3.13.23 \(\int \frac {(c+d x)^{10}}{(a+b x)^{18}} \, dx\)

Optimal. Leaf size=213 \[ -\frac {d^6 (c+d x)^{11}}{136136 (a+b x)^{11} (b c-a d)^7}+\frac {d^5 (c+d x)^{11}}{12376 (a+b x)^{12} (b c-a d)^6}-\frac {3 d^4 (c+d x)^{11}}{6188 (a+b x)^{13} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{476 (a+b x)^{14} (b c-a d)^4}-\frac {d^2 (c+d x)^{11}}{136 (a+b x)^{15} (b c-a d)^3}+\frac {3 d (c+d x)^{11}}{136 (a+b x)^{16} (b c-a d)^2}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)} \]

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Rubi [A]  time = 0.08, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, 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^6 (c+d x)^{11}}{136136 (a+b x)^{11} (b c-a d)^7}+\frac {d^5 (c+d x)^{11}}{12376 (a+b x)^{12} (b c-a d)^6}-\frac {3 d^4 (c+d x)^{11}}{6188 (a+b x)^{13} (b c-a d)^5}+\frac {d^3 (c+d x)^{11}}{476 (a+b x)^{14} (b c-a d)^4}-\frac {d^2 (c+d x)^{11}}{136 (a+b x)^{15} (b c-a d)^3}+\frac {3 d (c+d x)^{11}}{136 (a+b x)^{16} (b c-a d)^2}-\frac {(c+d x)^{11}}{17 (a+b x)^{17} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^18,x]

[Out]

-(c + d*x)^11/(17*(b*c - a*d)*(a + b*x)^17) + (3*d*(c + d*x)^11)/(136*(b*c - a*d)^2*(a + b*x)^16) - (d^2*(c +
d*x)^11)/(136*(b*c - a*d)^3*(a + b*x)^15) + (d^3*(c + d*x)^11)/(476*(b*c - a*d)^4*(a + b*x)^14) - (3*d^4*(c +
d*x)^11)/(6188*(b*c - a*d)^5*(a + b*x)^13) + (d^5*(c + d*x)^11)/(12376*(b*c - a*d)^6*(a + b*x)^12) - (d^6*(c +
d*x)^11)/(136136*(b*c - a*d)^7*(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)^{18}} \, dx &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}-\frac {(6 d) \int \frac {(c+d x)^{10}}{(a+b x)^{17}} \, dx}{17 (b c-a d)}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}+\frac {\left (15 d^2\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{16}} \, dx}{136 (b c-a d)^2}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}-\frac {d^3 \int \frac {(c+d x)^{10}}{(a+b x)^{15}} \, dx}{34 (b c-a d)^3}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}+\frac {\left (3 d^4\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx}{476 (b c-a d)^4}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}-\frac {\left (3 d^5\right ) \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx}{3094 (b c-a d)^5}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}+\frac {d^5 (c+d x)^{11}}{12376 (b c-a d)^6 (a+b x)^{12}}+\frac {d^6 \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx}{12376 (b c-a d)^6}\\ &=-\frac {(c+d x)^{11}}{17 (b c-a d) (a+b x)^{17}}+\frac {3 d (c+d x)^{11}}{136 (b c-a d)^2 (a+b x)^{16}}-\frac {d^2 (c+d x)^{11}}{136 (b c-a d)^3 (a+b x)^{15}}+\frac {d^3 (c+d x)^{11}}{476 (b c-a d)^4 (a+b x)^{14}}-\frac {3 d^4 (c+d x)^{11}}{6188 (b c-a d)^5 (a+b x)^{13}}+\frac {d^5 (c+d x)^{11}}{12376 (b c-a d)^6 (a+b x)^{12}}-\frac {d^6 (c+d x)^{11}}{136136 (b c-a d)^7 (a+b x)^{11}}\\ \end {align*}

