3.1313 \(\int \frac{(c+d x)^{10}}{(a+b x)^2} \, dx\)

Optimal. Leaf size=258 \[ \frac{5 d^9 (a+b x)^8 (b c-a d)}{4 b^{11}}+\frac{45 d^8 (a+b x)^7 (b c-a d)^2}{7 b^{11}}+\frac{20 d^7 (a+b x)^6 (b c-a d)^3}{b^{11}}+\frac{42 d^6 (a+b x)^5 (b c-a d)^4}{b^{11}}+\frac{63 d^5 (a+b x)^4 (b c-a d)^5}{b^{11}}+\frac{70 d^4 (a+b x)^3 (b c-a d)^6}{b^{11}}+\frac{60 d^3 (a+b x)^2 (b c-a d)^7}{b^{11}}+\frac{45 d^2 x (b c-a d)^8}{b^{10}}-\frac{(b c-a d)^{10}}{b^{11} (a+b x)}+\frac{10 d (b c-a d)^9 \log (a+b x)}{b^{11}}+\frac{d^{10} (a+b x)^9}{9 b^{11}} \]

[Out]

(45*d^2*(b*c - a*d)^8*x)/b^10 - (b*c - a*d)^10/(b^11*(a + b*x)) + (60*d^3*(b*c - a*d)^7*(a + b*x)^2)/b^11 + (7
0*d^4*(b*c - a*d)^6*(a + b*x)^3)/b^11 + (63*d^5*(b*c - a*d)^5*(a + b*x)^4)/b^11 + (42*d^6*(b*c - a*d)^4*(a + b
*x)^5)/b^11 + (20*d^7*(b*c - a*d)^3*(a + b*x)^6)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^7)/(7*b^11) + (5*d^9*(
b*c - a*d)*(a + b*x)^8)/(4*b^11) + (d^10*(a + b*x)^9)/(9*b^11) + (10*d*(b*c - a*d)^9*Log[a + b*x])/b^11

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Rubi [A]  time = 0.472624, 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} \[ \frac{5 d^9 (a+b x)^8 (b c-a d)}{4 b^{11}}+\frac{45 d^8 (a+b x)^7 (b c-a d)^2}{7 b^{11}}+\frac{20 d^7 (a+b x)^6 (b c-a d)^3}{b^{11}}+\frac{42 d^6 (a+b x)^5 (b c-a d)^4}{b^{11}}+\frac{63 d^5 (a+b x)^4 (b c-a d)^5}{b^{11}}+\frac{70 d^4 (a+b x)^3 (b c-a d)^6}{b^{11}}+\frac{60 d^3 (a+b x)^2 (b c-a d)^7}{b^{11}}+\frac{45 d^2 x (b c-a d)^8}{b^{10}}-\frac{(b c-a d)^{10}}{b^{11} (a+b x)}+\frac{10 d (b c-a d)^9 \log (a+b x)}{b^{11}}+\frac{d^{10} (a+b x)^9}{9 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^2,x]

[Out]

