3.1330 \(\int \frac{(c+d x)^{10}}{(a+b x)^{19}} \, dx\)

Optimal. Leaf size=244 \[ \frac{d^7 (c+d x)^{11}}{350064 (a+b x)^{11} (b c-a d)^8}-\frac{d^6 (c+d x)^{11}}{31824 (a+b x)^{12} (b c-a d)^7}+\frac{d^5 (c+d x)^{11}}{5304 (a+b x)^{13} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{1224 (a+b x)^{14} (b c-a d)^5}+\frac{7 d^3 (c+d x)^{11}}{2448 (a+b x)^{15} (b c-a d)^4}-\frac{7 d^2 (c+d x)^{11}}{816 (a+b x)^{16} (b c-a d)^3}+\frac{7 d (c+d x)^{11}}{306 (a+b x)^{17} (b c-a d)^2}-\frac{(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)} \]

[Out]

-(c + d*x)^11/(18*(b*c - a*d)*(a + b*x)^18) + (7*d*(c + d*x)^11)/(306*(b*c - a*d)^2*(a + b*x)^17) - (7*d^2*(c
+ d*x)^11)/(816*(b*c - a*d)^3*(a + b*x)^16) + (7*d^3*(c + d*x)^11)/(2448*(b*c - a*d)^4*(a + b*x)^15) - (d^4*(c
+ d*x)^11)/(1224*(b*c - a*d)^5*(a + b*x)^14) + (d^5*(c + d*x)^11)/(5304*(b*c - a*d)^6*(a + b*x)^13) - (d^6*(c
+ d*x)^11)/(31824*(b*c - a*d)^7*(a + b*x)^12) + (d^7*(c + d*x)^11)/(350064*(b*c - a*d)^8*(a + b*x)^11)

________________________________________________________________________________________

Rubi [A]  time = 0.104976, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {45, 37} \[ \frac{d^7 (c+d x)^{11}}{350064 (a+b x)^{11} (b c-a d)^8}-\frac{d^6 (c+d x)^{11}}{31824 (a+b x)^{12} (b c-a d)^7}+\frac{d^5 (c+d x)^{11}}{5304 (a+b x)^{13} (b c-a d)^6}-\frac{d^4 (c+d x)^{11}}{1224 (a+b x)^{14} (b c-a d)^5}+\frac{7 d^3 (c+d x)^{11}}{2448 (a+b x)^{15} (b c-a d)^4}-\frac{7 d^2 (c+d x)^{11}}{816 (a+b x)^{16} (b c-a d)^3}+\frac{7 d (c+d x)^{11}}{306 (a+b x)^{17} (b c-a d)^2}-\frac{(c+d x)^{11}}{18 (a+b x)^{18} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^19,x]

[Out]

