3.1320 \(\int \frac{(c+d x)^{10}}{(a+b x)^9} \, dx\)

Optimal. Leaf size=258 \[ \frac{d^9 x (10 b c-9 a d)}{b^{10}}-\frac{120 d^7 (b c-a d)^3}{b^{11} (a+b x)}-\frac{105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac{84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac{105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac{24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac{15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}+\frac{45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}-\frac{10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac{(b c-a d)^{10}}{8 b^{11} (a+b x)^8}+\frac{d^{10} x^2}{2 b^9} \]

[Out]

(d^9*(10*b*c - 9*a*d)*x)/b^10 + (d^10*x^2)/(2*b^9) - (b*c - a*d)^10/(8*b^11*(a + b*x)^8) - (10*d*(b*c - a*d)^9
)/(7*b^11*(a + b*x)^7) - (15*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^6) - (24*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^
5) - (105*d^4*(b*c - a*d)^6)/(2*b^11*(a + b*x)^4) - (84*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^3) - (105*d^6*(b*c
- a*d)^4)/(b^11*(a + b*x)^2) - (120*d^7*(b*c - a*d)^3)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*Log[a + b*x])/
b^11

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Rubi [A]  time = 0.340847, antiderivative size = 258, 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} \[ \frac{d^9 x (10 b c-9 a d)}{b^{10}}-\frac{120 d^7 (b c-a d)^3}{b^{11} (a+b x)}-\frac{105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac{84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac{105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac{24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac{15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}+\frac{45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}-\frac{10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac{(b c-a d)^{10}}{8 b^{11} (a+b x)^8}+\frac{d^{10} x^2}{2 b^9} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^9,x]

[Out]

(d^9*(10*b*c - 9*a*d)*x)/b^10 + (d^10*x^2)/(2*b^9) - (b*c - a*d)^10/(8*b^11*(a + b*x)^8) - (10*d*(b*c - a*d)^9
)/(7*b^11*(a + b*x)^7) - (15*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^6) - (24*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^
5) - (105*d^4*(b*c - a*d)^6)/(2*b^11*(a + b*x)^4) - (84*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^3) - (105*d^6*(b*c
- a*d)^4)/(b^11*(a + b*x)^2) - (120*d^7*(b*c - a*d)^3)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*Log[a + b*x])/
b^11

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{(c+d x)^{10}}{(a+b x)^9} \, dx &=\int \left (\frac{d^9 (10 b c-9 a d)}{b^{10}}+\frac{d^{10} x}{b^9}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)^9}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)^8}+\frac{45 d^2 (b c-a d)^8}{b^{10} (a+b x)^7}+\frac{120 d^3 (b c-a d)^7}{b^{10} (a+b x)^6}+\frac{210 d^4 (b c-a d)^6}{b^{10} (a+b x)^5}+\frac{252 d^5 (b c-a d)^5}{b^{10} (a+b x)^4}+\frac{210 d^6 (b c-a d)^4}{b^{10} (a+b x)^3}+\frac{120 d^7 (b c-a d)^3}{b^{10} (a+b x)^2}+\frac{45 d^8 (b c-a d)^2}{b^{10} (a+b x)}\right ) \, dx\\ &=\frac{d^9 (10 b c-9 a d) x}{b^{10}}+\frac{d^{10} x^2}{2 b^9}-\frac{(b c-a d)^{10}}{8 b^{11} (a+b x)^8}-\frac{10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac{15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}-\frac{24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac{105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac{84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac{105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac{120 d^7 (b c-a d)^3}{b^{11} (a+b x)}+\frac{45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.313807, size = 712, normalized size = 2.76 \[ \frac{-a^2 b^8 d^2 \left (1176 c^6 d^2 x^2+4704 c^5 d^3 x^3+14700 c^4 d^4 x^4+47040 c^3 d^5 x^5-105840 c^2 d^6 x^6+192 c^7 d x+15 c^8+4480 c d^7 x^7+3248 d^8 x^8\right )-8 a^3 b^7 d^3 \left (294 c^5 d^2 x^2+1470 c^4 d^3 x^3+7350 c^3 d^4 x^4-32340 c^2 d^5 x^5+42 c^6 d x+3 c^7+10780 c d^6 x^6+728 d^7 x^7\right )-14 a^4 b^6 d^4 \left (420 c^4 d^2 x^2+3360 c^3 d^3 x^3-26250 c^2 d^4 x^4+48 c^5 d x+3 c^6+19040 c d^5 x^5-1064 d^6 x^6\right )-28 a^5 b^5 d^5 \left (840 c^3 d^2 x^2-11508 c^2 d^3 x^3+60 c^4 d x+3 c^5+15050 c d^4 x^4-2744 d^5 x^5\right )+14 a^6 b^4 d^6 \left (12348 c^2 d^2 x^2-480 c^3 d x-15 c^4-28112 c d^3 x^3+10010 d^4 x^4\right )+8 a^7 b^3 d^7 \left (6534 c^2 d x-105 c^3-27538 c d^2 x^2+17542 d^3 x^3\right )+a^8 b^2 d^8 \left (6849 c^2-68704 c d x+81928 d^2 x^2\right )+2 a^9 b d^9 (13144 d x-4609 c)+3601 a^{10} d^{10}-2 a b^9 d \left (336 c^7 d^2 x^2+1176 c^6 d^3 x^3+2940 c^5 d^4 x^4+5880 c^4 d^5 x^5+11760 c^3 d^6 x^6-10080 c^2 d^7 x^7+60 c^8 d x+5 c^9-2240 c d^8 x^8+140 d^9 x^9\right )+2520 d^8 (a+b x)^8 (b c-a d)^2 \log (a+b x)+b^{10} \left (-\left (420 c^8 d^2 x^2+1344 c^7 d^3 x^3+2940 c^6 d^4 x^4+4704 c^5 d^5 x^5+5880 c^4 d^6 x^6+6720 c^3 d^7 x^7+80 c^9 d x+7 c^{10}-560 c d^9 x^9-28 d^{10} x^{10}\right )\right )}{56 b^{11} (a+b x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^9,x]