(45*d^2*(b*c - a*d)^8*x)/b^10 - (b*c - a*d)^10/(b^11*(a + b*x)) + (60*d^3*(b*c - a*d)^7*(a + b*x)^2)/b^11 + (7
0*d^4*(b*c - a*d)^6*(a + b*x)^3)/b^11 + (63*d^5*(b*c - a*d)^5*(a + b*x)^4)/b^11 + (42*d^6*(b*c - a*d)^4*(a + b
*x)^5)/b^11 + (20*d^7*(b*c - a*d)^3*(a + b*x)^6)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^7)/(7*b^11) + (5*d^9*(
b*c - a*d)*(a + b*x)^8)/(4*b^11) + (d^10*(a + b*x)^9)/(9*b^11) + (10*d*(b*c - a*d)^9*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)^2} \, dx &=\int \left (\frac{45 d^2 (b c-a d)^8}{b^{10}}+\frac{(b c-a d)^{10}}{b^{10} (a+b x)^2}+\frac{10 d (b c-a d)^9}{b^{10} (a+b x)}+\frac{120 d^3 (b c-a d)^7 (a+b x)}{b^{10}}+\frac{210 d^4 (b c-a d)^6 (a+b x)^2}{b^{10}}+\frac{252 d^5 (b c-a d)^5 (a+b x)^3}{b^{10}}+\frac{210 d^6 (b c-a d)^4 (a+b x)^4}{b^{10}}+\frac{120 d^7 (b c-a d)^3 (a+b x)^5}{b^{10}}+\frac{45 d^8 (b c-a d)^2 (a+b x)^6}{b^{10}}+\frac{10 d^9 (b c-a d) (a+b x)^7}{b^{10}}+\frac{d^{10} (a+b x)^8}{b^{10}}\right ) \, dx\\ &=\frac{45 d^2 (b c-a d)^8 x}{b^{10}}-\frac{(b c-a d)^{10}}{b^{11} (a+b x)}+\frac{60 d^3 (b c-a d)^7 (a+b x)^2}{b^{11}}+\frac{70 d^4 (b c-a d)^6 (a+b x)^3}{b^{11}}+\frac{63 d^5 (b c-a d)^5 (a+b x)^4}{b^{11}}+\frac{42 d^6 (b c-a d)^4 (a+b x)^5}{b^{11}}+\frac{20 d^7 (b c-a d)^3 (a+b x)^6}{b^{11}}+\frac{45 d^8 (b c-a d)^2 (a+b x)^7}{7 b^{11}}+\frac{5 d^9 (b c-a d) (a+b x)^8}{4 b^{11}}+\frac{d^{10} (a+b x)^9}{9 b^{11}}+\frac{10 d (b c-a d)^9 \log (a+b x)}{b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.232005, size = 708, normalized size = 2.74 \[ \frac{9 a^2 b^8 d^2 \left (11760 c^6 d^2 x^2+5880 c^5 d^3 x^3+2940 c^4 d^4 x^4+1176 c^3 d^5 x^5+336 c^2 d^6 x^6-6720 c^7 d x-1260 c^8+60 c d^7 x^7+5 d^8 x^8\right )-12 a^3 b^7 d^3 \left (13230 c^5 d^2 x^2+4410 c^4 d^3 x^3+1470 c^3 d^4 x^4+378 c^2 d^5 x^5-13230 c^6 d x-2520 c^7+63 c d^6 x^6+5 d^7 x^7\right )+42 a^4 b^6 d^4 \left (3780 c^4 d^2 x^2+840 c^3 d^3 x^3+180 c^2 d^4 x^4-6048 c^5 d x-1260 c^6+27 c d^5 x^5+2 d^6 x^6\right )-126 a^5 b^5 d^5 \left (840 c^3 d^2 x^2+120 c^2 d^3 x^3-2100 c^4 d x-504 c^5+15 c d^4 x^4+d^5 x^5\right )+210 a^6 b^4 d^6 \left (216 c^2 d^2 x^2-864 c^3 d x-252 c^4+18 c d^3 x^3+d^4 x^4\right )-420 a^7 b^3 d^7 \left (-189 c^2 d x-72 c^3+27 c d^2 x^2+d^3 x^3\right )+1260 a^8 b^2 d^8 \left (-9 c^2-16 c d x+d^2 x^2\right )+252 a^9 b d^9 (10 c+9 d x)-252 a^{10} d^{10}-a b^9 d \left (45360 c^7 d^2 x^2+35280 c^6 d^3 x^3+26460 c^5 d^4 x^4+15876 c^4 d^5 x^5+7056 c^3 d^6 x^6+2160 c^2 d^7 x^7-11340 c^8 d x-2520 c^9+405 c d^8 x^8+35 d^9 x^9\right )-2520 d (a+b x) (a d-b c)^9 \log (a+b x)+b^{10} \left (11340 c^8 d^2 x^2+15120 c^7 d^3 x^3+17640 c^6 d^4 x^4+15876 c^5 d^5 x^5+10584 c^4 d^6 x^6+5040 c^3 d^7 x^7+1620 c^2 d^8 x^8-252 c^{10}+315 c d^9 x^9+28 d^{10} x^{10}\right )}{252 b^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^2,x]

[Out]

