3.14.30 \(\int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx\) [1330]

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

[Out]

-1/18*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^18+7/306*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x+a)^17-7/816*d^2*(d*x+c)^11/(-a*d+b
*c)^3/(b*x+a)^16+7/2448*d^3*(d*x+c)^11/(-a*d+b*c)^4/(b*x+a)^15-1/1224*d^4*(d*x+c)^11/(-a*d+b*c)^5/(b*x+a)^14+1
/5304*d^5*(d*x+c)^11/(-a*d+b*c)^6/(b*x+a)^13-1/31824*d^6*(d*x+c)^11/(-a*d+b*c)^7/(b*x+a)^12+1/350064*d^7*(d*x+
c)^11/(-a*d+b*c)^8/(b*x+a)^11

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Rubi [A]
time = 0.07, antiderivative size = 244, normalized size of antiderivative = 1.00, 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 = {47, 37} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^19,x]

[Out]

-1/18*(c + d*x)^11/((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 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]

Rule 47

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])

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(694\) vs. \(2(244)=488\).
time = 0.17, size = 694, normalized size = 2.84 \begin {gather*} -\frac {a^{10} d^{10}+2 a^9 b d^9 (4 c+9 d x)+9 a^8 b^2 d^8 \left (4 c^2+16 c d x+17 d^2 x^2\right )+24 a^7 b^3 d^7 \left (5 c^3+27 c^2 d x+51 c d^2 x^2+34 d^3 x^3\right )+6 a^6 b^4 d^6 \left (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\right )+36 a^5 b^5 d^5 \left (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\right )+6 a^4 b^6 d^4 \left (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\right )+24 a^3 b^7 d^3 \left (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\right )+9 a^2 b^8 d^2 \left (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\right )+2 a b^9 d \left (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\right )+b^{10} \left (19448 c^{10}+205920 c^9 d x+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+388960 c d^9 x^9+43758 d^{10} x^{10}\right )}{350064 b^{11} (a+b x)^{18}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/350064*(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 + 68
0*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 + 18
360*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 + 742
56*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 + 17
5032*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
+ 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
+ 388960*c*d^9*x^9 + 43758*d^10*x^10))/(b^11*(a + b*x)^18)

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs. \(2(228)=456\).
time = 0.16, size = 867, normalized size = 3.55