-(c + d*x)^11/(18*(b*c - a*d)*(a + b*x)^18) + (7*d*(c + d*x)^11)/(306*(b*c - a*d)^2*(a + b*x)^17) - (7*d^2*(c
+ d*x)^11)/(816*(b*c - a*d)^3*(a + b*x)^16) + (7*d^3*(c + d*x)^11)/(2448*(b*c - a*d)^4*(a + b*x)^15) - (d^4*(c
+ d*x)^11)/(1224*(b*c - a*d)^5*(a + b*x)^14) + (d^5*(c + d*x)^11)/(5304*(b*c - a*d)^6*(a + b*x)^13) - (d^6*(c
+ d*x)^11)/(31824*(b*c - a*d)^7*(a + b*x)^12) + (d^7*(c + d*x)^11)/(350064*(b*c - a*d)^8*(a + b*x)^11)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^{10}}{(a+b x)^{19}} \, dx &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}-\frac{(7 d) \int \frac{(c+d x)^{10}}{(a+b x)^{18}} \, dx}{18 (b c-a d)}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}+\frac{\left (7 d^2\right ) \int \frac{(c+d x)^{10}}{(a+b x)^{17}} \, dx}{51 (b c-a d)^2}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}-\frac{\left (35 d^3\right ) \int \frac{(c+d x)^{10}}{(a+b x)^{16}} \, dx}{816 (b c-a d)^3}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac{7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}+\frac{\left (7 d^4\right ) \int \frac{(c+d x)^{10}}{(a+b x)^{15}} \, dx}{612 (b c-a d)^4}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac{7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}-\frac{d^4 (c+d x)^{11}}{1224 (b c-a d)^5 (a+b x)^{14}}-\frac{d^5 \int \frac{(c+d x)^{10}}{(a+b x)^{14}} \, dx}{408 (b c-a d)^5}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac{7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}-\frac{d^4 (c+d x)^{11}}{1224 (b c-a d)^5 (a+b x)^{14}}+\frac{d^5 (c+d x)^{11}}{5304 (b c-a d)^6 (a+b x)^{13}}+\frac{d^6 \int \frac{(c+d x)^{10}}{(a+b x)^{13}} \, dx}{2652 (b c-a d)^6}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac{7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}-\frac{d^4 (c+d x)^{11}}{1224 (b c-a d)^5 (a+b x)^{14}}+\frac{d^5 (c+d x)^{11}}{5304 (b c-a d)^6 (a+b x)^{13}}-\frac{d^6 (c+d x)^{11}}{31824 (b c-a d)^7 (a+b x)^{12}}-\frac{d^7 \int \frac{(c+d x)^{10}}{(a+b x)^{12}} \, dx}{31824 (b c-a d)^7}\\ &=-\frac{(c+d x)^{11}}{18 (b c-a d) (a+b x)^{18}}+\frac{7 d (c+d x)^{11}}{306 (b c-a d)^2 (a+b x)^{17}}-\frac{7 d^2 (c+d x)^{11}}{816 (b c-a d)^3 (a+b x)^{16}}+\frac{7 d^3 (c+d x)^{11}}{2448 (b c-a d)^4 (a+b x)^{15}}-\frac{d^4 (c+d x)^{11}}{1224 (b c-a d)^5 (a+b x)^{14}}+\frac{d^5 (c+d x)^{11}}{5304 (b c-a d)^6 (a+b x)^{13}}-\frac{d^6 (c+d x)^{11}}{31824 (b c-a d)^7 (a+b x)^{12}}+\frac{d^7 (c+d x)^{11}}{350064 (b c-a d)^8 (a+b x)^{11}}\\ \end{align*}

Mathematica [B]  time = 0.287515, size = 694, normalized size = 2.84 \[ -\frac{9 a^2 b^8 d^2 \left (29172 c^6 d^2 x^2+71808 c^5 d^3 x^3+112200 c^4 d^4 x^4+114240 c^3 d^5 x^5+74256 c^2 d^6 x^6+6864 c^7 d x+715 c^8+28288 c d^7 x^7+4862 d^8 x^8\right )+24 a^3 b^7 d^3 \left (5049 c^5 d^2 x^2+11220 c^4 d^3 x^3+15300 c^3 d^4 x^4+12852 c^2 d^5 x^5+1287 c^6 d x+143 c^7+6188 c d^6 x^6+1326 d^7 x^7\right )+6 a^4 b^6 d^4 \left (8415 c^4 d^2 x^2+16320 c^3 d^3 x^3+18360 c^2 d^4 x^4+2376 c^5 d x+286 c^6+11424 c d^5 x^5+3094 d^6 x^6\right )+36 a^5 b^5 d^5 \left (510 c^3 d^2 x^2+816 c^2 d^3 x^3+165 c^4 d x+22 c^5+680 c d^4 x^4+238 d^5 x^5\right )+6 a^6 b^4 d^6 \left (918 c^2 d^2 x^2+360 c^3 d x+55 c^4+1088 c d^3 x^3+510 d^4 x^4\right )+24 a^7 b^3 d^7 \left (27 c^2 d x+5 c^3+51 c d^2 x^2+34 d^3 x^3\right )+9 a^8 b^2 d^8 \left (4 c^2+16 c d x+17 d^2 x^2\right )+2 a^9 b d^9 (4 c+9 d x)+a^{10} d^{10}+2 a b^9 d \left (262548 c^7 d^2 x^2+700128 c^6 d^3 x^3+1211760 c^5 d^4 x^4+1413720 c^4 d^5 x^5+1113840 c^3 d^6 x^6+572832 c^2 d^7 x^7+57915 c^8 d x+5720 c^9+175032 c d^8 x^8+24310 d^9 x^9\right )+b^{10} \left (984555 c^8 d^2 x^2+2800512 c^7 d^3 x^3+5250960 c^6 d^4 x^4+6785856 c^5 d^5 x^5+6126120 c^4 d^6 x^6+3818880 c^3 d^7 x^7+1575288 c^2 d^8 x^8+205920 c^9 d x+19448 c^{10}+388960 c d^9 x^9+43758 d^{10} x^{10}\right )}{350064 b^{11} (a+b x)^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^19,x]