(-252*a^10*d^10 + 252*a^9*b*d^9*(10*c + 9*d*x) + 1260*a^8*b^2*d^8*(-9*c^2 - 16*c*d*x + d^2*x^2) - 420*a^7*b^3*
d^7*(-72*c^3 - 189*c^2*d*x + 27*c*d^2*x^2 + d^3*x^3) + 210*a^6*b^4*d^6*(-252*c^4 - 864*c^3*d*x + 216*c^2*d^2*x
^2 + 18*c*d^3*x^3 + d^4*x^4) - 126*a^5*b^5*d^5*(-504*c^5 - 2100*c^4*d*x + 840*c^3*d^2*x^2 + 120*c^2*d^3*x^3 +
15*c*d^4*x^4 + d^5*x^5) + 42*a^4*b^6*d^4*(-1260*c^6 - 6048*c^5*d*x + 3780*c^4*d^2*x^2 + 840*c^3*d^3*x^3 + 180*
c^2*d^4*x^4 + 27*c*d^5*x^5 + 2*d^6*x^6) - 12*a^3*b^7*d^3*(-2520*c^7 - 13230*c^6*d*x + 13230*c^5*d^2*x^2 + 4410
*c^4*d^3*x^3 + 1470*c^3*d^4*x^4 + 378*c^2*d^5*x^5 + 63*c*d^6*x^6 + 5*d^7*x^7) + 9*a^2*b^8*d^2*(-1260*c^8 - 672
0*c^7*d*x + 11760*c^6*d^2*x^2 + 5880*c^5*d^3*x^3 + 2940*c^4*d^4*x^4 + 1176*c^3*d^5*x^5 + 336*c^2*d^6*x^6 + 60*
c*d^7*x^7 + 5*d^8*x^8) - a*b^9*d*(-2520*c^9 - 11340*c^8*d*x + 45360*c^7*d^2*x^2 + 35280*c^6*d^3*x^3 + 26460*c^
5*d^4*x^4 + 15876*c^4*d^5*x^5 + 7056*c^3*d^6*x^6 + 2160*c^2*d^7*x^7 + 405*c*d^8*x^8 + 35*d^9*x^9) + b^10*(-252
*c^10 + 11340*c^8*d^2*x^2 + 15120*c^7*d^3*x^3 + 17640*c^6*d^4*x^4 + 15876*c^5*d^5*x^5 + 10584*c^4*d^6*x^6 + 50
40*c^3*d^7*x^7 + 1620*c^2*d^8*x^8 + 315*c*d^9*x^9 + 28*d^10*x^10) - 2520*d*(-(b*c) + a*d)^9*(a + b*x)*Log[a +
b*x])/(252*b^11*(a + b*x))

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Maple [B]  time = 0.014, size = 1066, normalized size = 4.1 \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)^2,x)

[Out]