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+8 a^{9} b c \,d^{9}+36 a^{8} b^{2} c^{2} d^{8}+120 a^{7} b^{3} c^{3} d^{7}+330 a^{6} b^{4} c^{4} d^{6}+792 a^{5} b^{5} c^{5} d^{5}+1716 a^{4} b^{6} c^{6} d^{4}+3432 a^{3} b^{7} c^{7} d^{3}+6435 a^{2} b^{8} c^{8} d^{2}+11440 a \,b^{9} c^{9} d +19448 b^{10} c^{10}}{350064 b^{11}}-\frac {d \left (a^{9} d^{9}+8 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}+120 a^{6} b^{3} c^{3} d^{6}+330 a^{5} b^{4} c^{4} d^{5}+792 a^{4} b^{5} c^{5} d^{4}+1716 a^{3} b^{6} c^{6} d^{3}+3432 a^{2} b^{7} c^{7} d^{2}+6435 a \,b^{8} c^{8} d +11440 b^{9} c^{9}\right ) x}{19448 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+8 a^{7} b c \,d^{7}+36 a^{6} b^{2} c^{2} d^{6}+120 a^{5} b^{3} c^{3} d^{5}+330 a^{4} b^{4} c^{4} d^{4}+792 a^{3} b^{5} c^{5} d^{3}+1716 a^{2} b^{6} c^{6} d^{2}+3432 a \,b^{7} c^{7} d +6435 b^{8} c^{8}\right ) x^{2}}{2288 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+8 a^{6} b c \,d^{6}+36 a^{5} b^{2} c^{2} d^{5}+120 a^{4} b^{3} c^{3} d^{4}+330 a^{3} b^{4} c^{4} d^{3}+792 a^{2} b^{5} c^{5} d^{2}+1716 a \,b^{6} c^{6} d +3432 b^{7} c^{7}\right ) x^{3}}{429 b^{8}}-\frac {5 d^{4} \left (a^{6} d^{6}+8 a^{5} b c \,d^{5}+36 a^{4} b^{2} c^{2} d^{4}+120 a^{3} b^{3} c^{3} d^{3}+330 a^{2} b^{4} c^{4} d^{2}+792 a \,b^{5} c^{5} d +1716 b^{6} c^{6}\right ) x^{4}}{572 b^{7}}-\frac {7 d^{5} \left (a^{5} d^{5}+8 a^{4} b c \,d^{4}+36 a^{3} b^{2} c^{2} d^{3}+120 a^{2} b^{3} c^{3} d^{2}+330 a \,b^{4} c^{4} d +792 b^{5} c^{5}\right ) x^{5}}{286 b^{6}}-\frac {7 d^{6} \left (a^{4} d^{4}+8 a^{3} b c \,d^{3}+36 a^{2} b^{2} c^{2} d^{2}+120 a \,b^{3} c^{3} d +330 b^{4} c^{4}\right ) x^{6}}{132 b^{5}}-\frac {d^{7} \left (a^{3} d^{3}+8 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d +120 b^{3} c^{3}\right ) x^{7}}{11 b^{4}}-\frac {d^{8} \left (a^{2} d^{2}+8 a b c d +36 b^{2} c^{2}\right ) x^{8}}{8 b^{3}}-\frac {5 d^{9} \left (a d +8 b c \right ) x^{9}}{36 b^{2}}-\frac {d^{10} x^{10}}{8 b}}{\left (b x +a \right )^{18}}\)	\(831\)
default	\(\frac {252 d^{5} \left (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}\right )}{13 b^{11} \left (b x +a \right )^{13}}-\frac {35 d^{6} \left (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}\right )}{2 b^{11} \left (b x +a \right )^{12}}-\frac {15 d^{4} \left (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}\right )}{b^{11} \left (b x +a \right )^{14}}+\frac {10 d^{9} \left (a d -b c \right )}{9 b^{11} \left (b x +a \right )^{9}}-\frac {d^{10}}{8 b^{11} \left (b x +a \right )^{8}}-\frac {9 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{11} \left (b x +a \right )^{10}}+\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{11 b^{11} \left (b x +a \right )^{11}}-\frac {45 d^{2} \left (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}\right )}{16 b^{11} \left (b x +a \right )^{16}}+\frac {8 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 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}\right )}{b^{11} \left (b x +a \right )^{15}}-\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}}{18 b^{11} \left (b x +a \right )^{18}}+\frac {10 d \left (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}\right )}{17 b^{11} \left (b x +a \right )^{17}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{7} d^{10}-8 a^{9} b^{8} c \,d^{9}-36 a^{8} b^{9} c^{2} d^{8}-120 a^{7} b^{10} c^{3} d^{7}-330 a^{6} b^{11} c^{4} d^{6}-792 a^{5} b^{12} c^{5} d^{5}-1716 a^{4} b^{13} c^{6} d^{4}-3432 a^{3} b^{14} c^{7} d^{3}-6435 a^{2} b^{15} c^{8} d^{2}-11440 a \,b^{16} c^{9} d -19448 b^{17} c^{10}}{350064 b^{18}}+\frac {\left (-a^{9} b^{7} d^{10}-8 a^{8} b^{8} c \,d^{9}-36 a^{7} b^{9} c^{2} d^{8}-120 a^{6} b^{10} c^{3} d^{7}-330 a^{5} b^{11} c^{4} d^{6}-792 a^{4} b^{12} c^{5} d^{5}-1716 a^{3} b^{13} c^{6} d^{4}-3432 a^{2} b^{14} c^{7} d^{3}-6435 a \,b^{15} c^{8} d^{2}-11440 b^{16} c^{9} d \right ) x}{19448 b^{17}}+\frac {\left (-a^{8} b^{7} d^{10}-8 a^{7} b^{8} c \,d^{9}-36 a^{6} b^{9} c^{2} d^{8}-120 a^{5} b^{10} c^{3} d^{7}-330 a^{4} b^{11} c^{4} d^{6}-792 a^{3} b^{12} c^{5} d^{5}-1716 a^{2} b^{13} c^{6} d^{4}-3432 a \,b^{14} c^{7} d^{3}-6435 b^{15} c^{8} d^{2}\right ) x^{2}}{2288 b^{16}}+\frac {\left (-a^{7} b^{7} d^{10}-8 a^{6} b^{8} c \,d^{9}-36 a^{5} b^{9} c^{2} d^{8}-120 a^{4} b^{10} c^{3} d^{7}-330 a^{3} b^{11} c^{4} d^{6}-792 a^{2} b^{12} c^{5} d^{5}-1716 a \,b^{13} c^{6} d^{4}-3432 b^{14} c^{7} d^{3}\right ) x^{3}}{429 b^{15}}+\frac {5 \left (-a^{6} b^{7} d^{10}-8 a^{5} b^{8} c \,d^{9}-36 a^{4} b^{9} c^{2} d^{8}-120 a^{3} b^{10} c^{3} d^{7}-330 a^{2} b^{11} c^{4} d^{6}-792 a \,c^{5} d^{5} b^{12}-1716 b^{13} c^{6} d^{4}\right ) x^{4}}{572 b^{14}}+\frac {7 \left (-a^{5} b^{7} d^{10}-8 a^{4} b^{8} c \,d^{9}-36 a^{3} b^{9} c^{2} d^{8}-120 a^{2} b^{10} c^{3} d^{7}-330 a \,b^{11} c^{4} d^{6}-792 c^{5} d^{5} b^{12}\right ) x^{5}}{286 b^{13}}+\frac {7 \left (-a^{4} b^{7} d^{10}-8 a^{3} b^{8} c \,d^{9}-36 a^{2} b^{9} c^{2} d^{8}-120 a \,b^{10} c^{3} d^{7}-330 b^{11} c^{4} d^{6}\right ) x^{6}}{132 b^{12}}+\frac {\left (-a^{3} b^{7} d^{10}-8 a^{2} b^{8} c \,d^{9}-36 a \,b^{9} c^{2} d^{8}-120 b^{10} c^{3} d^{7}\right ) x^{7}}{11 b^{11}}+\frac {\left (-a^{2} b^{7} d^{10}-8 a \,b^{8} c \,d^{9}-36 b^{9} c^{2} d^{8}\right ) x^{8}}{8 b^{10}}+\frac {5 \left (-a \,b^{7} d^{10}-8 b^{8} c \,d^{9}\right ) x^{9}}{36 b^{9}}-\frac {d^{10} x^{10}}{8 b}}{\left (b x +a \right )^{18}}\)	\(909\)
gosper	\(-\frac {43758 d^{10} x^{10} b^{10}+48620 a \,b^{9} d^{10} x^{9}+388960 b^{10} c \,d^{9} x^{9}+43758 a^{2} b^{8} d^{10} x^{8}+350064 a \,b^{9} c \,d^{9} x^{8}+1575288 b^{10} c^{2} d^{8} x^{8}+31824 a^{3} b^{7} d^{10} x^{7}+254592 a^{2} b^{8} c \,d^{9} x^{7}+1145664 a \,b^{9} c^{2} d^{8} x^{7}+3818880 b^{10} c^{3} d^{7} x^{7}+18564 a^{4} b^{6} d^{10} x^{6}+148512 a^{3} b^{7} c \,d^{9} x^{6}+668304 a^{2} b^{8} c^{2} d^{8} x^{6}+2227680 a \,b^{9} c^{3} d^{7} x^{6}+6126120 b^{10} c^{4} d^{6} x^{6}+8568 a^{5} b^{5} d^{10} x^{5}+68544 a^{4} b^{6} c \,d^{9} x^{5}+308448 a^{3} b^{7} c^{2} d^{8} x^{5}+1028160 a^{2} b^{8} c^{3} d^{7} x^{5}+2827440 a \,b^{9} c^{4} d^{6} x^{5}+6785856 b^{10} c^{5} d^{5} x^{5}+3060 a^{6} b^{4} d^{10} x^{4}+24480 a^{5} b^{5} c \,d^{9} x^{4}+110160 a^{4} b^{6} c^{2} d^{8} x^{4}+367200 a^{3} b^{7} c^{3} d^{7} x^{4}+1009800 a^{2} b^{8} c^{4} d^{6} x^{4}+2423520 a \,b^{9} c^{5} d^{5} x^{4}+5250960 b^{10} c^{6} d^{4} x^{4}+816 a^{7} b^{3} d^{10} x^{3}+6528 a^{6} b^{4} c \,d^{9} x^{3}+29376 a^{5} b^{5} c^{2} d^{8} x^{3}+97920 a^{4} b^{6} c^{3} d^{7} x^{3}+269280 a^{3} b^{7} c^{4} d^{6} x^{3}+646272 a^{2} b^{8} c^{5} d^{5} x^{3}+1400256 a \,b^{9} c^{6} d^{4} x^{3}+2800512 b^{10} c^{7} d^{3} x^{3}+153 a^{8} b^{2} d^{10} x^{2}+1224 a^{7} b^{3} c \,d^{9} x^{2}+5508 a^{6} b^{4} c^{2} d^{8} x^{2}+18360 a^{5} b^{5} c^{3} d^{7} x^{2}+50490 a^{4} b^{6} c^{4} d^{6} x^{2}+121176 a^{3} b^{7} c^{5} d^{5} x^{2}+262548 a^{2} b^{8} c^{6} d^{4} x^{2}+525096 a \,b^{9} c^{7} d^{3} x^{2}+984555 b^{10} c^{8} d^{2} x^{2}+18 a^{9} b \,d^{10} x +144 a^{8} b^{2} c \,d^{9} x +648 a^{7} b^{3} c^{2} d^{8} x +2160 a^{6} b^{4} c^{3} d^{7} x +5940 a^{5} b^{5} c^{4} d^{6} x +14256 a^{4} b^{6} c^{5} d^{5} x +30888 a^{3} b^{7} c^{6} d^{4} x +61776 a^{2} b^{8} c^{7} d^{3} x +115830 a \,b^{9} c^{8} d^{2} x +205920 b^{10} c^{9} d x +a^{10} d^{10}+8 a^{9} b c \,d^{9}+36 a^{8} b^{2} c^{2} d^{8}+120 a^{7} b^{3} c^{3} d^{7}+330 a^{6} b^{4} c^{4} d^{6}+792 a^{5} b^{5} c^{5} d^{5}+1716 a^{4} b^{6} c^{6} d^{4}+3432 a^{3} b^{7} c^{7} d^{3}+6435 a^{2} b^{8} c^{8} d^{2}+11440 a \,b^{9} c^{9} d +19448 b^{10} c^{10}}{350064 b^{11} \left (b x +a \right )^{18}}\)	\(962\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^19,x,method=_RETURNVERBOSE)