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Mathematica [B]  time = 0.32, size = 690, normalized size = 3.24 \begin {gather*} -\frac {a^{10} d^{10}+a^9 b d^9 (7 c+17 d x)+a^8 b^2 d^8 \left (28 c^2+119 c d x+136 d^2 x^2\right )+4 a^7 b^3 d^7 \left (21 c^3+119 c^2 d x+238 c d^2 x^2+170 d^3 x^3\right )+14 a^6 b^4 d^6 \left (15 c^4+102 c^3 d x+272 c^2 d^2 x^2+340 c d^3 x^3+170 d^4 x^4\right )+14 a^5 b^5 d^5 \left (33 c^5+255 c^4 d x+816 c^3 d^2 x^2+1360 c^2 d^3 x^3+1190 c d^4 x^4+442 d^5 x^5\right )+14 a^4 b^6 d^4 \left (66 c^6+561 c^5 d x+2040 c^4 d^2 x^2+4080 c^3 d^3 x^3+4760 c^2 d^4 x^4+3094 c d^5 x^5+884 d^6 x^6\right )+4 a^3 b^7 d^3 \left (429 c^7+3927 c^6 d x+15708 c^5 d^2 x^2+35700 c^4 d^3 x^3+49980 c^3 d^4 x^4+43316 c^2 d^5 x^5+21658 c d^6 x^6+4862 d^7 x^7\right )+a^2 b^8 d^2 \left (3003 c^8+29172 c^7 d x+125664 c^6 d^2 x^2+314160 c^5 d^3 x^3+499800 c^4 d^4 x^4+519792 c^3 d^5 x^5+346528 c^2 d^6 x^6+136136 c d^7 x^7+24310 d^8 x^8\right )+a b^9 d \left (5005 c^9+51051 c^8 d x+233376 c^7 d^2 x^2+628320 c^6 d^3 x^3+1099560 c^5 d^4 x^4+1299480 c^4 d^5 x^5+1039584 c^3 d^6 x^6+544544 c^2 d^7 x^7+170170 c d^8 x^8+24310 d^9 x^9\right )+b^{10} \left (8008 c^{10}+85085 c^9 d x+408408 c^8 d^2 x^2+1166880 c^7 d^3 x^3+2199120 c^6 d^4 x^4+2858856 c^5 d^5 x^5+2598960 c^4 d^6 x^6+1633632 c^3 d^7 x^7+680680 c^2 d^8 x^8+170170 c d^9 x^9+19448 d^{10} x^{10}\right )}{136136 b^{11} (a+b x)^{17}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^18,x]

[Out]

-1/136136*(a^10*d^10 + a^9*b*d^9*(7*c + 17*d*x) + a^8*b^2*d^8*(28*c^2 + 119*c*d*x + 136*d^2*x^2) + 4*a^7*b^3*d
^7*(21*c^3 + 119*c^2*d*x + 238*c*d^2*x^2 + 170*d^3*x^3) + 14*a^6*b^4*d^6*(15*c^4 + 102*c^3*d*x + 272*c^2*d^2*x
^2 + 340*c*d^3*x^3 + 170*d^4*x^4) + 14*a^5*b^5*d^5*(33*c^5 + 255*c^4*d*x + 816*c^3*d^2*x^2 + 1360*c^2*d^3*x^3
+ 1190*c*d^4*x^4 + 442*d^5*x^5) + 14*a^4*b^6*d^4*(66*c^6 + 561*c^5*d*x + 2040*c^4*d^2*x^2 + 4080*c^3*d^3*x^3 +
4760*c^2*d^4*x^4 + 3094*c*d^5*x^5 + 884*d^6*x^6) + 4*a^3*b^7*d^3*(429*c^7 + 3927*c^6*d*x + 15708*c^5*d^2*x^2
+ 35700*c^4*d^3*x^3 + 49980*c^3*d^4*x^4 + 43316*c^2*d^5*x^5 + 21658*c*d^6*x^6 + 4862*d^7*x^7) + a^2*b^8*d^2*(3
003*c^8 + 29172*c^7*d*x + 125664*c^6*d^2*x^2 + 314160*c^5*d^3*x^3 + 499800*c^4*d^4*x^4 + 519792*c^3*d^5*x^5 +
346528*c^2*d^6*x^6 + 136136*c*d^7*x^7 + 24310*d^8*x^8) + a*b^9*d*(5005*c^9 + 51051*c^8*d*x + 233376*c^7*d^2*x^
2 + 628320*c^6*d^3*x^3 + 1099560*c^5*d^4*x^4 + 1299480*c^4*d^5*x^5 + 1039584*c^3*d^6*x^6 + 544544*c^2*d^7*x^7
+ 170170*c*d^8*x^8 + 24310*d^9*x^9) + b^10*(8008*c^10 + 85085*c^9*d*x + 408408*c^8*d^2*x^2 + 1166880*c^7*d^3*x
^3 + 2199120*c^6*d^4*x^4 + 2858856*c^5*d^5*x^5 + 2598960*c^4*d^6*x^6 + 1633632*c^3*d^7*x^7 + 680680*c^2*d^8*x^
8 + 170170*c*d^9*x^9 + 19448*d^10*x^10))/(b^11*(a + b*x)^17)