[Out]

(3601*a^10*d^10 + 2*a^9*b*d^9*(-4609*c + 13144*d*x) + a^8*b^2*d^8*(6849*c^2 - 68704*c*d*x + 81928*d^2*x^2) + 8
*a^7*b^3*d^7*(-105*c^3 + 6534*c^2*d*x - 27538*c*d^2*x^2 + 17542*d^3*x^3) + 14*a^6*b^4*d^6*(-15*c^4 - 480*c^3*d
*x + 12348*c^2*d^2*x^2 - 28112*c*d^3*x^3 + 10010*d^4*x^4) - 28*a^5*b^5*d^5*(3*c^5 + 60*c^4*d*x + 840*c^3*d^2*x
^2 - 11508*c^2*d^3*x^3 + 15050*c*d^4*x^4 - 2744*d^5*x^5) - 14*a^4*b^6*d^4*(3*c^6 + 48*c^5*d*x + 420*c^4*d^2*x^
2 + 3360*c^3*d^3*x^3 - 26250*c^2*d^4*x^4 + 19040*c*d^5*x^5 - 1064*d^6*x^6) - 8*a^3*b^7*d^3*(3*c^7 + 42*c^6*d*x
+ 294*c^5*d^2*x^2 + 1470*c^4*d^3*x^3 + 7350*c^3*d^4*x^4 - 32340*c^2*d^5*x^5 + 10780*c*d^6*x^6 + 728*d^7*x^7)
- a^2*b^8*d^2*(15*c^8 + 192*c^7*d*x + 1176*c^6*d^2*x^2 + 4704*c^5*d^3*x^3 + 14700*c^4*d^4*x^4 + 47040*c^3*d^5*
x^5 - 105840*c^2*d^6*x^6 + 4480*c*d^7*x^7 + 3248*d^8*x^8) - 2*a*b^9*d*(5*c^9 + 60*c^8*d*x + 336*c^7*d^2*x^2 +
1176*c^6*d^3*x^3 + 2940*c^5*d^4*x^4 + 5880*c^4*d^5*x^5 + 11760*c^3*d^6*x^6 - 10080*c^2*d^7*x^7 - 2240*c*d^8*x^
8 + 140*d^9*x^9) - b^10*(7*c^10 + 80*c^9*d*x + 420*c^8*d^2*x^2 + 1344*c^7*d^3*x^3 + 2940*c^6*d^4*x^4 + 4704*c^
5*d^5*x^5 + 5880*c^4*d^6*x^6 + 6720*c^3*d^7*x^7 - 560*c*d^9*x^9 - 28*d^10*x^10) + 2520*d^8*(b*c - a*d)^2*(a +
b*x)^8*Log[a + b*x])/(56*b^11*(a + b*x)^8)