[Out]

252/13/b^11*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*x+a)^13
-35/2/b^11*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*x+a)^12-15/b^11*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*x+a)^14+10/9
/b^11*d^9*(a*d-b*c)/(b*x+a)^9-1/8*d^10/b^11/(b*x+a)^8-9/2/b^11*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)^10+120/
11/b^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*x+a)^11-45/16/b^11*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*x+a)^16+8/b^11*d^3*(a^7*d^7-7*a^6*b*c*d^6+21*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*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^2-1
0*a*b^9*c^9*d+b^10*c^10)/b^11/(b*x+a)^18+10/17/b^11*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*x
+a)^17

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (228) = 456\).
time = 0.33, size = 1052, normalized size = 4.31 \begin {gather*} -\frac {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 \, {\left (8 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 43758 \, {\left (36 \, b^{10} c^{2} d^{8} + 8 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 31824 \, {\left (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}\right )} x^{7} + 18564 \, {\left (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}\right )} x^{6} + 8568 \, {\left (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}\right )} x^{5} + 3060 \, {\left (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}\right )} x^{4} + 816 \, {\left (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}\right )} x^{3} + 153 \, {\left (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}\right )} x^{2} + 18 \, {\left (11440 \, b^{10} c^{9} d + 6435 \, 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} + 120 \, 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}\right )} x}{350064 \, {\left (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}\right )}} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (228) = 456\).
time = 0.31, size = 1052, normalized size = 4.31 \begin {gather*} -\frac {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 \, {\left (8 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 43758 \, {\left (36 \, b^{10} c^{2} d^{8} + 8 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 31824 \, {\left (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}\right )} x^{7} + 18564 \, {\left (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}\right )} x^{6} + 8568 \, {\left (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}\right )} x^{5} + 3060 \, {\left (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}\right )} x^{4} + 816 \, {\left (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}\right )} x^{3} + 153 \, {\left (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}\right )} x^{2} + 18 \, {\left (11440 \, b^{10} c^{9} d + 6435 \, 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} + 120 \, 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}\right )} x}{350064 \, {\left (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}\right )}} \end {gather*}