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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)^{18}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^18,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^18, x]

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fricas [B]  time = 1.31, size = 1041, normalized size = 4.89 \begin {gather*} -\frac {19448 \, b^{10} d^{10} x^{10} + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10} + 24310 \, {\left (7 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 24310 \, {\left (28 \, b^{10} c^{2} d^{8} + 7 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 19448 \, {\left (84 \, b^{10} c^{3} d^{7} + 28 \, a b^{9} c^{2} d^{8} + 7 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 12376 \, {\left (210 \, b^{10} c^{4} d^{6} + 84 \, a b^{9} c^{3} d^{7} + 28 \, a^{2} b^{8} c^{2} d^{8} + 7 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 6188 \, {\left (462 \, b^{10} c^{5} d^{5} + 210 \, a b^{9} c^{4} d^{6} + 84 \, a^{2} b^{8} c^{3} d^{7} + 28 \, a^{3} b^{7} c^{2} d^{8} + 7 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 2380 \, {\left (924 \, b^{10} c^{6} d^{4} + 462 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} + 84 \, a^{3} b^{7} c^{3} d^{7} + 28 \, a^{4} b^{6} c^{2} d^{8} + 7 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 680 \, {\left (1716 \, b^{10} c^{7} d^{3} + 924 \, a b^{9} c^{6} d^{4} + 462 \, a^{2} b^{8} c^{5} d^{5} + 210 \, a^{3} b^{7} c^{4} d^{6} + 84 \, a^{4} b^{6} c^{3} d^{7} + 28 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 136 \, {\left (3003 \, b^{10} c^{8} d^{2} + 1716 \, a b^{9} c^{7} d^{3} + 924 \, a^{2} b^{8} c^{6} d^{4} + 462 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 84 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} + 7 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 17 \, {\left (5005 \, b^{10} c^{9} d + 3003 \, a b^{9} c^{8} d^{2} + 1716 \, a^{2} b^{8} c^{7} d^{3} + 924 \, a^{3} b^{7} c^{6} d^{4} + 462 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} + 28 \, a^{7} b^{3} c^{2} d^{8} + 7 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{136136 \, {\left (b^{28} x^{17} + 17 \, a b^{27} x^{16} + 136 \, a^{2} b^{26} x^{15} + 680 \, a^{3} b^{25} x^{14} + 2380 \, a^{4} b^{24} x^{13} + 6188 \, a^{5} b^{23} x^{12} + 12376 \, a^{6} b^{22} x^{11} + 19448 \, a^{7} b^{21} x^{10} + 24310 \, a^{8} b^{20} x^{9} + 24310 \, a^{9} b^{19} x^{8} + 19448 \, a^{10} b^{18} x^{7} + 12376 \, a^{11} b^{17} x^{6} + 6188 \, a^{12} b^{16} x^{5} + 2380 \, a^{13} b^{15} x^{4} + 680 \, a^{14} b^{14} x^{3} + 136 \, a^{15} b^{13} x^{2} + 17 \, a^{16} b^{12} x + a^{17} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="fricas")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*
d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^
8 + 7*a^9*b*c*d^9 + a^10*d^10 + 24310*(7*b^10*c*d^9 + a*b^9*d^10)*x^9 + 24310*(28*b^10*c^2*d^8 + 7*a*b^9*c*d^9
+ a^2*b^8*d^10)*x^8 + 19448*(84*b^10*c^3*d^7 + 28*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 12376
*(210*b^10*c^4*d^6 + 84*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 + 7*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6188*(462*b
^10*c^5*d^5 + 210*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 + 28*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^
5 + 2380*(924*b^10*c^6*d^4 + 462*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 + 84*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8
+ 7*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 680*(1716*b^10*c^7*d^3 + 924*a*b^9*c^6*d^4 + 462*a^2*b^8*c^5*d^5 + 21
0*a^3*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 + 28*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 136*(3003*
b^10*c^8*d^2 + 1716*a*b^9*c^7*d^3 + 924*a^2*b^8*c^6*d^4 + 462*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 84*a^5*b
^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 + 7*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 17*(5005*b^10*c^9*d + 3003*a*b^9*c^8*d
^2 + 1716*a^2*b^8*c^7*d^3 + 924*a^3*b^7*c^6*d^4 + 462*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d
^7 + 28*a^7*b^3*c^2*d^8 + 7*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^28*x^17 + 17*a*b^27*x^16 + 136*a^2*b^26*x^15 + 6
80*a^3*b^25*x^14 + 2380*a^4*b^24*x^13 + 6188*a^5*b^23*x^12 + 12376*a^6*b^22*x^11 + 19448*a^7*b^21*x^10 + 24310
*a^8*b^20*x^9 + 24310*a^9*b^19*x^8 + 19448*a^10*b^18*x^7 + 12376*a^11*b^17*x^6 + 6188*a^12*b^16*x^5 + 2380*a^1
3*b^15*x^4 + 680*a^14*b^14*x^3 + 136*a^15*b^13*x^2 + 17*a^16*b^12*x + a^17*b^11)

