3.1312 \(\int \frac{(c+d x)^{10}}{a+b x} \, dx\)

Optimal. Leaf size=241 \[ \frac{d x (b c-a d)^9}{b^{10}}+\frac{(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac{(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac{(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac{(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac{(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac{(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac{(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac{(c+d x)^9 (b c-a d)}{9 b^2}+\frac{(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac{(c+d x)^{10}}{10 b} \]

[Out]

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

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Rubi [A]  time = 0.097776, antiderivative size = 241, 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} \[ \frac{d x (b c-a d)^9}{b^{10}}+\frac{(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac{(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac{(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac{(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac{(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac{(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac{(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac{(c+d x)^9 (b c-a d)}{9 b^2}+\frac{(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac{(c+d x)^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x),x]

[Out]

(d*(b*c - a*d)^9*x)/b^10 + ((b*c - a*d)^8*(c + d*x)^2)/(2*b^9) + ((b*c - a*d)^7*(c + d*x)^3)/(3*b^8) + ((b*c -
a*d)^6*(c + d*x)^4)/(4*b^7) + ((b*c - a*d)^5*(c + d*x)^5)/(5*b^6) + ((b*c - a*d)^4*(c + d*x)^6)/(6*b^5) + ((b
*c - a*d)^3*(c + d*x)^7)/(7*b^4) + ((b*c - a*d)^2*(c + d*x)^8)/(8*b^3) + ((b*c - a*d)*(c + d*x)^9)/(9*b^2) + (
c + d*x)^10/(10*b) + ((b*c - a*d)^10*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} \, dx &=\int \left (\frac{d (b c-a d)^9}{b^{10}}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)}+\frac{d (b c-a d)^8 (c+d x)}{b^9}+\frac{d (b c-a d)^7 (c+d x)^2}{b^8}+\frac{d (b c-a d)^6 (c+d x)^3}{b^7}+\frac{d (b c-a d)^5 (c+d x)^4}{b^6}+\frac{d (b c-a d)^4 (c+d x)^5}{b^5}+\frac{d (b c-a d)^3 (c+d x)^6}{b^4}+\frac{d (b c-a d)^2 (c+d x)^7}{b^3}+\frac{d (b c-a d) (c+d x)^8}{b^2}+\frac{d (c+d x)^9}{b}\right ) \, dx\\ &=\frac{d (b c-a d)^9 x}{b^{10}}+\frac{(b c-a d)^8 (c+d x)^2}{2 b^9}+\frac{(b c-a d)^7 (c+d x)^3}{3 b^8}+\frac{(b c-a d)^6 (c+d x)^4}{4 b^7}+\frac{(b c-a d)^5 (c+d x)^5}{5 b^6}+\frac{(b c-a d)^4 (c+d x)^6}{6 b^5}+\frac{(b c-a d)^3 (c+d x)^7}{7 b^4}+\frac{(b c-a d)^2 (c+d x)^8}{8 b^3}+\frac{(b c-a d) (c+d x)^9}{9 b^2}+\frac{(c+d x)^{10}}{10 b}+\frac{(b c-a d)^{10} \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.297758, size = 591, normalized size = 2.45 \[ \frac{d x \left (45 a^2 b^7 d^2 \left (4704 c^5 d^2 x^2+2940 c^4 d^3 x^3+1344 c^3 d^4 x^4+420 c^2 d^5 x^5+5880 c^6 d x+6720 c^7+80 c d^6 x^6+7 d^7 x^7\right )-120 a^3 b^6 d^3 \left (1470 c^4 d^2 x^2+630 c^3 d^3 x^3+189 c^2 d^4 x^4+2646 c^5 d x+4410 c^6+35 c d^5 x^5+3 d^6 x^6\right )+210 a^4 b^5 d^4 \left (480 c^3 d^2 x^2+135 c^2 d^3 x^3+1260 c^4 d x+3024 c^5+24 c d^4 x^4+2 d^5 x^5\right )-252 a^5 b^4 d^5 \left (150 c^2 d^2 x^2+600 c^3 d x+2100 c^4+25 c d^3 x^3+2 d^4 x^4\right )+210 a^6 b^3 d^6 \left (270 c^2 d x+1440 c^3+40 c d^2 x^2+3 d^3 x^3\right )-840 a^7 b^2 d^7 \left (135 c^2+15 c d x+d^2 x^2\right )+1260 a^8 b d^8 (20 c+d x)-2520 a^9 d^9-10 a b^8 d \left (17640 c^6 d^2 x^2+15876 c^5 d^3 x^3+10584 c^4 d^4 x^4+5040 c^3 d^5 x^5+1620 c^2 d^6 x^6+15120 c^7 d x+11340 c^8+315 c d^7 x^7+28 d^8 x^8\right )+b^9 \left (100800 c^7 d^2 x^2+132300 c^6 d^3 x^3+127008 c^5 d^4 x^4+88200 c^4 d^5 x^5+43200 c^3 d^6 x^6+14175 c^2 d^7 x^7+56700 c^8 d x+25200 c^9+2800 c d^8 x^8+252 d^9 x^9\right )\right )}{2520 b^{10}}+\frac{(b c-a d)^{10} \log (a+b x)}{b^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x),x]