[Out]

-(a^10*d^10 + 2*a^9*b*d^9*(4*c + 9*d*x) + 9*a^8*b^2*d^8*(4*c^2 + 16*c*d*x + 17*d^2*x^2) + 24*a^7*b^3*d^7*(5*c^
3 + 27*c^2*d*x + 51*c*d^2*x^2 + 34*d^3*x^3) + 6*a^6*b^4*d^6*(55*c^4 + 360*c^3*d*x + 918*c^2*d^2*x^2 + 1088*c*d
^3*x^3 + 510*d^4*x^4) + 36*a^5*b^5*d^5*(22*c^5 + 165*c^4*d*x + 510*c^3*d^2*x^2 + 816*c^2*d^3*x^3 + 680*c*d^4*x
^4 + 238*d^5*x^5) + 6*a^4*b^6*d^4*(286*c^6 + 2376*c^5*d*x + 8415*c^4*d^2*x^2 + 16320*c^3*d^3*x^3 + 18360*c^2*d
^4*x^4 + 11424*c*d^5*x^5 + 3094*d^6*x^6) + 24*a^3*b^7*d^3*(143*c^7 + 1287*c^6*d*x + 5049*c^5*d^2*x^2 + 11220*c
^4*d^3*x^3 + 15300*c^3*d^4*x^4 + 12852*c^2*d^5*x^5 + 6188*c*d^6*x^6 + 1326*d^7*x^7) + 9*a^2*b^8*d^2*(715*c^8 +
6864*c^7*d*x + 29172*c^6*d^2*x^2 + 71808*c^5*d^3*x^3 + 112200*c^4*d^4*x^4 + 114240*c^3*d^5*x^5 + 74256*c^2*d^
6*x^6 + 28288*c*d^7*x^7 + 4862*d^8*x^8) + 2*a*b^9*d*(5720*c^9 + 57915*c^8*d*x + 262548*c^7*d^2*x^2 + 700128*c^
6*d^3*x^3 + 1211760*c^5*d^4*x^4 + 1413720*c^4*d^5*x^5 + 1113840*c^3*d^6*x^6 + 572832*c^2*d^7*x^7 + 175032*c*d^
8*x^8 + 24310*d^9*x^9) + b^10*(19448*c^10 + 205920*c^9*d*x + 984555*c^8*d^2*x^2 + 2800512*c^7*d^3*x^3 + 525096
0*c^6*d^4*x^4 + 6785856*c^5*d^5*x^5 + 6126120*c^4*d^6*x^6 + 3818880*c^3*d^7*x^7 + 1575288*c^2*d^8*x^8 + 388960
*c*d^9*x^9 + 43758*d^10*x^10))/(350064*b^11*(a + b*x)^18)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 867, normalized size = 3.6 \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)^19,x)

[Out]