-1/b/(b*x+a)*c^10-1/b^11/(b*x+a)*a^10*d^10-10/b^11*d^10*ln(b*x+a)*a^9+10/b^2*d*ln(b*x+a)*c^9+9*d^10/b^10*a^8*x
+70*d^4/b^2*x^3*c^6-4*d^10/b^9*x^2*a^7+60*d^3/b^2*x^2*c^7+5/4*d^9/b^2*x^8*c+3/7*d^10/b^4*x^7*a^2+45/7*d^8/b^2*
x^7*c^2-2/3*d^10/b^5*x^6*a^3+20*d^7/b^2*x^6*c^3+42*d^6/b^2*x^5*c^4-3/2*d^10/b^7*x^4*a^5+63*d^5/b^2*x^4*c^5+7/3
*d^10/b^8*x^3*a^6-1/4*d^10/b^3*x^8*a+45*d^2/b^2*c^8*x+d^10/b^6*x^5*a^4-8*d^9/b^5*x^5*a^3*c-20/7*d^9/b^3*x^7*a*
c+5*d^9/b^4*x^6*a^2*c-15*d^8/b^3*x^6*a*c^2+10/b^10/(b*x+a)*a^9*c*d^9-45/b^9/(b*x+a)*a^8*c^2*d^8+120/b^8/(b*x+a
)*a^7*c^3*d^7-210/b^7/(b*x+a)*a^6*c^4*d^6+252/b^6/(b*x+a)*a^5*c^5*d^5-210/b^5/(b*x+a)*a^4*c^6*d^4+120/b^4/(b*x
+a)*a^3*c^7*d^3-45/b^3/(b*x+a)*a^2*c^8*d^2-168*d^5/b^3*x^3*a*c^5+35*d^9/b^8*x^2*a^6*c+315*d^8/b^8*a^6*c^2*x-72
0*d^7/b^7*a^5*c^3*x+1050*d^6/b^6*a^4*c^4*x-1008*d^5/b^5*a^3*c^5*x+630*d^4/b^4*a^2*c^6*x-240*d^3/b^3*a*c^7*x+21
0*d^6/b^4*x^3*a^2*c^4-20*d^9/b^7*x^3*a^5*c+75*d^8/b^6*x^3*a^4*c^2-160*d^7/b^5*x^3*a^3*c^3+90*d^7/b^4*x^4*a^2*c
^3-105*d^6/b^3*x^4*a*c^4-48*d^7/b^3*x^5*a*c^3+25/2*d^9/b^6*x^4*a^4*c-45*d^8/b^5*x^4*a^3*c^2+27*d^8/b^4*x^5*a^2
*c^2-360/b^9*d^8*ln(b*x+a)*a^7*c^2+840/b^8*d^7*ln(b*x+a)*a^6*c^3-1260/b^7*d^6*ln(b*x+a)*a^5*c^4+1260/b^6*d^5*l
n(b*x+a)*a^4*c^5-840/b^5*d^4*ln(b*x+a)*a^3*c^6+360/b^4*d^3*ln(b*x+a)*a^2*c^7-90/b^3*d^2*ln(b*x+a)*a*c^8-135*d^
8/b^7*x^2*a^5*c^2+300*d^7/b^6*x^2*a^4*c^3-420*d^6/b^5*x^2*a^3*c^4+378*d^5/b^4*x^2*a^2*c^5-210*d^4/b^3*x^2*a*c^
6-80*d^9/b^9*a^7*c*x+1/9*d^10/b^2*x^9+10/b^2/(b*x+a)*a*c^9*d+90/b^10*d^9*ln(b*x+a)*a^8*c

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Maxima [B]  time = 0.985015, size = 1180, normalized size = 4.57 \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)^2,x, algorithm="maxima")

[Out]

-(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)/(b^12*x +
a*b^11) + 1/252*(28*b^8*d^10*x^9 + 63*(5*b^8*c*d^9 - a*b^7*d^10)*x^8 + 36*(45*b^8*c^2*d^8 - 20*a*b^7*c*d^9 +
3*a^2*b^6*d^10)*x^7 + 84*(60*b^8*c^3*d^7 - 45*a*b^7*c^2*d^8 + 15*a^2*b^6*c*d^9 - 2*a^3*b^5*d^10)*x^6 + 252*(42
*b^8*c^4*d^6 - 48*a*b^7*c^3*d^7 + 27*a^2*b^6*c^2*d^8 - 8*a^3*b^5*c*d^9 + a^4*b^4*d^10)*x^5 + 126*(126*b^8*c^5*
d^5 - 210*a*b^7*c^4*d^6 + 180*a^2*b^6*c^3*d^7 - 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 - 3*a^5*b^3*d^10)*x^4 +
84*(210*b^8*c^6*d^4 - 504*a*b^7*c^5*d^5 + 630*a^2*b^6*c^4*d^6 - 480*a^3*b^5*c^3*d^7 + 225*a^4*b^4*c^2*d^8 - 60
*a^5*b^3*c*d^9 + 7*a^6*b^2*d^10)*x^3 + 252*(60*b^8*c^7*d^3 - 210*a*b^7*c^6*d^4 + 378*a^2*b^6*c^5*d^5 - 420*a^3
*b^5*c^4*d^6 + 300*a^4*b^4*c^3*d^7 - 135*a^5*b^3*c^2*d^8 + 35*a^6*b^2*c*d^9 - 4*a^7*b*d^10)*x^2 + 252*(45*b^8*
c^8*d^2 - 240*a*b^7*c^7*d^3 + 630*a^2*b^6*c^6*d^4 - 1008*a^3*b^5*c^5*d^5 + 1050*a^4*b^4*c^4*d^6 - 720*a^5*b^3*
c^3*d^7 + 315*a^6*b^2*c^2*d^8 - 80*a^7*b*c*d^9 + 9*a^8*d^10)*x)/b^10 + 10*(b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^
2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 126*a^4*b^5*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b
^2*c^2*d^8 + 9*a^8*b*c*d^9 - a^9*d^10)*log(b*x + a)/b^11

