5 d^4 x^4+12495 c^4 d^5 x^5-4655 c^3 d^6 x^6-3185 c^2 d^7 x^7+20923 c^8 d x+3349 c^9-315 c d^8 x^8+35 d^9 x^9\right )\right )}{70 d^{10} (c+d x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^9/(c + d*x)^8,x]

[Out]

-(10*a^9*d^9 + 15*a^8*b*d^8*(c + 7*d*x) + 24*a^7*b^2*d^7*(c^2 + 7*c*d*x + 21*d^2*x^2) + 42*a^6*b^3*d^6*(c^3 +
7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) + 84*a^5*b^4*d^5*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*
d^4*x^4) + 210*a^4*b^5*d^4*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 21*d^5*x^5) + 8
40*a^3*b^6*d^3*(c^6 + 7*c^5*d*x + 21*c^4*d^2*x^2 + 35*c^3*d^3*x^3 + 35*c^2*d^4*x^4 + 21*c*d^5*x^5 + 7*d^6*x^6)
- 6*a^2*b^7*c*d^2*(1089*c^6 + 7203*c^5*d*x + 20139*c^4*d^2*x^2 + 30625*c^3*d^3*x^3 + 26950*c^2*d^4*x^4 + 1323
0*c*d^5*x^5 + 2940*d^6*x^6) + 6*a*b^8*d*(1443*c^8 + 9261*c^7*d*x + 24843*c^6*d^2*x^2 + 35525*c^5*d^3*x^3 + 281
75*c^4*d^4*x^4 + 11025*c^3*d^5*x^5 + 735*c^2*d^6*x^6 - 735*c*d^7*x^7 - 105*d^8*x^8) - b^9*(3349*c^9 + 20923*c^
8*d*x + 53949*c^7*d^2*x^2 + 72275*c^6*d^3*x^3 + 50225*c^5*d^4*x^4 + 12495*c^4*d^5*x^5 - 4655*c^3*d^6*x^6 - 318
5*c^2*d^7*x^7 - 315*c*d^8*x^8 + 35*d^9*x^9) - 2520*b^7*(b*c - a*d)^2*(c + d*x)^7*Log[c + d*x])/(70*d^10*(c + d
*x)^7)

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Maple [B]  time = 0.015, size = 1035, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^9/(d*x+c)^8,x)

[Out]