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giac [B]  time = 1.32, size = 961, normalized size = 4.51 \begin {gather*} -\frac {19448 \, b^{10} d^{10} x^{10} + 170170 \, b^{10} c d^{9} x^{9} + 24310 \, a b^{9} d^{10} x^{9} + 680680 \, b^{10} c^{2} d^{8} x^{8} + 170170 \, a b^{9} c d^{9} x^{8} + 24310 \, a^{2} b^{8} d^{10} x^{8} + 1633632 \, b^{10} c^{3} d^{7} x^{7} + 544544 \, a b^{9} c^{2} d^{8} x^{7} + 136136 \, a^{2} b^{8} c d^{9} x^{7} + 19448 \, a^{3} b^{7} d^{10} x^{7} + 2598960 \, b^{10} c^{4} d^{6} x^{6} + 1039584 \, a b^{9} c^{3} d^{7} x^{6} + 346528 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 86632 \, a^{3} b^{7} c d^{9} x^{6} + 12376 \, a^{4} b^{6} d^{10} x^{6} + 2858856 \, b^{10} c^{5} d^{5} x^{5} + 1299480 \, a b^{9} c^{4} d^{6} x^{5} + 519792 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 173264 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 43316 \, a^{4} b^{6} c d^{9} x^{5} + 6188 \, a^{5} b^{5} d^{10} x^{5} + 2199120 \, b^{10} c^{6} d^{4} x^{4} + 1099560 \, a b^{9} c^{5} d^{5} x^{4} + 499800 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 199920 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 66640 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 16660 \, a^{5} b^{5} c d^{9} x^{4} + 2380 \, a^{6} b^{4} d^{10} x^{4} + 1166880 \, b^{10} c^{7} d^{3} x^{3} + 628320 \, a b^{9} c^{6} d^{4} x^{3} + 314160 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 142800 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 57120 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 19040 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 4760 \, a^{6} b^{4} c d^{9} x^{3} + 680 \, a^{7} b^{3} d^{10} x^{3} + 408408 \, b^{10} c^{8} d^{2} x^{2} + 233376 \, a b^{9} c^{7} d^{3} x^{2} + 125664 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 62832 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 28560 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 11424 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 3808 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 952 \, a^{7} b^{3} c d^{9} x^{2} + 136 \, a^{8} b^{2} d^{10} x^{2} + 85085 \, b^{10} c^{9} d x + 51051 \, a b^{9} c^{8} d^{2} x + 29172 \, a^{2} b^{8} c^{7} d^{3} x + 15708 \, a^{3} b^{7} c^{6} d^{4} x + 7854 \, a^{4} b^{6} c^{5} d^{5} x + 3570 \, a^{5} b^{5} c^{4} d^{6} x + 1428 \, a^{6} b^{4} c^{3} d^{7} x + 476 \, a^{7} b^{3} c^{2} d^{8} x + 119 \, a^{8} b^{2} c d^{9} x + 17 \, a^{9} b d^{10} x + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10}}{136136 \, {\left (b x + a\right )}^{17} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="giac")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 170170*b^10*c*d^9*x^9 + 24310*a*b^9*d^10*x^9 + 680680*b^10*c^2*d^8*x^8 + 170
170*a*b^9*c*d^9*x^8 + 24310*a^2*b^8*d^10*x^8 + 1633632*b^10*c^3*d^7*x^7 + 544544*a*b^9*c^2*d^8*x^7 + 136136*a^
2*b^8*c*d^9*x^7 + 19448*a^3*b^7*d^10*x^7 + 2598960*b^10*c^4*d^6*x^6 + 1039584*a*b^9*c^3*d^7*x^6 + 346528*a^2*b
^8*c^2*d^8*x^6 + 86632*a^3*b^7*c*d^9*x^6 + 12376*a^4*b^6*d^10*x^6 + 2858856*b^10*c^5*d^5*x^5 + 1299480*a*b^9*c
^4*d^6*x^5 + 519792*a^2*b^8*c^3*d^7*x^5 + 173264*a^3*b^7*c^2*d^8*x^5 + 43316*a^4*b^6*c*d^9*x^5 + 6188*a^5*b^5*
d^10*x^5 + 2199120*b^10*c^6*d^4*x^4 + 1099560*a*b^9*c^5*d^5*x^4 + 499800*a^2*b^8*c^4*d^6*x^4 + 199920*a^3*b^7*
c^3*d^7*x^4 + 66640*a^4*b^6*c^2*d^8*x^4 + 16660*a^5*b^5*c*d^9*x^4 + 2380*a^6*b^4*d^10*x^4 + 1166880*b^10*c^7*d
^3*x^3 + 628320*a*b^9*c^6*d^4*x^3 + 314160*a^2*b^8*c^5*d^5*x^3 + 142800*a^3*b^7*c^4*d^6*x^3 + 57120*a^4*b^6*c^
3*d^7*x^3 + 19040*a^5*b^5*c^2*d^8*x^3 + 4760*a^6*b^4*c*d^9*x^3 + 680*a^7*b^3*d^10*x^3 + 408408*b^10*c^8*d^2*x^
2 + 233376*a*b^9*c^7*d^3*x^2 + 125664*a^2*b^8*c^6*d^4*x^2 + 62832*a^3*b^7*c^5*d^5*x^2 + 28560*a^4*b^6*c^4*d^6*
x^2 + 11424*a^5*b^5*c^3*d^7*x^2 + 3808*a^6*b^4*c^2*d^8*x^2 + 952*a^7*b^3*c*d^9*x^2 + 136*a^8*b^2*d^10*x^2 + 85
085*b^10*c^9*d*x + 51051*a*b^9*c^8*d^2*x + 29172*a^2*b^8*c^7*d^3*x + 15708*a^3*b^7*c^6*d^4*x + 7854*a^4*b^6*c^
5*d^5*x + 3570*a^5*b^5*c^4*d^6*x + 1428*a^6*b^4*c^3*d^7*x + 476*a^7*b^3*c^2*d^8*x + 119*a^8*b^2*c*d^9*x + 17*a
^9*b*d^10*x + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*d^3 + 924*a^4*b^6*c^
6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 + 7*a^9*b*c*d^9 +
a^10*d^10)/((b*x + a)^17*b^11)