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Fricas [B]  time = 1.95732, size = 2402, normalized size = 9.31 \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)^2,x, algorithm="fricas")

[Out]

1/252*(28*b^10*d^10*x^10 - 252*b^10*c^10 + 2520*a*b^9*c^9*d - 11340*a^2*b^8*c^8*d^2 + 30240*a^3*b^7*c^7*d^3 -
52920*a^4*b^6*c^6*d^4 + 63504*a^5*b^5*c^5*d^5 - 52920*a^6*b^4*c^4*d^6 + 30240*a^7*b^3*c^3*d^7 - 11340*a^8*b^2*
c^2*d^8 + 2520*a^9*b*c*d^9 - 252*a^10*d^10 + 35*(9*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(36*b^10*c^2*d^8 - 9*a*b^
9*c*d^9 + a^2*b^8*d^10)*x^8 + 60*(84*b^10*c^3*d^7 - 36*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 8
4*(126*b^10*c^4*d^6 - 84*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 - 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 126*(126*b
^10*c^5*d^5 - 126*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 - 36*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^
5 + 210*(84*b^10*c^6*d^4 - 126*a*b^9*c^5*d^5 + 126*a^2*b^8*c^4*d^6 - 84*a^3*b^7*c^3*d^7 + 36*a^4*b^6*c^2*d^8 -
9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 420*(36*b^10*c^7*d^3 - 84*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 - 126*a^3
*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 - 36*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 1260*(9*b^10*c^
8*d^2 - 36*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 - 126*a^3*b^7*c^5*d^5 + 126*a^4*b^6*c^4*d^6 - 84*a^5*b^5*c^3*d^7
+ 36*a^6*b^4*c^2*d^8 - 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 252*(45*a*b^9*c^8*d^2 - 240*a^2*b^8*c^7*d^3 + 63
0*a^3*b^7*c^6*d^4 - 1008*a^4*b^6*c^5*d^5 + 1050*a^5*b^5*c^4*d^6 - 720*a^6*b^4*c^3*d^7 + 315*a^7*b^3*c^2*d^8 -
80*a^8*b^2*c*d^9 + 9*a^9*b*d^10)*x + 2520*(a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c
^6*d^4 + 126*a^5*b^5*c^5*d^5 - 126*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 - 36*a^8*b^2*c^2*d^8 + 9*a^9*b*c*d^9 -
a^10*d^10 + (b^10*c^9*d - 9*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 - 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 - 1
26*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d^7 - 36*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 - a^9*b*d^10)*x)*log(b*x + a))/
(b^12*x + a*b^11)