1/2*b^9*x^2/d^8-72*b^8/d^9*ln(d*x+c)*a*c+9/7/d^2/(d*x+c)^7*a^8*b*c-36/7/d^3/(d*x+c)^7*c^2*a^7*b^2+12/d^4/(d*x+
c)^7*c^3*a^6*b^3-18/d^5/(d*x+c)^7*a^5*b^4*c^4-315*b^5/d^6/(d*x+c)^4*a^4*c^2+420*b^6/d^7/(d*x+c)^4*a^3*c^3-315*
b^7/d^8/(d*x+c)^4*a^2*c^4+126*b^8/d^9/(d*x+c)^4*a*c^5+12*b^2/d^3/(d*x+c)^6*a^7*c-42*b^3/d^4/(d*x+c)^6*a^6*c^2+
84*b^4/d^5/(d*x+c)^6*a^5*c^3-105*b^5/d^6/(d*x+c)^6*a^4*c^4+84*b^6/d^7/(d*x+c)^6*a^3*c^5-42*b^7/d^8/(d*x+c)^6*a
^2*c^6+12*b^8/d^9/(d*x+c)^6*a*c^7+252*b^6/d^7/(d*x+c)^2*a^3*c-378*b^7/d^8/(d*x+c)^2*a^2*c^2+252*b^8/d^9/(d*x+c
)^2*a*c^3+126*b^4/d^5/(d*x+c)^4*a^5*c-252*b^8/d^9/(d*x+c)*a*c^2+252/5*b^3/d^4/(d*x+c)^5*a^6*c-756/5*b^4/d^5/(d
*x+c)^5*a^5*c^2+252*b^5/d^6/(d*x+c)^5*a^4*c^3-252*b^6/d^7/(d*x+c)^5*a^3*c^4+756/5*b^7/d^8/(d*x+c)^5*a^2*c^5-25
2/5*b^8/d^9/(d*x+c)^5*a*c^6+9*b^8/d^8*a*x-8*b^9/d^9*x*c-21*b^3/d^4/(d*x+c)^4*a^6-21*b^9/d^10/(d*x+c)^4*c^6+36/
5*b^9/d^10/(d*x+c)^5*c^7-63*b^5/d^6/(d*x+c)^2*a^4-63*b^9/d^10/(d*x+c)^2*c^4-42*b^4/d^5/(d*x+c)^3*a^5+42*b^9/d^
10/(d*x+c)^3*c^5-3/2*b/d^2/(d*x+c)^6*a^8-3/2*b^9/d^10/(d*x+c)^6*c^8-36/5*b^2/d^3/(d*x+c)^5*a^7+1/7/d^10/(d*x+c
)^7*b^9*c^9-84*b^6/d^7/(d*x+c)*a^3+84*b^9/d^10/(d*x+c)*c^3+36*b^7/d^8*ln(d*x+c)*a^2+36*b^9/d^10*ln(d*x+c)*c^2-
1/7/d/(d*x+c)^7*a^9+210*b^5/d^6/(d*x+c)^3*a^4*c-420*b^6/d^7/(d*x+c)^3*a^3*c^2+420*b^7/d^8/(d*x+c)^3*a^2*c^3-21
0*b^8/d^9/(d*x+c)^3*a*c^4+252*b^7/d^8/(d*x+c)*a^2*c+18/d^6/(d*x+c)^7*a^4*b^5*c^5-12/d^7/(d*x+c)^7*a^3*b^6*c^6+
36/7/d^8/(d*x+c)^7*a^2*b^7*c^7-9/7/d^9/(d*x+c)^7*a*b^8*c^8

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Maxima [B]  time = 1.26394, size = 1061, normalized size = 4.57 \begin{align*} \frac{3349 \, b^{9} c^{9} - 8658 \, a b^{8} c^{8} d + 6534 \, a^{2} b^{7} c^{7} d^{2} - 840 \, a^{3} b^{6} c^{6} d^{3} - 210 \, a^{4} b^{5} c^{5} d^{4} - 84 \, a^{5} b^{4} c^{4} d^{5} - 42 \, a^{6} b^{3} c^{3} d^{6} - 24 \, a^{7} b^{2} c^{2} d^{7} - 15 \, a^{8} b c d^{8} - 10 \, a^{9} d^{9} + 5880 \,{\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{6} + 4410 \,{\left (7 \, b^{9} c^{4} d^{5} - 20 \, a b^{8} c^{3} d^{6} + 18 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x^{5} + 1470 \,{\left (47 \, b^{9} c^{5} d^{4} - 130 \, a b^{8} c^{4} d^{5} + 110 \, a^{2} b^{7} c^{3} d^{6} - 20 \, a^{3} b^{6} c^{2} d^{7} - 5 \, a^{4} b^{5} c d^{8} - 2 \, a^{5} b^{4} d^{9}\right )} x^{4} + 1470 \,{\left (57 \, b^{9} c^{6} d^{3} - 154 \, a b^{8} c^{5} d^{4} + 125 \, a^{2} b^{7} c^{4} d^{5} - 20 \, a^{3} b^{6} c^{3} d^{6} - 5 \, a^{4} b^{5} c^{2} d^{7} - 2 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + 126 \,{\left (459 \, b^{9} c^{7} d^{2} - 1218 \, a b^{8} c^{6} d^{3} + 959 \, a^{2} b^{7} c^{5} d^{4} - 140 \, a^{3} b^{6} c^{4} d^{5} - 35 \, a^{4} b^{5} c^{3} d^{6} - 14 \, a^{5} b^{4} c^{2} d^{7} - 7 \, a^{6} b^{3} c d^{8} - 4 \, a^{7} b^{2} d^{9}\right )} x^{2} + 21 \,{\left (1023 \, b^{9} c^{8} d - 2676 \, a b^{8} c^{7} d^{2} + 2058 \, a^{2} b^{7} c^{6} d^{3} - 280 \, a^{3} b^{6} c^{5} d^{4} - 70 \, a^{4} b^{5} c^{4} d^{5} - 28 \, a^{5} b^{4} c^{3} d^{6} - 14 \, a^{6} b^{3} c^{2} d^{7} - 8 \, a^{7} b^{2} c d^{8} - 5 \, a^{8} b d^{9}\right )} x}{70 \,{\left (d^{17} x^{7} + 7 \, c d^{16} x^{6} + 21 \, c^{2} d^{15} x^{5} + 35 \, c^{3} d^{14} x^{4} + 35 \, c^{4} d^{13} x^{3} + 21 \, c^{5} d^{12} x^{2} + 7 \, c^{6} d^{11} x + c^{7} d^{10}\right )}} + \frac{b^{9} d x^{2} - 2 \,{\left (8 \, b^{9} c - 9 \, a b^{8} d\right )} x}{2 \, d^{9}} + \frac{36 \,{\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} \log \left (d x + c\right )}{d^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="maxima")