-9/2*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^10-1/8*d^10/b^11/(b*x+a)^8-15*d^4*(a^6*d^6-6*a^5*b*c*d^5+15*
a^4*b^2*c^2*d^4-20*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2-6*a*b^5*c^5*d+b^6*c^6)/b^11/(b*x+a)^14-45/16*d^2*(a^8*d^
8-8*a^7*b*c*d^7+28*a^6*b^2*c^2*d^6-56*a^5*b^3*c^3*d^5+70*a^4*b^4*c^4*d^4-56*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2
-8*a*b^7*c^7*d+b^8*c^8)/b^11/(b*x+a)^16+120/11*d^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^11/(b*x+a)^
11+10/17*d*(a^9*d^9-9*a^8*b*c*d^8+36*a^7*b^2*c^2*d^7-84*a^6*b^3*c^3*d^6+126*a^5*b^4*c^4*d^5-126*a^4*b^5*c^5*d^
4+84*a^3*b^6*c^6*d^3-36*a^2*b^7*c^7*d^2+9*a*b^8*c^8*d-b^9*c^9)/b^11/(b*x+a)^17+252/13*d^5*(a^5*d^5-5*a^4*b*c*d
^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/b^11/(b*x+a)^13+8*d^3*(a^7*d^7-7*a^6*b*c*d^6+2
1*a^5*b^2*c^2*d^5-35*a^4*b^3*c^3*d^4+35*a^3*b^4*c^4*d^3-21*a^2*b^5*c^5*d^2+7*a*b^6*c^6*d-b^7*c^7)/b^11/(b*x+a)
^15-1/18*(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)/b^11/(b*x+a)^18-35/2*
d^6*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/b^11/(b*x+a)^12+10/9*d^9*(a*d-b*c)/b^11/(b
*x+a)^9

________________________________________________________________________________________

Maxima [B]  time = 1.31229, size = 1420, normalized size = 5.82 \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)^19,x, algorithm="maxima")

[Out]

-1/350064*(43758*b^10*d^10*x^10 + 19448*b^10*c^10 + 11440*a*b^9*c^9*d + 6435*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^
7*d^3 + 1716*a^4*b^6*c^6*d^4 + 792*a^5*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7*b^3*c^3*d^7 + 36*a^8*b^2*c^
2*d^8 + 8*a^9*b*c*d^9 + a^10*d^10 + 48620*(8*b^10*c*d^9 + a*b^9*d^10)*x^9 + 43758*(36*b^10*c^2*d^8 + 8*a*b^9*c
*d^9 + a^2*b^8*d^10)*x^8 + 31824*(120*b^10*c^3*d^7 + 36*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 +
18564*(330*b^10*c^4*d^6 + 120*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 + 8*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 8568*
(792*b^10*c^5*d^5 + 330*a*b^9*c^4*d^6 + 120*a^2*b^8*c^3*d^7 + 36*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 + a^5*b^5*d
^10)*x^5 + 3060*(1716*b^10*c^6*d^4 + 792*a*b^9*c^5*d^5 + 330*a^2*b^8*c^4*d^6 + 120*a^3*b^7*c^3*d^7 + 36*a^4*b^
6*c^2*d^8 + 8*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 816*(3432*b^10*c^7*d^3 + 1716*a*b^9*c^6*d^4 + 792*a^2*b^8*c^
5*d^5 + 330*a^3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 + 36*a^5*b^5*c^2*d^8 + 8*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 +
153*(6435*b^10*c^8*d^2 + 3432*a*b^9*c^7*d^3 + 1716*a^2*b^8*c^6*d^4 + 792*a^3*b^7*c^5*d^5 + 330*a^4*b^6*c^4*d^
6 + 120*a^5*b^5*c^3*d^7 + 36*a^6*b^4*c^2*d^8 + 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 18*(11440*b^10*c^9*d + 64
35*a*b^9*c^8*d^2 + 3432*a^2*b^8*c^7*d^3 + 1716*a^3*b^7*c^6*d^4 + 792*a^4*b^6*c^5*d^5 + 330*a^5*b^5*c^4*d^6 + 1
20*a^6*b^4*c^3*d^7 + 36*a^7*b^3*c^2*d^8 + 8*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^29*x^18 + 18*a*b^28*x^17 + 153*a
^2*b^27*x^16 + 816*a^3*b^26*x^15 + 3060*a^4*b^25*x^14 + 8568*a^5*b^24*x^13 + 18564*a^6*b^23*x^12 + 31824*a^7*b
^22*x^11 + 43758*a^8*b^21*x^10 + 48620*a^9*b^20*x^9 + 43758*a^10*b^19*x^8 + 31824*a^11*b^18*x^7 + 18564*a^12*b
^17*x^6 + 8568*a^13*b^16*x^5 + 3060*a^14*b^15*x^4 + 816*a^15*b^14*x^3 + 153*a^16*b^13*x^2 + 18*a^17*b^12*x + a
^18*b^11)