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Sympy [B]  time = 2.83658, size = 796, normalized size = 3.09 \begin{align*} - \frac{a^{10} d^{10} - 10 a^{9} b c d^{9} + 45 a^{8} b^{2} c^{2} d^{8} - 120 a^{7} b^{3} c^{3} d^{7} + 210 a^{6} b^{4} c^{4} d^{6} - 252 a^{5} b^{5} c^{5} d^{5} + 210 a^{4} b^{6} c^{6} d^{4} - 120 a^{3} b^{7} c^{7} d^{3} + 45 a^{2} b^{8} c^{8} d^{2} - 10 a b^{9} c^{9} d + b^{10} c^{10}}{a b^{11} + b^{12} x} + \frac{d^{10} x^{9}}{9 b^{2}} - \frac{x^{8} \left (a d^{10} - 5 b c d^{9}\right )}{4 b^{3}} + \frac{x^{7} \left (3 a^{2} d^{10} - 20 a b c d^{9} + 45 b^{2} c^{2} d^{8}\right )}{7 b^{4}} - \frac{x^{6} \left (2 a^{3} d^{10} - 15 a^{2} b c d^{9} + 45 a b^{2} c^{2} d^{8} - 60 b^{3} c^{3} d^{7}\right )}{3 b^{5}} + \frac{x^{5} \left (a^{4} d^{10} - 8 a^{3} b c d^{9} + 27 a^{2} b^{2} c^{2} d^{8} - 48 a b^{3} c^{3} d^{7} + 42 b^{4} c^{4} d^{6}\right )}{b^{6}} - \frac{x^{4} \left (3 a^{5} d^{10} - 25 a^{4} b c d^{9} + 90 a^{3} b^{2} c^{2} d^{8} - 180 a^{2} b^{3} c^{3} d^{7} + 210 a b^{4} c^{4} d^{6} - 126 b^{5} c^{5} d^{5}\right )}{2 b^{7}} + \frac{x^{3} \left (7 a^{6} d^{10} - 60 a^{5} b c d^{9} + 225 a^{4} b^{2} c^{2} d^{8} - 480 a^{3} b^{3} c^{3} d^{7} + 630 a^{2} b^{4} c^{4} d^{6} - 504 a b^{5} c^{5} d^{5} + 210 b^{6} c^{6} d^{4}\right )}{3 b^{8}} - \frac{x^{2} \left (4 a^{7} d^{10} - 35 a^{6} b c d^{9} + 135 a^{5} b^{2} c^{2} d^{8} - 300 a^{4} b^{3} c^{3} d^{7} + 420 a^{3} b^{4} c^{4} d^{6} - 378 a^{2} b^{5} c^{5} d^{5} + 210 a b^{6} c^{6} d^{4} - 60 b^{7} c^{7} d^{3}\right )}{b^{9}} + \frac{x \left (9 a^{8} d^{10} - 80 a^{7} b c d^{9} + 315 a^{6} b^{2} c^{2} d^{8} - 720 a^{5} b^{3} c^{3} d^{7} + 1050 a^{4} b^{4} c^{4} d^{6} - 1008 a^{3} b^{5} c^{5} d^{5} + 630 a^{2} b^{6} c^{6} d^{4} - 240 a b^{7} c^{7} d^{3} + 45 b^{8} c^{8} d^{2}\right )}{b^{10}} - \frac{10 d \left (a d - b c\right )^{9} \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)**2,x)

[Out]

-(a**10*d**10 - 10*a**9*b*c*d**9 + 45*a**8*b**2*c**2*d**8 - 120*a**7*b**3*c**3*d**7 + 210*a**6*b**4*c**4*d**6
- 252*a**5*b**5*c**5*d**5 + 210*a**4*b**6*c**6*d**4 - 120*a**3*b**7*c**7*d**3 + 45*a**2*b**8*c**8*d**2 - 10*a*
b**9*c**9*d + b**10*c**10)/(a*b**11 + b**12*x) + d**10*x**9/(9*b**2) - x**8*(a*d**10 - 5*b*c*d**9)/(4*b**3) +
x**7*(3*a**2*d**10 - 20*a*b*c*d**9 + 45*b**2*c**2*d**8)/(7*b**4) - x**6*(2*a**3*d**10 - 15*a**2*b*c*d**9 + 45*
a*b**2*c**2*d**8 - 60*b**3*c**3*d**7)/(3*b**5) + x**5*(a**4*d**10 - 8*a**3*b*c*d**9 + 27*a**2*b**2*c**2*d**8 -
48*a*b**3*c**3*d**7 + 42*b**4*c**4*d**6)/b**6 - x**4*(3*a**5*d**10 - 25*a**4*b*c*d**9 + 90*a**3*b**2*c**2*d**
8 - 180*a**2*b**3*c**3*d**7 + 210*a*b**4*c**4*d**6 - 126*b**5*c**5*d**5)/(2*b**7) + x**3*(7*a**6*d**10 - 60*a*
*5*b*c*d**9 + 225*a**4*b**2*c**2*d**8 - 480*a**3*b**3*c**3*d**7 + 630*a**2*b**4*c**4*d**6 - 504*a*b**5*c**5*d*
*5 + 210*b**6*c**6*d**4)/(3*b**8) - x**2*(4*a**7*d**10 - 35*a**6*b*c*d**9 + 135*a**5*b**2*c**2*d**8 - 300*a**4
*b**3*c**3*d**7 + 420*a**3*b**4*c**4*d**6 - 378*a**2*b**5*c**5*d**5 + 210*a*b**6*c**6*d**4 - 60*b**7*c**7*d**3
)/b**9 + x*(9*a**8*d**10 - 80*a**7*b*c*d**9 + 315*a**6*b**2*c**2*d**8 - 720*a**5*b**3*c**3*d**7 + 1050*a**4*b*
*4*c**4*d**6 - 1008*a**3*b**5*c**5*d**5 + 630*a**2*b**6*c**6*d**4 - 240*a*b**7*c**7*d**3 + 45*b**8*c**8*d**2)/
b**10 - 10*d*(a*d - b*c)**9*log(a + b*x)/b**11