[Out]

1/70*(3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*b^5*c^5*d^4 - 84*
a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 + 5880*(b^9*c^3*d^6 -
3*a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^6 + 4410*(7*b^9*c^4*d^5 - 20*a*b^8*c^3*d^6 + 18*a^2*b^7*c^2
*d^7 - 4*a^3*b^6*c*d^8 - a^4*b^5*d^9)*x^5 + 1470*(47*b^9*c^5*d^4 - 130*a*b^8*c^4*d^5 + 110*a^2*b^7*c^3*d^6 - 2
0*a^3*b^6*c^2*d^7 - 5*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 + 1470*(57*b^9*c^6*d^3 - 154*a*b^8*c^5*d^4 + 125*a^2*
b^7*c^4*d^5 - 20*a^3*b^6*c^3*d^6 - 5*a^4*b^5*c^2*d^7 - 2*a^5*b^4*c*d^8 - a^6*b^3*d^9)*x^3 + 126*(459*b^9*c^7*d
^2 - 1218*a*b^8*c^6*d^3 + 959*a^2*b^7*c^5*d^4 - 140*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 14*a^5*b^4*c^2*d^7
- 7*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 + 21*(1023*b^9*c^8*d - 2676*a*b^8*c^7*d^2 + 2058*a^2*b^7*c^6*d^3 - 280*
a^3*b^6*c^5*d^4 - 70*a^4*b^5*c^4*d^5 - 28*a^5*b^4*c^3*d^6 - 14*a^6*b^3*c^2*d^7 - 8*a^7*b^2*c*d^8 - 5*a^8*b*d^9
)*x)/(d^17*x^7 + 7*c*d^16*x^6 + 21*c^2*d^15*x^5 + 35*c^3*d^14*x^4 + 35*c^4*d^13*x^3 + 21*c^5*d^12*x^2 + 7*c^6*
d^11*x + c^7*d^10) + 1/2*(b^9*d*x^2 - 2*(8*b^9*c - 9*a*b^8*d)*x)/d^9 + 36*(b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2
)*log(d*x + c)/d^10

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Fricas [B]  time = 1.89062, size = 2298, normalized size = 9.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="fricas")

[Out]