[Out]

(d*x*(-2520*a^9*d^9 + 1260*a^8*b*d^8*(20*c + d*x) - 840*a^7*b^2*d^7*(135*c^2 + 15*c*d*x + d^2*x^2) + 210*a^6*b
^3*d^6*(1440*c^3 + 270*c^2*d*x + 40*c*d^2*x^2 + 3*d^3*x^3) - 252*a^5*b^4*d^5*(2100*c^4 + 600*c^3*d*x + 150*c^2
*d^2*x^2 + 25*c*d^3*x^3 + 2*d^4*x^4) + 210*a^4*b^5*d^4*(3024*c^5 + 1260*c^4*d*x + 480*c^3*d^2*x^2 + 135*c^2*d^
3*x^3 + 24*c*d^4*x^4 + 2*d^5*x^5) - 120*a^3*b^6*d^3*(4410*c^6 + 2646*c^5*d*x + 1470*c^4*d^2*x^2 + 630*c^3*d^3*
x^3 + 189*c^2*d^4*x^4 + 35*c*d^5*x^5 + 3*d^6*x^6) + 45*a^2*b^7*d^2*(6720*c^7 + 5880*c^6*d*x + 4704*c^5*d^2*x^2
+ 2940*c^4*d^3*x^3 + 1344*c^3*d^4*x^4 + 420*c^2*d^5*x^5 + 80*c*d^6*x^6 + 7*d^7*x^7) - 10*a*b^8*d*(11340*c^8 +
15120*c^7*d*x + 17640*c^6*d^2*x^2 + 15876*c^5*d^3*x^3 + 10584*c^4*d^4*x^4 + 5040*c^3*d^5*x^5 + 1620*c^2*d^6*x
^6 + 315*c*d^7*x^7 + 28*d^8*x^8) + b^9*(25200*c^9 + 56700*c^8*d*x + 100800*c^7*d^2*x^2 + 132300*c^6*d^3*x^3 +
127008*c^5*d^4*x^4 + 88200*c^4*d^5*x^5 + 43200*c^3*d^6*x^6 + 14175*c^2*d^7*x^7 + 2800*c*d^8*x^8 + 252*d^9*x^9)
))/(2520*b^10) + ((b*c - a*d)^10*Log[a + b*x])/b^11

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Maple [B]  time = 0.007, size = 1022, normalized size = 4.2 \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),x)

[Out]