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Giac [B]  time = 1.08716, size = 1366, normalized size = 5.29 \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)^2,x, algorithm="giac")

[Out]

1/252*(28*d^10 + 315*(b^2*c*d^9 - a*b*d^10)/((b*x + a)*b) + 1620*(b^4*c^2*d^8 - 2*a*b^3*c*d^9 + a^2*b^2*d^10)/
((b*x + a)^2*b^2) + 5040*(b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)/((b*x + a)^3*b^3) +
10584*(b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)/((b*x + a)^4*b^4) +
15876*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d
^10)/((b*x + a)^5*b^5) + 17640*(b^12*c^6*d^4 - 6*a*b^11*c^5*d^5 + 15*a^2*b^10*c^4*d^6 - 20*a^3*b^9*c^3*d^7 + 1
5*a^4*b^8*c^2*d^8 - 6*a^5*b^7*c*d^9 + a^6*b^6*d^10)/((b*x + a)^6*b^6) + 15120*(b^14*c^7*d^3 - 7*a*b^13*c^6*d^4
+ 21*a^2*b^12*c^5*d^5 - 35*a^3*b^11*c^4*d^6 + 35*a^4*b^10*c^3*d^7 - 21*a^5*b^9*c^2*d^8 + 7*a^6*b^8*c*d^9 - a^
7*b^7*d^10)/((b*x + a)^7*b^7) + 11340*(b^16*c^8*d^2 - 8*a*b^15*c^7*d^3 + 28*a^2*b^14*c^6*d^4 - 56*a^3*b^13*c^5
*d^5 + 70*a^4*b^12*c^4*d^6 - 56*a^5*b^11*c^3*d^7 + 28*a^6*b^10*c^2*d^8 - 8*a^7*b^9*c*d^9 + a^8*b^8*d^10)/((b*x
+ a)^8*b^8))*(b*x + a)^9/b^11 - 10*(b^9*c^9*d - 9*a*b^8*c^8*d^2 + 36*a^2*b^7*c^7*d^3 - 84*a^3*b^6*c^6*d^4 + 1
26*a^4*b^5*c^5*d^5 - 126*a^5*b^4*c^4*d^6 + 84*a^6*b^3*c^3*d^7 - 36*a^7*b^2*c^2*d^8 + 9*a^8*b*c*d^9 - a^9*d^10)
*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^11 - (b^19*c^10/(b*x + a) - 10*a*b^18*c^9*d/(b*x + a) + 45*a^2*b^17*
c^8*d^2/(b*x + a) - 120*a^3*b^16*c^7*d^3/(b*x + a) + 210*a^4*b^15*c^6*d^4/(b*x + a) - 252*a^5*b^14*c^5*d^5/(b*
x + a) + 210*a^6*b^13*c^4*d^6/(b*x + a) - 120*a^7*b^12*c^3*d^7/(b*x + a) + 45*a^8*b^11*c^2*d^8/(b*x + a) - 10*
a^9*b^10*c*d^9/(b*x + a) + a^10*b^9*d^10/(b*x + a))/b^20