________________________________________________________________________________________

Fricas [B]  time = 1.94583, size = 2391, normalized size = 9.8 \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)^19,x, algorithm="fricas")

[Out]

-1/350064*(43758*b^10*d^10*x^10 + 19448*b^10*c^10 + 11440*a*b^9*c^9*d + 6435*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^
7*d^3 + 1716*a^4*b^6*c^6*d^4 + 792*a^5*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7*b^3*c^3*d^7 + 36*a^8*b^2*c^
2*d^8 + 8*a^9*b*c*d^9 + a^10*d^10 + 48620*(8*b^10*c*d^9 + a*b^9*d^10)*x^9 + 43758*(36*b^10*c^2*d^8 + 8*a*b^9*c
*d^9 + a^2*b^8*d^10)*x^8 + 31824*(120*b^10*c^3*d^7 + 36*a*b^9*c^2*d^8 + 8*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 +
18564*(330*b^10*c^4*d^6 + 120*a*b^9*c^3*d^7 + 36*a^2*b^8*c^2*d^8 + 8*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 8568*
(792*b^10*c^5*d^5 + 330*a*b^9*c^4*d^6 + 120*a^2*b^8*c^3*d^7 + 36*a^3*b^7*c^2*d^8 + 8*a^4*b^6*c*d^9 + a^5*b^5*d
^10)*x^5 + 3060*(1716*b^10*c^6*d^4 + 792*a*b^9*c^5*d^5 + 330*a^2*b^8*c^4*d^6 + 120*a^3*b^7*c^3*d^7 + 36*a^4*b^
6*c^2*d^8 + 8*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 816*(3432*b^10*c^7*d^3 + 1716*a*b^9*c^6*d^4 + 792*a^2*b^8*c^
5*d^5 + 330*a^3*b^7*c^4*d^6 + 120*a^4*b^6*c^3*d^7 + 36*a^5*b^5*c^2*d^8 + 8*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 +
153*(6435*b^10*c^8*d^2 + 3432*a*b^9*c^7*d^3 + 1716*a^2*b^8*c^6*d^4 + 792*a^3*b^7*c^5*d^5 + 330*a^4*b^6*c^4*d^
6 + 120*a^5*b^5*c^3*d^7 + 36*a^6*b^4*c^2*d^8 + 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 18*(11440*b^10*c^9*d + 64
35*a*b^9*c^8*d^2 + 3432*a^2*b^8*c^7*d^3 + 1716*a^3*b^7*c^6*d^4 + 792*a^4*b^6*c^5*d^5 + 330*a^5*b^5*c^4*d^6 + 1
20*a^6*b^4*c^3*d^7 + 36*a^7*b^3*c^2*d^8 + 8*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^29*x^18 + 18*a*b^28*x^17 + 153*a
^2*b^27*x^16 + 816*a^3*b^26*x^15 + 3060*a^4*b^25*x^14 + 8568*a^5*b^24*x^13 + 18564*a^6*b^23*x^12 + 31824*a^7*b
^22*x^11 + 43758*a^8*b^21*x^10 + 48620*a^9*b^20*x^9 + 43758*a^10*b^19*x^8 + 31824*a^11*b^18*x^7 + 18564*a^12*b
^17*x^6 + 8568*a^13*b^16*x^5 + 3060*a^14*b^15*x^4 + 816*a^15*b^14*x^3 + 153*a^16*b^13*x^2 + 18*a^17*b^12*x + a
^18*b^11)