1/8*d^10/b^3*x^8*a^2+45/8*d^8/b*x^8*c^2-1/7*d^10/b^4*x^7*a^3+120/7*d^7/b*x^7*c^3+10*d/b*c^9*x-d^10/b^10*a^9*x+
1/b^11*ln(b*x+a)*a^10*d^10+1/4*d^10/b^7*x^4*a^6+105/2*d^4/b*x^4*c^6-1/3*d^10/b^8*x^3*a^7+40*d^3/b*x^3*c^7+1/2*
d^10/b^9*x^2*a^8+45/2*d^2/b*x^2*c^8+252/5*d^5/b*x^5*c^5+35*d^6/b*x^6*c^4-1/5*d^10/b^6*x^5*a^5+1/6*d^10/b^5*x^6
*a^4-1/9*d^10/b^2*x^9*a+10/9*d^9/b*x^9*c-45*d^8/b^8*a^7*c^2*x+120*d^7/b^7*a^6*c^3*x-45*d^2/b^2*a*c^8*x-60*d^3/
b^2*x^2*a*c^7-30*d^7/b^4*x^4*a^3*c^3+10*d^9/b^9*a^8*c*x+120*d^3/b^3*a^2*c^7*x-126*d^5/b^4*x^2*a^3*c^5+105*d^4/
b^3*x^2*a^2*c^6+45/2*d^8/b^7*x^2*a^6*c^2+45/4*d^8/b^5*x^4*a^4*c^2+1/b*ln(b*x+a)*c^10+1/10*d^10/b*x^10-5/4*d^9/
b^2*x^8*a*c-210*d^6/b^6*a^5*c^4*x+252*d^5/b^5*a^4*c^5*x-210*d^4/b^4*a^3*c^6*x+105/2*d^6/b^3*x^4*a^2*c^4-20*d^7
/b^2*x^6*a*c^3-5/3*d^9/b^4*x^6*a^3*c+15/2*d^8/b^3*x^6*a^2*c^2-9*d^8/b^4*x^5*a^3*c^2+10/7*d^9/b^3*x^7*a^2*c-45/
7*d^8/b^2*x^7*a*c^2+210/b^5*ln(b*x+a)*a^4*c^6*d^4-120/b^4*ln(b*x+a)*a^3*c^7*d^3+45/b^3*ln(b*x+a)*a^2*c^8*d^2-1
0/b^2*ln(b*x+a)*a*c^9*d-60*d^7/b^6*x^2*a^5*c^3+105*d^6/b^5*x^2*a^4*c^4-5*d^9/b^8*x^2*a^7*c+84*d^5/b^3*x^3*a^2*
c^5-70*d^4/b^2*x^3*a*c^6-5/2*d^9/b^6*x^4*a^5*c-63*d^5/b^2*x^4*a*c^5+10/3*d^9/b^7*x^3*a^6*c-15*d^8/b^6*x^3*a^5*
c^2+40*d^7/b^5*x^3*a^4*c^3-70*d^6/b^4*x^3*a^3*c^4+2*d^9/b^5*x^5*a^4*c-42*d^6/b^2*x^5*a*c^4+24*d^7/b^3*x^5*a^2*
c^3-252/b^6*ln(b*x+a)*a^5*c^5*d^5-120/b^8*ln(b*x+a)*a^7*c^3*d^7+210/b^7*ln(b*x+a)*a^6*c^4*d^6+45/b^9*ln(b*x+a)
*a^8*c^2*d^8-10/b^10*ln(b*x+a)*a^9*c*d^9

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Maxima [B]  time = 0.978517, size = 1169, normalized size = 4.85 \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),x, algorithm="maxima")

[Out]