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Maple [B]  time = 0.017, size = 1256, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^9,x)

[Out]

1/2*d^10*x^2/b^9-9*d^10/b^10*a*x+10*d^9/b^9*x*c+120/b^11*d^10/(b*x+a)*a^3-120/b^8*d^7/(b*x+a)*c^3-1/8/b^11/(b*
x+a)^8*a^10*d^10-15/2/b^11*d^10/(b*x+a)^6*a^8-15/2/b^3*d^2/(b*x+a)^6*c^8-105/b^11*d^10/(b*x+a)^2*a^4+10/7/b^11
*d^10/(b*x+a)^7*a^9-10/7/b^2*d/(b*x+a)^7*c^9+45/b^11*d^10*ln(b*x+a)*a^2+45/b^9*d^8*ln(b*x+a)*c^2-105/2/b^11*d^
10/(b*x+a)^4*a^6-105/2/b^5*d^4/(b*x+a)^4*c^6-105/b^7*d^6/(b*x+a)^2*c^4+84/b^11*d^10/(b*x+a)^3*a^5-84/b^6*d^5/(
b*x+a)^3*c^5+24/b^11*d^10/(b*x+a)^5*a^7-24/b^4*d^3/(b*x+a)^5*c^7-180/b^6*d^5/(b*x+a)^7*a^4*c^5+120/b^5*d^4/(b*
x+a)^7*a^3*c^6-1/8/b/(b*x+a)^8*c^10+15/b^8/(b*x+a)^8*a^7*c^3*d^7-105/4/b^7/(b*x+a)^8*a^6*c^4*d^6+63/2/b^6/(b*x
+a)^8*a^5*c^5*d^5-105/4/b^5/(b*x+a)^8*a^4*c^6*d^4+15/b^4/(b*x+a)^8*a^3*c^7*d^3-45/8/b^3/(b*x+a)^8*a^2*c^8*d^2+
5/4/b^2/(b*x+a)^8*a*c^9*d+60/b^10*d^9/(b*x+a)^6*a^7*c-210/b^9*d^8/(b*x+a)^6*a^6*c^2+360/7/b^9*d^8/(b*x+a)^7*a^
7*c^2-120/b^8*d^7/(b*x+a)^7*a^6*c^3+180/b^7*d^6/(b*x+a)^7*a^5*c^4-360/7/b^4*d^3/(b*x+a)^7*a^2*c^7+90/7/b^3*d^2
/(b*x+a)^7*a*c^8+420/b^8*d^7/(b*x+a)^6*a^5*c^3-525/b^7*d^6/(b*x+a)^6*a^4*c^4+420/b^6*d^5/(b*x+a)^6*a^3*c^5-157
5/2/b^9*d^8/(b*x+a)^4*a^4*c^2+1050/b^8*d^7/(b*x+a)^4*a^3*c^3-1575/2/b^7*d^6/(b*x+a)^4*a^2*c^4+315/b^10*d^9/(b*
x+a)^4*a^5*c-90/b^10*d^9*ln(b*x+a)*a*c+315/b^6*d^5/(b*x+a)^4*a*c^5-90/7/b^10*d^9/(b*x+a)^7*a^8*c+420/b^7*d^6/(
b*x+a)^3*a*c^4-168/b^10*d^9/(b*x+a)^5*a^6*c+504/b^9*d^8/(b*x+a)^5*a^5*c^2-840/b^8*d^7/(b*x+a)^5*a^4*c^3-360/b^
10*d^9/(b*x+a)*a^2*c+360/b^9*d^8/(b*x+a)*a*c^2+5/4/b^10/(b*x+a)^8*a^9*c*d^9-45/8/b^9/(b*x+a)^8*a^8*c^2*d^8+168
/b^5*d^4/(b*x+a)^5*a*c^6-210/b^5*d^4/(b*x+a)^6*a^2*c^6+60/b^4*d^3/(b*x+a)^6*a*c^7+420/b^10*d^9/(b*x+a)^2*a^3*c
-630/b^9*d^8/(b*x+a)^2*a^2*c^2+420/b^8*d^7/(b*x+a)^2*a*c^3-420/b^10*d^9/(b*x+a)^3*a^4*c+840/b^9*d^8/(b*x+a)^3*
a^3*c^2-840/b^8*d^7/(b*x+a)^3*a^2*c^3+840/b^7*d^6/(b*x+a)^5*a^3*c^4-504/b^6*d^5/(b*x+a)^5*a^2*c^5