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)

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Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**19,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (228) = 456\).
time = 0.00, size = 1029, normalized size = 4.22 \begin {gather*} \frac {-43758 x^{10} d^{10} b^{10}-48620 x^{9} d^{10} b^{9} a-388960 x^{9} d^{9} c b^{10}-43758 x^{8} d^{10} b^{8} a^{2}-350064 x^{8} d^{9} c b^{9} a-1575288 x^{8} d^{8} c^{2} b^{10}-31824 x^{7} d^{10} b^{7} a^{3}-254592 x^{7} d^{9} c b^{8} a^{2}-1145664 x^{7} d^{8} c^{2} b^{9} a-3818880 x^{7} d^{7} c^{3} b^{10}-18564 x^{6} d^{10} b^{6} a^{4}-148512 x^{6} d^{9} c b^{7} a^{3}-668304 x^{6} d^{8} c^{2} b^{8} a^{2}-2227680 x^{6} d^{7} c^{3} b^{9} a-6126120 x^{6} d^{6} c^{4} b^{10}-8568 x^{5} d^{10} b^{5} a^{5}-68544 x^{5} d^{9} c b^{6} a^{4}-308448 x^{5} d^{8} c^{2} b^{7} a^{3}-1028160 x^{5} d^{7} c^{3} b^{8} a^{2}-2827440 x^{5} d^{6} c^{4} b^{9} a-6785856 x^{5} d^{5} c^{5} b^{10}-3060 x^{4} d^{10} b^{4} a^{6}-24480 x^{4} d^{9} c b^{5} a^{5}-110160 x^{4} d^{8} c^{2} b^{6} a^{4}-367200 x^{4} d^{7} c^{3} b^{7} a^{3}-1009800 x^{4} d^{6} c^{4} b^{8} a^{2}-2423520 x^{4} d^{5} c^{5} b^{9} a-5250960 x^{4} d^{4} c^{6} b^{10}-816 x^{3} d^{10} b^{3} a^{7}-6528 x^{3} d^{9} c b^{4} a^{6}-29376 x^{3} d^{8} c^{2} b^{5} a^{5}-97920 x^{3} d^{7} c^{3} b^{6} a^{4}-269280 x^{3} d^{6} c^{4} b^{7} a^{3}-646272 x^{3} d^{5} c^{5} b^{8} a^{2}-1400256 x^{3} d^{4} c^{6} b^{9} a-2800512 x^{3} d^{3} c^{7} b^{10}-153 x^{2} d^{10} b^{2} a^{8}-1224 x^{2} d^{9} c b^{3} a^{7}-5508 x^{2} d^{8} c^{2} b^{4} a^{6}-18360 x^{2} d^{7} c^{3} b^{5} a^{5}-50490 x^{2} d^{6} c^{4} b^{6} a^{4}-121176 x^{2} d^{5} c^{5} b^{7} a^{3}-262548 x^{2} d^{4} c^{6} b^{8} a^{2}-525096 x^{2} d^{3} c^{7} b^{9} a-984555 x^{2} d^{2} c^{8} b^{10}-18 x d^{10} b a^{9}-144 x d^{9} c b^{2} a^{8}-648 x d^{8} c^{2} b^{3} a^{7}-2160 x d^{7} c^{3} b^{4} a^{6}-5940 x d^{6} c^{4} b^{5} a^{5}-14256 x d^{5} c^{5} b^{6} a^{4}-30888 x d^{4} c^{6} b^{7} a^{3}-61776 x d^{3} c^{7} b^{8} a^{2}-115830 x d^{2} c^{8} b^{9} a-205920 x d c^{9} b^{10}-d^{10} a^{10}-8 d^{9} c b a^{9}-36 d^{8} c^{2} b^{2} a^{8}-120 d^{7} c^{3} b^{3} a^{7}-330 d^{6} c^{4} b^{4} a^{6}-792 d^{5} c^{5} b^{5} a^{5}-1716 d^{4} c^{6} b^{6} a^{4}-3432 d^{3} c^{7} b^{7} a^{3}-6435 d^{2} c^{8} b^{8} a^{2}-11440 d c^{9} b^{9} a-19448 c^{10} b^{10}}{350064 b^{11} \left (x b+a\right )^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^19,x)