1/2520*(252*b^9*d^10*x^10 + 280*(10*b^9*c*d^9 - a*b^8*d^10)*x^9 + 315*(45*b^9*c^2*d^8 - 10*a*b^8*c*d^9 + a^2*b
^7*d^10)*x^8 + 360*(120*b^9*c^3*d^7 - 45*a*b^8*c^2*d^8 + 10*a^2*b^7*c*d^9 - a^3*b^6*d^10)*x^7 + 420*(210*b^9*c
^4*d^6 - 120*a*b^8*c^3*d^7 + 45*a^2*b^7*c^2*d^8 - 10*a^3*b^6*c*d^9 + a^4*b^5*d^10)*x^6 + 504*(252*b^9*c^5*d^5
- 210*a*b^8*c^4*d^6 + 120*a^2*b^7*c^3*d^7 - 45*a^3*b^6*c^2*d^8 + 10*a^4*b^5*c*d^9 - a^5*b^4*d^10)*x^5 + 630*(2
10*b^9*c^6*d^4 - 252*a*b^8*c^5*d^5 + 210*a^2*b^7*c^4*d^6 - 120*a^3*b^6*c^3*d^7 + 45*a^4*b^5*c^2*d^8 - 10*a^5*b
^4*c*d^9 + a^6*b^3*d^10)*x^4 + 840*(120*b^9*c^7*d^3 - 210*a*b^8*c^6*d^4 + 252*a^2*b^7*c^5*d^5 - 210*a^3*b^6*c^
4*d^6 + 120*a^4*b^5*c^3*d^7 - 45*a^5*b^4*c^2*d^8 + 10*a^6*b^3*c*d^9 - a^7*b^2*d^10)*x^3 + 1260*(45*b^9*c^8*d^2
- 120*a*b^8*c^7*d^3 + 210*a^2*b^7*c^6*d^4 - 252*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 - 120*a^5*b^4*c^3*d^7 +
45*a^6*b^3*c^2*d^8 - 10*a^7*b^2*c*d^9 + a^8*b*d^10)*x^2 + 2520*(10*b^9*c^9*d - 45*a*b^8*c^8*d^2 + 120*a^2*b^7
*c^7*d^3 - 210*a^3*b^6*c^6*d^4 + 252*a^4*b^5*c^5*d^5 - 210*a^5*b^4*c^4*d^6 + 120*a^6*b^3*c^3*d^7 - 45*a^7*b^2*
c^2*d^8 + 10*a^8*b*c*d^9 - a^9*d^10)*x)/b^10 + (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)*log(b*x + a)/b^11

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Fricas [B]  time = 1.76077, size = 1879, normalized size = 7.8 \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),x, algorithm="fricas")

[Out]

1/2520*(252*b^10*d^10*x^10 + 280*(10*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(45*b^10*c^2*d^8 - 10*a*b^9*c*d^9 + a^
2*b^8*d^10)*x^8 + 360*(120*b^10*c^3*d^7 - 45*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(210*b
^10*c^4*d^6 - 120*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 - 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 504*(252*b^10*c^
5*d^5 - 210*a*b^9*c^4*d^6 + 120*a^2*b^8*c^3*d^7 - 45*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 +
630*(210*b^10*c^6*d^4 - 252*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 - 120*a^3*b^7*c^3*d^7 + 45*a^4*b^6*c^2*d^8 - 1
0*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 840*(120*b^10*c^7*d^3 - 210*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 - 210*a^
3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 - 45*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 1260*(45*b^1
0*c^8*d^2 - 120*a*b^9*c^7*d^3 + 210*a^2*b^8*c^6*d^4 - 252*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 120*a^5*b^5*
c^3*d^7 + 45*a^6*b^4*c^2*d^8 - 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 2520*(10*b^10*c^9*d - 45*a*b^9*c^8*d^2 +
120*a^2*b^8*c^7*d^3 - 210*a^3*b^7*c^6*d^4 + 252*a^4*b^6*c^5*d^5 - 210*a^5*b^5*c^4*d^6 + 120*a^6*b^4*c^3*d^7 -
45*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 - a^9*b*d^10)*x + 2520*(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)*log(b*x + a))/b^11