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maple [B]  time = 0.01, size = 867, normalized size = 4.07 \begin {gather*} -\frac {d^{10}}{7 \left (b x +a \right )^{7} b^{11}}+\frac {5 \left (a d -b c \right ) d^{9}}{4 \left (b x +a \right )^{8} b^{11}}-\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{8}}{\left (b x +a \right )^{9} b^{11}}+\frac {12 \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}}{\left (b x +a \right )^{10} b^{11}}-\frac {210 \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}}{11 \left (b x +a \right )^{11} b^{11}}+\frac {21 \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}}{\left (b x +a \right )^{12} b^{11}}-\frac {210 \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}}{13 \left (b x +a \right )^{13} b^{11}}+\frac {60 \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}}{7 \left (b x +a \right )^{14} b^{11}}-\frac {3 \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}}{\left (b x +a \right )^{15} b^{11}}+\frac {5 \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}{8 \left (b x +a \right )^{16} 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}}{17 \left (b x +a \right )^{17} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^18,x)

[Out]

5/4*d^9*(a*d-b*c)/b^11/(b*x+a)^8-210/13*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)^13-1/7*d^10/b^11/(b*x+a)^7-5*d^8*(a^2*d^2-2*a*b*c*d+b^2*c^2)
/b^11/(b*x+a)^9-1/17*(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)^17-210/11*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^11/(b*x+a)^11-3*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)^15+21*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)^12+12*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)^10+60/
7*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)^14+5/8*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)^16