1/70*(35*b^9*d^9*x^9 + 3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*
b^5*c^5*d^4 - 84*a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 - 315
*(b^9*c*d^8 - 2*a*b^8*d^9)*x^8 - 245*(13*b^9*c^2*d^7 - 18*a*b^8*c*d^8)*x^7 - 245*(19*b^9*c^3*d^6 + 18*a*b^8*c^
2*d^7 - 72*a^2*b^7*c*d^8 + 24*a^3*b^6*d^9)*x^6 + 735*(17*b^9*c^4*d^5 - 90*a*b^8*c^3*d^6 + 108*a^2*b^7*c^2*d^7
- 24*a^3*b^6*c*d^8 - 6*a^4*b^5*d^9)*x^5 + 245*(205*b^9*c^5*d^4 - 690*a*b^8*c^4*d^5 + 660*a^2*b^7*c^3*d^6 - 120
*a^3*b^6*c^2*d^7 - 30*a^4*b^5*c*d^8 - 12*a^5*b^4*d^9)*x^4 + 245*(295*b^9*c^6*d^3 - 870*a*b^8*c^5*d^4 + 750*a^2
*b^7*c^4*d^5 - 120*a^3*b^6*c^3*d^6 - 30*a^4*b^5*c^2*d^7 - 12*a^5*b^4*c*d^8 - 6*a^6*b^3*d^9)*x^3 + 21*(2569*b^9
*c^7*d^2 - 7098*a*b^8*c^6*d^3 + 5754*a^2*b^7*c^5*d^4 - 840*a^3*b^6*c^4*d^5 - 210*a^4*b^5*c^3*d^6 - 84*a^5*b^4*
c^2*d^7 - 42*a^6*b^3*c*d^8 - 24*a^7*b^2*d^9)*x^2 + 7*(2989*b^9*c^8*d - 7938*a*b^8*c^7*d^2 + 6174*a^2*b^7*c^6*d
^3 - 840*a^3*b^6*c^5*d^4 - 210*a^4*b^5*c^4*d^5 - 84*a^5*b^4*c^3*d^6 - 42*a^6*b^3*c^2*d^7 - 24*a^7*b^2*c*d^8 -
15*a^8*b*d^9)*x + 2520*(b^9*c^9 - 2*a*b^8*c^8*d + a^2*b^7*c^7*d^2 + (b^9*c^2*d^7 - 2*a*b^8*c*d^8 + a^2*b^7*d^9
)*x^7 + 7*(b^9*c^3*d^6 - 2*a*b^8*c^2*d^7 + a^2*b^7*c*d^8)*x^6 + 21*(b^9*c^4*d^5 - 2*a*b^8*c^3*d^6 + a^2*b^7*c^
2*d^7)*x^5 + 35*(b^9*c^5*d^4 - 2*a*b^8*c^4*d^5 + a^2*b^7*c^3*d^6)*x^4 + 35*(b^9*c^6*d^3 - 2*a*b^8*c^5*d^4 + a^
2*b^7*c^4*d^5)*x^3 + 21*(b^9*c^7*d^2 - 2*a*b^8*c^6*d^3 + a^2*b^7*c^5*d^4)*x^2 + 7*(b^9*c^8*d - 2*a*b^8*c^7*d^2
+ a^2*b^7*c^6*d^3)*x)*log(d*x + c))/(d^17*x^7 + 7*c*d^16*x^6 + 21*c^2*d^15*x^5 + 35*c^3*d^14*x^4 + 35*c^4*d^1
3*x^3 + 21*c^5*d^12*x^2 + 7*c^6*d^11*x + c^7*d^10)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**9/(d*x+c)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.06169, size = 976, normalized size = 4.21 \begin{align*} \frac{36 \,{\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{10}} + \frac{b^{9} d^{8} x^{2} - 16 \, b^{9} c d^{7} x + 18 \, a b^{8} d^{8} x}{2 \, d^{16}} + \frac{3349 \, b^{9} c^{9} - 8658 \, a b^{8} c^{8} d + 6534 \, a^{2} b^{7} c^{7} d^{2} - 840 \, a^{3} b^{6} c^{6} d^{3} - 210 \, a^{4} b^{5} c^{5} d^{4} - 84 \, a^{5} b^{4} c^{4} d^{5} - 42 \, a^{6} b^{3} c^{3} d^{6} - 24 \, a^{7} b^{2} c^{2} d^{7} - 15 \, a^{8} b c d^{8} - 10 \, a^{9} d^{9} + 5880 \,{\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{6} + 4410 \,{\left (7 \, b^{9} c^{4} d^{5} - 20 \, a b^{8} c^{3} d^{6} + 18 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x^{5} + 1470 \,{\left (47 \, b^{9} c^{5} d^{4} - 130 \, a b^{8} c^{4} d^{5} + 110 \, a^{2} b^{7} c^{3} d^{6} - 20 \, a^{3} b^{6} c^{2} d^{7} - 5 \, a^{4} b^{5} c d^{8} - 2 \, a^{5} b^{4} d^{9}\right )} x^{4} + 1470 \,{\left (57 \, b^{9} c^{6} d^{3} - 154 \, a b^{8} c^{5} d^{4} + 125 \, a^{2} b^{7} c^{4} d^{5} - 20 \, a^{3} b^{6} c^{3} d^{6} - 5 \, a^{4} b^{5} c^{2} d^{7} - 2 \, a^{5} b^{4} c d^{8} - a^{6} b^{3} d^{9}\right )} x^{3} + 126 \,{\left (459 \, b^{9} c^{7} d^{2} - 1218 \, a b^{8} c^{6} d^{3} + 959 \, a^{2} b^{7} c^{5} d^{4} - 140 \, a^{3} b^{6} c^{4} d^{5} - 35 \, a^{4} b^{5} c^{3} d^{6} - 14 \, a^{5} b^{4} c^{2} d^{7} - 7 \, a^{6} b^{3} c d^{8} - 4 \, a^{7} b^{2} d^{9}\right )} x^{2} + 21 \,{\left (1023 \, b^{9} c^{8} d - 2676 \, a b^{8} c^{7} d^{2} + 2058 \, a^{2} b^{7} c^{6} d^{3} - 280 \, a^{3} b^{6} c^{5} d^{4} - 70 \, a^{4} b^{5} c^{4} d^{5} - 28 \, a^{5} b^{4} c^{3} d^{6} - 14 \, a^{6} b^{3} c^{2} d^{7} - 8 \, a^{7} b^{2} c d^{8} - 5 \, a^{8} b d^{9}\right )} x}{70 \,{\left (d x + c\right )}^{7} d^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/(d*x+c)^8,x, algorithm="giac")