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Sympy [B]  time = 1.43806, size = 772, normalized size = 3.2 \begin{align*} \frac{d^{10} x^{10}}{10 b} - \frac{x^{9} \left (a d^{10} - 10 b c d^{9}\right )}{9 b^{2}} + \frac{x^{8} \left (a^{2} d^{10} - 10 a b c d^{9} + 45 b^{2} c^{2} d^{8}\right )}{8 b^{3}} - \frac{x^{7} \left (a^{3} d^{10} - 10 a^{2} b c d^{9} + 45 a b^{2} c^{2} d^{8} - 120 b^{3} c^{3} d^{7}\right )}{7 b^{4}} + \frac{x^{6} \left (a^{4} d^{10} - 10 a^{3} b c d^{9} + 45 a^{2} b^{2} c^{2} d^{8} - 120 a b^{3} c^{3} d^{7} + 210 b^{4} c^{4} d^{6}\right )}{6 b^{5}} - \frac{x^{5} \left (a^{5} d^{10} - 10 a^{4} b c d^{9} + 45 a^{3} b^{2} c^{2} d^{8} - 120 a^{2} b^{3} c^{3} d^{7} + 210 a b^{4} c^{4} d^{6} - 252 b^{5} c^{5} d^{5}\right )}{5 b^{6}} + \frac{x^{4} \left (a^{6} d^{10} - 10 a^{5} b c d^{9} + 45 a^{4} b^{2} c^{2} d^{8} - 120 a^{3} b^{3} c^{3} d^{7} + 210 a^{2} b^{4} c^{4} d^{6} - 252 a b^{5} c^{5} d^{5} + 210 b^{6} c^{6} d^{4}\right )}{4 b^{7}} - \frac{x^{3} \left (a^{7} d^{10} - 10 a^{6} b c d^{9} + 45 a^{5} b^{2} c^{2} d^{8} - 120 a^{4} b^{3} c^{3} d^{7} + 210 a^{3} b^{4} c^{4} d^{6} - 252 a^{2} b^{5} c^{5} d^{5} + 210 a b^{6} c^{6} d^{4} - 120 b^{7} c^{7} d^{3}\right )}{3 b^{8}} + \frac{x^{2} \left (a^{8} d^{10} - 10 a^{7} b c d^{9} + 45 a^{6} b^{2} c^{2} d^{8} - 120 a^{5} b^{3} c^{3} d^{7} + 210 a^{4} b^{4} c^{4} d^{6} - 252 a^{3} b^{5} c^{5} d^{5} + 210 a^{2} b^{6} c^{6} d^{4} - 120 a b^{7} c^{7} d^{3} + 45 b^{8} c^{8} d^{2}\right )}{2 b^{9}} - \frac{x \left (a^{9} d^{10} - 10 a^{8} b c d^{9} + 45 a^{7} b^{2} c^{2} d^{8} - 120 a^{6} b^{3} c^{3} d^{7} + 210 a^{5} b^{4} c^{4} d^{6} - 252 a^{4} b^{5} c^{5} d^{5} + 210 a^{3} b^{6} c^{6} d^{4} - 120 a^{2} b^{7} c^{7} d^{3} + 45 a b^{8} c^{8} d^{2} - 10 b^{9} c^{9} d\right )}{b^{10}} + \frac{\left (a d - b c\right )^{10} \log{\left (a + b x \right )}}{b^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**10*x**10/(10*b) - x**9*(a*d**10 - 10*b*c*d**9)/(9*b**2) + x**8*(a**2*d**10 - 10*a*b*c*d**9 + 45*b**2*c**2*d
**8)/(8*b**3) - x**7*(a**3*d**10 - 10*a**2*b*c*d**9 + 45*a*b**2*c**2*d**8 - 120*b**3*c**3*d**7)/(7*b**4) + x**
6*(a**4*d**10 - 10*a**3*b*c*d**9 + 45*a**2*b**2*c**2*d**8 - 120*a*b**3*c**3*d**7 + 210*b**4*c**4*d**6)/(6*b**5
) - x**5*(a**5*d**10 - 10*a**4*b*c*d**9 + 45*a**3*b**2*c**2*d**8 - 120*a**2*b**3*c**3*d**7 + 210*a*b**4*c**4*d
**6 - 252*b**5*c**5*d**5)/(5*b**6) + x**4*(a**6*d**10 - 10*a**5*b*c*d**9 + 45*a**4*b**2*c**2*d**8 - 120*a**3*b
**3*c**3*d**7 + 210*a**2*b**4*c**4*d**6 - 252*a*b**5*c**5*d**5 + 210*b**6*c**6*d**4)/(4*b**7) - x**3*(a**7*d**
10 - 10*a**6*b*c*d**9 + 45*a**5*b**2*c**2*d**8 - 120*a**4*b**3*c**3*d**7 + 210*a**3*b**4*c**4*d**6 - 252*a**2*
b**5*c**5*d**5 + 210*a*b**6*c**6*d**4 - 120*b**7*c**7*d**3)/(3*b**8) + x**2*(a**8*d**10 - 10*a**7*b*c*d**9 + 4
5*a**6*b**2*c**2*d**8 - 120*a**5*b**3*c**3*d**7 + 210*a**4*b**4*c**4*d**6 - 252*a**3*b**5*c**5*d**5 + 210*a**2
*b**6*c**6*d**4 - 120*a*b**7*c**7*d**3 + 45*b**8*c**8*d**2)/(2*b**9) - x*(a**9*d**10 - 10*a**8*b*c*d**9 + 45*a
**7*b**2*c**2*d**8 - 120*a**6*b**3*c**3*d**7 + 210*a**5*b**4*c**4*d**6 - 252*a**4*b**5*c**5*d**5 + 210*a**3*b*
*6*c**6*d**4 - 120*a**2*b**7*c**7*d**3 + 45*a*b**8*c**8*d**2 - 10*b**9*c**9*d)/b**10 + (a*d - b*c)**10*log(a +
b*x)/b**11