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maxima [B]  time = 2.29, size = 1041, normalized size = 4.89 \begin {gather*} -\frac {19448 \, b^{10} d^{10} x^{10} + 8008 \, b^{10} c^{10} + 5005 \, a b^{9} c^{9} d + 3003 \, a^{2} b^{8} c^{8} d^{2} + 1716 \, a^{3} b^{7} c^{7} d^{3} + 924 \, a^{4} b^{6} c^{6} d^{4} + 462 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 84 \, a^{7} b^{3} c^{3} d^{7} + 28 \, a^{8} b^{2} c^{2} d^{8} + 7 \, a^{9} b c d^{9} + a^{10} d^{10} + 24310 \, {\left (7 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 24310 \, {\left (28 \, b^{10} c^{2} d^{8} + 7 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 19448 \, {\left (84 \, b^{10} c^{3} d^{7} + 28 \, a b^{9} c^{2} d^{8} + 7 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 12376 \, {\left (210 \, b^{10} c^{4} d^{6} + 84 \, a b^{9} c^{3} d^{7} + 28 \, a^{2} b^{8} c^{2} d^{8} + 7 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 6188 \, {\left (462 \, b^{10} c^{5} d^{5} + 210 \, a b^{9} c^{4} d^{6} + 84 \, a^{2} b^{8} c^{3} d^{7} + 28 \, a^{3} b^{7} c^{2} d^{8} + 7 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 2380 \, {\left (924 \, b^{10} c^{6} d^{4} + 462 \, a b^{9} c^{5} d^{5} + 210 \, a^{2} b^{8} c^{4} d^{6} + 84 \, a^{3} b^{7} c^{3} d^{7} + 28 \, a^{4} b^{6} c^{2} d^{8} + 7 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 680 \, {\left (1716 \, b^{10} c^{7} d^{3} + 924 \, a b^{9} c^{6} d^{4} + 462 \, a^{2} b^{8} c^{5} d^{5} + 210 \, a^{3} b^{7} c^{4} d^{6} + 84 \, a^{4} b^{6} c^{3} d^{7} + 28 \, a^{5} b^{5} c^{2} d^{8} + 7 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 136 \, {\left (3003 \, b^{10} c^{8} d^{2} + 1716 \, a b^{9} c^{7} d^{3} + 924 \, a^{2} b^{8} c^{6} d^{4} + 462 \, a^{3} b^{7} c^{5} d^{5} + 210 \, a^{4} b^{6} c^{4} d^{6} + 84 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} + 7 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 17 \, {\left (5005 \, b^{10} c^{9} d + 3003 \, a b^{9} c^{8} d^{2} + 1716 \, a^{2} b^{8} c^{7} d^{3} + 924 \, a^{3} b^{7} c^{6} d^{4} + 462 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 84 \, a^{6} b^{4} c^{3} d^{7} + 28 \, a^{7} b^{3} c^{2} d^{8} + 7 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{136136 \, {\left (b^{28} x^{17} + 17 \, a b^{27} x^{16} + 136 \, a^{2} b^{26} x^{15} + 680 \, a^{3} b^{25} x^{14} + 2380 \, a^{4} b^{24} x^{13} + 6188 \, a^{5} b^{23} x^{12} + 12376 \, a^{6} b^{22} x^{11} + 19448 \, a^{7} b^{21} x^{10} + 24310 \, a^{8} b^{20} x^{9} + 24310 \, a^{9} b^{19} x^{8} + 19448 \, a^{10} b^{18} x^{7} + 12376 \, a^{11} b^{17} x^{6} + 6188 \, a^{12} b^{16} x^{5} + 2380 \, a^{13} b^{15} x^{4} + 680 \, a^{14} b^{14} x^{3} + 136 \, a^{15} b^{13} x^{2} + 17 \, a^{16} b^{12} x + a^{17} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^18,x, algorithm="maxima")

[Out]

-1/136136*(19448*b^10*d^10*x^10 + 8008*b^10*c^10 + 5005*a*b^9*c^9*d + 3003*a^2*b^8*c^8*d^2 + 1716*a^3*b^7*c^7*
d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^
8 + 7*a^9*b*c*d^9 + a^10*d^10 + 24310*(7*b^10*c*d^9 + a*b^9*d^10)*x^9 + 24310*(28*b^10*c^2*d^8 + 7*a*b^9*c*d^9
+ a^2*b^8*d^10)*x^8 + 19448*(84*b^10*c^3*d^7 + 28*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 12376
*(210*b^10*c^4*d^6 + 84*a*b^9*c^3*d^7 + 28*a^2*b^8*c^2*d^8 + 7*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6188*(462*b
^10*c^5*d^5 + 210*a*b^9*c^4*d^6 + 84*a^2*b^8*c^3*d^7 + 28*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^
5 + 2380*(924*b^10*c^6*d^4 + 462*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 + 84*a^3*b^7*c^3*d^7 + 28*a^4*b^6*c^2*d^8
+ 7*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 680*(1716*b^10*c^7*d^3 + 924*a*b^9*c^6*d^4 + 462*a^2*b^8*c^5*d^5 + 21
0*a^3*b^7*c^4*d^6 + 84*a^4*b^6*c^3*d^7 + 28*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 136*(3003*
b^10*c^8*d^2 + 1716*a*b^9*c^7*d^3 + 924*a^2*b^8*c^6*d^4 + 462*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 84*a^5*b
^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 + 7*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 17*(5005*b^10*c^9*d + 3003*a*b^9*c^8*d
^2 + 1716*a^2*b^8*c^7*d^3 + 924*a^3*b^7*c^6*d^4 + 462*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 84*a^6*b^4*c^3*d
^7 + 28*a^7*b^3*c^2*d^8 + 7*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^28*x^17 + 17*a*b^27*x^16 + 136*a^2*b^26*x^15 + 6
80*a^3*b^25*x^14 + 2380*a^4*b^24*x^13 + 6188*a^5*b^23*x^12 + 12376*a^6*b^22*x^11 + 19448*a^7*b^21*x^10 + 24310
*a^8*b^20*x^9 + 24310*a^9*b^19*x^8 + 19448*a^10*b^18*x^7 + 12376*a^11*b^17*x^6 + 6188*a^12*b^16*x^5 + 2380*a^1
3*b^15*x^4 + 680*a^14*b^14*x^3 + 136*a^15*b^13*x^2 + 17*a^16*b^12*x + a^17*b^11)