[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)

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Mupad [B]
time = 12.02, size = 1153, normalized size = 4.73 \begin {gather*} -\frac {a^{10}\,d^{10}+8\,a^9\,b\,c\,d^9+18\,a^9\,b\,d^{10}\,x+36\,a^8\,b^2\,c^2\,d^8+144\,a^8\,b^2\,c\,d^9\,x+153\,a^8\,b^2\,d^{10}\,x^2+120\,a^7\,b^3\,c^3\,d^7+648\,a^7\,b^3\,c^2\,d^8\,x+1224\,a^7\,b^3\,c\,d^9\,x^2+816\,a^7\,b^3\,d^{10}\,x^3+330\,a^6\,b^4\,c^4\,d^6+2160\,a^6\,b^4\,c^3\,d^7\,x+5508\,a^6\,b^4\,c^2\,d^8\,x^2+6528\,a^6\,b^4\,c\,d^9\,x^3+3060\,a^6\,b^4\,d^{10}\,x^4+792\,a^5\,b^5\,c^5\,d^5+5940\,a^5\,b^5\,c^4\,d^6\,x+18360\,a^5\,b^5\,c^3\,d^7\,x^2+29376\,a^5\,b^5\,c^2\,d^8\,x^3+24480\,a^5\,b^5\,c\,d^9\,x^4+8568\,a^5\,b^5\,d^{10}\,x^5+1716\,a^4\,b^6\,c^6\,d^4+14256\,a^4\,b^6\,c^5\,d^5\,x+50490\,a^4\,b^6\,c^4\,d^6\,x^2+97920\,a^4\,b^6\,c^3\,d^7\,x^3+110160\,a^4\,b^6\,c^2\,d^8\,x^4+68544\,a^4\,b^6\,c\,d^9\,x^5+18564\,a^4\,b^6\,d^{10}\,x^6+3432\,a^3\,b^7\,c^7\,d^3+30888\,a^3\,b^7\,c^6\,d^4\,x+121176\,a^3\,b^7\,c^5\,d^5\,x^2+269280\,a^3\,b^7\,c^4\,d^6\,x^3+367200\,a^3\,b^7\,c^3\,d^7\,x^4+308448\,a^3\,b^7\,c^2\,d^8\,x^5+148512\,a^3\,b^7\,c\,d^9\,x^6+31824\,a^3\,b^7\,d^{10}\,x^7+6435\,a^2\,b^8\,c^8\,d^2+61776\,a^2\,b^8\,c^7\,d^3\,x+262548\,a^2\,b^8\,c^6\,d^4\,x^2+646272\,a^2\,b^8\,c^5\,d^5\,x^3+1009800\,a^2\,b^8\,c^4\,d^6\,x^4+1028160\,a^2\,b^8\,c^3\,d^7\,x^5+668304\,a^2\,b^8\,c^2\,d^8\,x^6+254592\,a^2\,b^8\,c\,d^9\,x^7+43758\,a^2\,b^8\,d^{10}\,x^8+11440\,a\,b^9\,c^9\,d+115830\,a\,b^9\,c^8\,d^2\,x+525096\,a\,b^9\,c^7\,d^3\,x^2+1400256\,a\,b^9\,c^6\,d^4\,x^3+2423520\,a\,b^9\,c^5\,d^5\,x^4+2827440\,a\,b^9\,c^4\,d^6\,x^5+2227680\,a\,b^9\,c^3\,d^7\,x^6+1145664\,a\,b^9\,c^2\,d^8\,x^7+350064\,a\,b^9\,c\,d^9\,x^8+48620\,a\,b^9\,d^{10}\,x^9+19448\,b^{10}\,c^{10}+205920\,b^{10}\,c^9\,d\,x+984555\,b^{10}\,c^8\,d^2\,x^2+2800512\,b^{10}\,c^7\,d^3\,x^3+5250960\,b^{10}\,c^6\,d^4\,x^4+6785856\,b^{10}\,c^5\,d^5\,x^5+6126120\,b^{10}\,c^4\,d^6\,x^6+3818880\,b^{10}\,c^3\,d^7\,x^7+1575288\,b^{10}\,c^2\,d^8\,x^8+388960\,b^{10}\,c\,d^9\,x^9+43758\,b^{10}\,d^{10}\,x^{10}}{350064\,a^{18}\,b^{11}+6301152\,a^{17}\,b^{12}\,x+53559792\,a^{16}\,b^{13}\,x^2+285652224\,a^{15}\,b^{14}\,x^3+1071195840\,a^{14}\,b^{15}\,x^4+2999348352\,a^{13}\,b^{16}\,x^5+6498588096\,a^{12}\,b^{17}\,x^6+11140436736\,a^{11}\,b^{18}\,x^7+15318100512\,a^{10}\,b^{19}\,x^8+17020111680\,a^9\,b^{20}\,x^9+15318100512\,a^8\,b^{21}\,x^{10}[phone]\,a^7\,b^{22}\,x^{11}[phone]\,a^6\,b^{23}\,x^{12}[phone]\,a^5\,b^{24}\,x^{13}[phone]\,a^4\,b^{25}\,x^{14}[phone]\,a^3\,b^{26}\,x^{15}[phone]\,a^2\,b^{27}\,x^{16}+6301152\,a\,b^{28}\,x^{17}+350064\,b^{29}\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^19,x)

