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

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

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Rubi [A]  time = 0.10, 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 = {45, 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]

-(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 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 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])

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]  time = 0.28, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^{10}}{(a+b x)^{19}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.19, 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)

________________________________________________________________________________________

giac [B]  time = 1.40, size = 961, normalized size = 3.94 \begin {gather*} -\frac {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} + 350064 \, 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 + 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 + 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}}{350064 \, {\left (b x + a\right )}^{18} b^{11}} \end {gather*}

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)

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maple [B]  time = 0.01, size = 867, normalized size = 3.55 \begin {gather*} -\frac {d^{10}}{8 \left (b x +a \right )^{8} b^{11}}+\frac {10 \left (a d -b c \right ) d^{9}}{9 \left (b x +a \right )^{9} b^{11}}-\frac {9 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{2 \left (b x +a \right )^{10} b^{11}}+\frac {120 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{7}}{11 \left (b x +a \right )^{11} b^{11}}-\frac {35 \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 ) d^{6}}{2 \left (b x +a \right )^{12} b^{11}}+\frac {252 \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 ) d^{5}}{13 \left (b x +a \right )^{13} b^{11}}-\frac {15 \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 ) d^{4}}{\left (b x +a \right )^{14} b^{11}}+\frac {8 \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 ) d^{3}}{\left (b x +a \right )^{15} b^{11}}-\frac {45 \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 ) d^{2}}{16 \left (b x +a \right )^{16} b^{11}}+\frac {10 \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 ) d}{17 \left (b x +a \right )^{17} b^{11}}-\frac {d^{10} a^{10}-10 c \,d^{9} a^{9} b +45 c^{2} d^{8} a^{8} b^{2}-120 a^{7} c^{3} d^{7} b^{3}+210 c^{4} d^{6} a^{6} b^{4}-252 a^{5} c^{5} d^{5} b^{5}+210 a^{4} c^{6} d^{4} b^{6}-120 a^{3} c^{7} d^{3} b^{7}+45 a^{2} c^{8} d^{2} b^{8}-10 a \,b^{9} c^{9} d +c^{10} b^{10}}{18 \left (b x +a \right )^{18} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*d^10/b^11/(b*x+a)^8+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+10/9*d^9*(a*d-b*c)/b^11/(b*x+a)^9+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+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+8*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^11/(b*x+a)^15-35/2*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)^1
2-1/18*(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)^18-9/2*d^8
*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^11/(b*x+a)^10-15*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

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maxima [B]  time = 2.54, 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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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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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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