[Out]

36*(b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*log(abs(d*x + c))/d^10 + 1/2*(b^9*d^8*x^2 - 16*b^9*c*d^7*x + 18*a*b^8
*d^8*x)/d^16 + 1/70*(3349*b^9*c^9 - 8658*a*b^8*c^8*d + 6534*a^2*b^7*c^7*d^2 - 840*a^3*b^6*c^6*d^3 - 210*a^4*b^
5*c^5*d^4 - 84*a^5*b^4*c^4*d^5 - 42*a^6*b^3*c^3*d^6 - 24*a^7*b^2*c^2*d^7 - 15*a^8*b*c*d^8 - 10*a^9*d^9 + 5880*
(b^9*c^3*d^6 - 3*a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^6 + 4410*(7*b^9*c^4*d^5 - 20*a*b^8*c^3*d^6 +
18*a^2*b^7*c^2*d^7 - 4*a^3*b^6*c*d^8 - a^4*b^5*d^9)*x^5 + 1470*(47*b^9*c^5*d^4 - 130*a*b^8*c^4*d^5 + 110*a^2*
b^7*c^3*d^6 - 20*a^3*b^6*c^2*d^7 - 5*a^4*b^5*c*d^8 - 2*a^5*b^4*d^9)*x^4 + 1470*(57*b^9*c^6*d^3 - 154*a*b^8*c^5
*d^4 + 125*a^2*b^7*c^4*d^5 - 20*a^3*b^6*c^3*d^6 - 5*a^4*b^5*c^2*d^7 - 2*a^5*b^4*c*d^8 - a^6*b^3*d^9)*x^3 + 126
*(459*b^9*c^7*d^2 - 1218*a*b^8*c^6*d^3 + 959*a^2*b^7*c^5*d^4 - 140*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 14*a
^5*b^4*c^2*d^7 - 7*a^6*b^3*c*d^8 - 4*a^7*b^2*d^9)*x^2 + 21*(1023*b^9*c^8*d - 2676*a*b^8*c^7*d^2 + 2058*a^2*b^7
*c^6*d^3 - 280*a^3*b^6*c^5*d^4 - 70*a^4*b^5*c^4*d^5 - 28*a^5*b^4*c^3*d^6 - 14*a^6*b^3*c^2*d^7 - 8*a^7*b^2*c*d^
8 - 5*a^8*b*d^9)*x)/((d*x + c)^7*d^10)