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Maxima [B]  time = 1.37796, size = 1276, normalized size = 4.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^9,x, algorithm="maxima")

[Out]

-1/56*(7*b^10*c^10 + 10*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 84*a^5*b^
5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 - 6849*a^8*b^2*c^2*d^8 + 9218*a^9*b*c*d^9 - 3601*a^10*d^
10 + 6720*(b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 5880*(b^10*c^4*d^6 + 4*a*b^9
*c^3*d^7 - 18*a^2*b^8*c^2*d^8 + 20*a^3*b^7*c*d^9 - 7*a^4*b^6*d^10)*x^6 + 2352*(2*b^10*c^5*d^5 + 5*a*b^9*c^4*d^
6 + 20*a^2*b^8*c^3*d^7 - 110*a^3*b^7*c^2*d^8 + 130*a^4*b^6*c*d^9 - 47*a^5*b^5*d^10)*x^5 + 2940*(b^10*c^6*d^4 +
2*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 - 125*a^4*b^6*c^2*d^8 + 154*a^5*b^5*c*d^9 - 57*a^6*b
^4*d^10)*x^4 + 336*(4*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 14*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 140*a^4*b^6*c
^3*d^7 - 959*a^5*b^5*c^2*d^8 + 1218*a^6*b^4*c*d^9 - 459*a^7*b^3*d^10)*x^3 + 84*(5*b^10*c^8*d^2 + 8*a*b^9*c^7*d
^3 + 14*a^2*b^8*c^6*d^4 + 28*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 - 2058*a^6*b^4*c^2*d^8
+ 2676*a^7*b^3*c*d^9 - 1023*a^8*b^2*d^10)*x^2 + 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 42
*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*b^3*c^2*d^8 + 865
8*a^8*b^2*c*d^9 - 3349*a^9*b*d^10)*x)/(b^19*x^8 + 8*a*b^18*x^7 + 28*a^2*b^17*x^6 + 56*a^3*b^16*x^5 + 70*a^4*b^
15*x^4 + 56*a^5*b^14*x^3 + 28*a^6*b^13*x^2 + 8*a^7*b^12*x + a^8*b^11) + 1/2*(b*d^10*x^2 + 2*(10*b*c*d^9 - 9*a*
d^10)*x)/b^10 + 45*(b^2*c^2*d^8 - 2*a*b*c*d^9 + a^2*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.8687, size = 2763, normalized size = 10.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^9,x, algorithm="fricas")

[Out]