________________________________________________________________________________________

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)**19,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.05882, size = 1297, normalized size = 5.32 \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)^19,x, algorithm="giac")

[Out]

-1/350064*(43758*b^10*d^10*x^10 + 388960*b^10*c*d^9*x^9 + 48620*a*b^9*d^10*x^9 + 1575288*b^10*c^2*d^8*x^8 + 35
0064*a*b^9*c*d^9*x^8 + 43758*a^2*b^8*d^10*x^8 + 3818880*b^10*c^3*d^7*x^7 + 1145664*a*b^9*c^2*d^8*x^7 + 254592*
a^2*b^8*c*d^9*x^7 + 31824*a^3*b^7*d^10*x^7 + 6126120*b^10*c^4*d^6*x^6 + 2227680*a*b^9*c^3*d^7*x^6 + 668304*a^2
*b^8*c^2*d^8*x^6 + 148512*a^3*b^7*c*d^9*x^6 + 18564*a^4*b^6*d^10*x^6 + 6785856*b^10*c^5*d^5*x^5 + 2827440*a*b^
9*c^4*d^6*x^5 + 1028160*a^2*b^8*c^3*d^7*x^5 + 308448*a^3*b^7*c^2*d^8*x^5 + 68544*a^4*b^6*c*d^9*x^5 + 8568*a^5*
b^5*d^10*x^5 + 5250960*b^10*c^6*d^4*x^4 + 2423520*a*b^9*c^5*d^5*x^4 + 1009800*a^2*b^8*c^4*d^6*x^4 + 367200*a^3
*b^7*c^3*d^7*x^4 + 110160*a^4*b^6*c^2*d^8*x^4 + 24480*a^5*b^5*c*d^9*x^4 + 3060*a^6*b^4*d^10*x^4 + 2800512*b^10
*c^7*d^3*x^3 + 1400256*a*b^9*c^6*d^4*x^3 + 646272*a^2*b^8*c^5*d^5*x^3 + 269280*a^3*b^7*c^4*d^6*x^3 + 97920*a^4
*b^6*c^3*d^7*x^3 + 29376*a^5*b^5*c^2*d^8*x^3 + 6528*a^6*b^4*c*d^9*x^3 + 816*a^7*b^3*d^10*x^3 + 984555*b^10*c^8
*d^2*x^2 + 525096*a*b^9*c^7*d^3*x^2 + 262548*a^2*b^8*c^6*d^4*x^2 + 121176*a^3*b^7*c^5*d^5*x^2 + 50490*a^4*b^6*
c^4*d^6*x^2 + 18360*a^5*b^5*c^3*d^7*x^2 + 5508*a^6*b^4*c^2*d^8*x^2 + 1224*a^7*b^3*c*d^9*x^2 + 153*a^8*b^2*d^10
*x^2 + 205920*b^10*c^9*d*x + 115830*a*b^9*c^8*d^2*x + 61776*a^2*b^8*c^7*d^3*x + 30888*a^3*b^7*c^6*d^4*x + 1425
6*a^4*b^6*c^5*d^5*x + 5940*a^5*b^5*c^4*d^6*x + 2160*a^6*b^4*c^3*d^7*x + 648*a^7*b^3*c^2*d^8*x + 144*a^8*b^2*c*
d^9*x + 18*a^9*b*d^10*x + 19448*b^10*c^10 + 11440*a*b^9*c^9*d + 6435*a^2*b^8*c^8*d^2 + 3432*a^3*b^7*c^7*d^3 +
1716*a^4*b^6*c^6*d^4 + 792*a^5*b^5*c^5*d^5 + 330*a^6*b^4*c^4*d^6 + 120*a^7*b^3*c^3*d^7 + 36*a^8*b^2*c^2*d^8 +
8*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^18*b^11)