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Giac [B]  time = 1.05812, size = 1297, normalized size = 5.38 \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),x, algorithm="giac")

[Out]

1/2520*(252*b^9*d^10*x^10 + 2800*b^9*c*d^9*x^9 - 280*a*b^8*d^10*x^9 + 14175*b^9*c^2*d^8*x^8 - 3150*a*b^8*c*d^9
*x^8 + 315*a^2*b^7*d^10*x^8 + 43200*b^9*c^3*d^7*x^7 - 16200*a*b^8*c^2*d^8*x^7 + 3600*a^2*b^7*c*d^9*x^7 - 360*a
^3*b^6*d^10*x^7 + 88200*b^9*c^4*d^6*x^6 - 50400*a*b^8*c^3*d^7*x^6 + 18900*a^2*b^7*c^2*d^8*x^6 - 4200*a^3*b^6*c
*d^9*x^6 + 420*a^4*b^5*d^10*x^6 + 127008*b^9*c^5*d^5*x^5 - 105840*a*b^8*c^4*d^6*x^5 + 60480*a^2*b^7*c^3*d^7*x^
5 - 22680*a^3*b^6*c^2*d^8*x^5 + 5040*a^4*b^5*c*d^9*x^5 - 504*a^5*b^4*d^10*x^5 + 132300*b^9*c^6*d^4*x^4 - 15876
0*a*b^8*c^5*d^5*x^4 + 132300*a^2*b^7*c^4*d^6*x^4 - 75600*a^3*b^6*c^3*d^7*x^4 + 28350*a^4*b^5*c^2*d^8*x^4 - 630
0*a^5*b^4*c*d^9*x^4 + 630*a^6*b^3*d^10*x^4 + 100800*b^9*c^7*d^3*x^3 - 176400*a*b^8*c^6*d^4*x^3 + 211680*a^2*b^
7*c^5*d^5*x^3 - 176400*a^3*b^6*c^4*d^6*x^3 + 100800*a^4*b^5*c^3*d^7*x^3 - 37800*a^5*b^4*c^2*d^8*x^3 + 8400*a^6
*b^3*c*d^9*x^3 - 840*a^7*b^2*d^10*x^3 + 56700*b^9*c^8*d^2*x^2 - 151200*a*b^8*c^7*d^3*x^2 + 264600*a^2*b^7*c^6*
d^4*x^2 - 317520*a^3*b^6*c^5*d^5*x^2 + 264600*a^4*b^5*c^4*d^6*x^2 - 151200*a^5*b^4*c^3*d^7*x^2 + 56700*a^6*b^3
*c^2*d^8*x^2 - 12600*a^7*b^2*c*d^9*x^2 + 1260*a^8*b*d^10*x^2 + 25200*b^9*c^9*d*x - 113400*a*b^8*c^8*d^2*x + 30
2400*a^2*b^7*c^7*d^3*x - 529200*a^3*b^6*c^6*d^4*x + 635040*a^4*b^5*c^5*d^5*x - 529200*a^5*b^4*c^4*d^6*x + 3024
00*a^6*b^3*c^3*d^7*x - 113400*a^7*b^2*c^2*d^8*x + 25200*a^8*b*c*d^9*x - 2520*a^9*d^10*x)/b^10 + (b^10*c^10 - 1
0*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)*log(abs(b*x + a))/b^11