1/56*(28*b^10*d^10*x^10 - 7*b^10*c^10 - 10*a*b^9*c^9*d - 15*a^2*b^8*c^8*d^2 - 24*a^3*b^7*c^7*d^3 - 42*a^4*b^6*
c^6*d^4 - 84*a^5*b^5*c^5*d^5 - 210*a^6*b^4*c^4*d^6 - 840*a^7*b^3*c^3*d^7 + 6849*a^8*b^2*c^2*d^8 - 9218*a^9*b*c
*d^9 + 3601*a^10*d^10 + 280*(2*b^10*c*d^9 - a*b^9*d^10)*x^9 + 112*(40*a*b^9*c*d^9 - 29*a^2*b^8*d^10)*x^8 - 448
*(15*b^10*c^3*d^7 - 45*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + 13*a^3*b^7*d^10)*x^7 - 392*(15*b^10*c^4*d^6 + 60*a*b
^9*c^3*d^7 - 270*a^2*b^8*c^2*d^8 + 220*a^3*b^7*c*d^9 - 38*a^4*b^6*d^10)*x^6 - 784*(6*b^10*c^5*d^5 + 15*a*b^9*c
^4*d^6 + 60*a^2*b^8*c^3*d^7 - 330*a^3*b^7*c^2*d^8 + 340*a^4*b^6*c*d^9 - 98*a^5*b^5*d^10)*x^5 - 980*(3*b^10*c^6
*d^4 + 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 60*a^3*b^7*c^3*d^7 - 375*a^4*b^6*c^2*d^8 + 430*a^5*b^5*c*d^9 - 1
43*a^6*b^4*d^10)*x^4 - 112*(12*b^10*c^7*d^3 + 21*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 + 42
0*a^4*b^6*c^3*d^7 - 2877*a^5*b^5*c^2*d^8 + 3514*a^6*b^4*c*d^9 - 1253*a^7*b^3*d^10)*x^3 - 28*(15*b^10*c^8*d^2 +
24*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 84*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 840*a^5*b^5*c^3*d^7 - 6174
*a^6*b^4*c^2*d^8 + 7868*a^7*b^3*c*d^9 - 2926*a^8*b^2*d^10)*x^2 - 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2*
b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*b
^3*c^2*d^8 + 8588*a^8*b^2*c*d^9 - 3286*a^9*b*d^10)*x + 2520*(a^8*b^2*c^2*d^8 - 2*a^9*b*c*d^9 + a^10*d^10 + (b^
10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 8*(a*b^9*c^2*d^8 - 2*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 28*(
a^2*b^8*c^2*d^8 - 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 56*(a^3*b^7*c^2*d^8 - 2*a^4*b^6*c*d^9 + a^5*b^5*d^10)*
x^5 + 70*(a^4*b^6*c^2*d^8 - 2*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 56*(a^5*b^5*c^2*d^8 - 2*a^6*b^4*c*d^9 + a^7*
b^3*d^10)*x^3 + 28*(a^6*b^4*c^2*d^8 - 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 8*(a^7*b^3*c^2*d^8 - 2*a^8*b^2*c*d
^9 + a^9*b*d^10)*x)*log(b*x + a))/(b^19*x^8 + 8*a*b^18*x^7 + 28*a^2*b^17*x^6 + 56*a^3*b^16*x^5 + 70*a^4*b^15*x
^4 + 56*a^5*b^14*x^3 + 28*a^6*b^13*x^2 + 8*a^7*b^12*x + a^8*b^11)

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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((d*x+c)**10/(b*x+a)**9,x)

[Out]

Timed out

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Giac [B]  time = 1.069, size = 1176, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^9,x, algorithm="giac")

[Out]

45*(b^2*c^2*d^8 - 2*a*b*c*d^9 + a^2*d^10)*log(abs(b*x + a))/b^11 + 1/2*(b^9*d^10*x^2 + 20*b^9*c*d^9*x - 18*a*b
^8*d^10*x)/b^18 - 1/56*(7*b^10*c^10 + 10*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^
6*d^4 + 84*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 - 6849*a^8*b^2*c^2*d^8 + 9218*a^9*b*c*d
^9 - 3601*a^10*d^10 + 6720*(b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 5880*(b^10*
c^4*d^6 + 4*a*b^9*c^3*d^7 - 18*a^2*b^8*c^2*d^8 + 20*a^3*b^7*c*d^9 - 7*a^4*b^6*d^10)*x^6 + 2352*(2*b^10*c^5*d^5
+ 5*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 - 110*a^3*b^7*c^2*d^8 + 130*a^4*b^6*c*d^9 - 47*a^5*b^5*d^10)*x^5 + 294
0*(b^10*c^6*d^4 + 2*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 - 125*a^4*b^6*c^2*d^8 + 154*a^5*b^5
*c*d^9 - 57*a^6*b^4*d^10)*x^4 + 336*(4*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 14*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^
6 + 140*a^4*b^6*c^3*d^7 - 959*a^5*b^5*c^2*d^8 + 1218*a^6*b^4*c*d^9 - 459*a^7*b^3*d^10)*x^3 + 84*(5*b^10*c^8*d^
2 + 8*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 28*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 - 205
8*a^6*b^4*c^2*d^8 + 2676*a^7*b^3*c*d^9 - 1023*a^8*b^2*d^10)*x^2 + 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2
*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*
b^3*c^2*d^8 + 8658*a^8*b^2*c*d^9 - 3349*a^9*b*d^10)*x)/((b*x + a)^8*b^11)