[Out]

-(a^10*d^10 + 19448*b^10*c^10 + 43758*b^10*d^10*x^10 + 48620*a*b^9*d^10*x^9 + 388960*b^10*c*d^9*x^9 + 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 + 153*a^8*b^2*d^10*x^2 + 816*a^7*b^3*d^10*x^3 + 3060*a^6*b^4*d^10*x^4 + 85
68*a^5*b^5*d^10*x^5 + 18564*a^4*b^6*d^10*x^6 + 31824*a^3*b^7*d^10*x^7 + 43758*a^2*b^8*d^10*x^8 + 984555*b^10*c
^8*d^2*x^2 + 2800512*b^10*c^7*d^3*x^3 + 5250960*b^10*c^6*d^4*x^4 + 6785856*b^10*c^5*d^5*x^5 + 6126120*b^10*c^4
*d^6*x^6 + 3818880*b^10*c^3*d^7*x^7 + 1575288*b^10*c^2*d^8*x^8 + 11440*a*b^9*c^9*d + 8*a^9*b*c*d^9 + 18*a^9*b*
d^10*x + 205920*b^10*c^9*d*x + 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 + 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 + 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 + 1028160*a^2*b^8*c^3*d^7*x^5 + 308448*a^3*b^7*c^2*d^8*x^5 + 668304*
a^2*b^8*c^2*d^8*x^6 + 115830*a*b^9*c^8*d^2*x + 144*a^8*b^2*c*d^9*x + 350064*a*b^9*c*d^9*x^8 + 61776*a^2*b^8*c^
7*d^3*x + 30888*a^3*b^7*c^6*d^4*x + 14256*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 + 525096*a*b^9*c^7*d^3*x^2 + 1224*a^7*b^3*c*d^9*x^2 + 1400256*a*b^9*c^6*d^4*x^3 + 6528
*a^6*b^4*c*d^9*x^3 + 2423520*a*b^9*c^5*d^5*x^4 + 24480*a^5*b^5*c*d^9*x^4 + 2827440*a*b^9*c^4*d^6*x^5 + 68544*a
^4*b^6*c*d^9*x^5 + 2227680*a*b^9*c^3*d^7*x^6 + 148512*a^3*b^7*c*d^9*x^6 + 1145664*a*b^9*c^2*d^8*x^7 + 254592*a
^2*b^8*c*d^9*x^7)/(350064*a^18*b^11 + 350064*b^29*x^18 + 6301152*a^17*b^12*x + 6301152*a*b^28*x^17 + 53559792*
a^16*b^13*x^2 + 285652224*a^15*b^14*x^3 + 1071195840*a^14*b^15*x^4 + 2999348352*a^13*b^16*x^5 + 6498588096*a^1
2*b^17*x^6 + 11140436736*a^11*b^18*x^7 + 15318100512*a^10*b^19*x^8 + 17020111680*a^9*b^20*x^9 + 15318100512*a^
8*b^21*x^10 + 11140436736*a^7*b^22*x^11 + 6498588096*a^6*b^23*x^12 + 2999348352*a^5*b^24*x^13 + 1071195840*a^4
*b^25*x^14 + 285652224*a^3*b^26*x^15 + 53559792*a^2*b^27*x^16)

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