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mupad [B]  time = 0.66, size = 1142, normalized size = 5.36 \begin {gather*} -\frac {a^{10}\,d^{10}+7\,a^9\,b\,c\,d^9+17\,a^9\,b\,d^{10}\,x+28\,a^8\,b^2\,c^2\,d^8+119\,a^8\,b^2\,c\,d^9\,x+136\,a^8\,b^2\,d^{10}\,x^2+84\,a^7\,b^3\,c^3\,d^7+476\,a^7\,b^3\,c^2\,d^8\,x+952\,a^7\,b^3\,c\,d^9\,x^2+680\,a^7\,b^3\,d^{10}\,x^3+210\,a^6\,b^4\,c^4\,d^6+1428\,a^6\,b^4\,c^3\,d^7\,x+3808\,a^6\,b^4\,c^2\,d^8\,x^2+4760\,a^6\,b^4\,c\,d^9\,x^3+2380\,a^6\,b^4\,d^{10}\,x^4+462\,a^5\,b^5\,c^5\,d^5+3570\,a^5\,b^5\,c^4\,d^6\,x+11424\,a^5\,b^5\,c^3\,d^7\,x^2+19040\,a^5\,b^5\,c^2\,d^8\,x^3+16660\,a^5\,b^5\,c\,d^9\,x^4+6188\,a^5\,b^5\,d^{10}\,x^5+924\,a^4\,b^6\,c^6\,d^4+7854\,a^4\,b^6\,c^5\,d^5\,x+28560\,a^4\,b^6\,c^4\,d^6\,x^2+57120\,a^4\,b^6\,c^3\,d^7\,x^3+66640\,a^4\,b^6\,c^2\,d^8\,x^4+43316\,a^4\,b^6\,c\,d^9\,x^5+12376\,a^4\,b^6\,d^{10}\,x^6+1716\,a^3\,b^7\,c^7\,d^3+15708\,a^3\,b^7\,c^6\,d^4\,x+62832\,a^3\,b^7\,c^5\,d^5\,x^2+142800\,a^3\,b^7\,c^4\,d^6\,x^3+199920\,a^3\,b^7\,c^3\,d^7\,x^4+173264\,a^3\,b^7\,c^2\,d^8\,x^5+86632\,a^3\,b^7\,c\,d^9\,x^6+19448\,a^3\,b^7\,d^{10}\,x^7+3003\,a^2\,b^8\,c^8\,d^2+29172\,a^2\,b^8\,c^7\,d^3\,x+125664\,a^2\,b^8\,c^6\,d^4\,x^2+314160\,a^2\,b^8\,c^5\,d^5\,x^3+499800\,a^2\,b^8\,c^4\,d^6\,x^4+519792\,a^2\,b^8\,c^3\,d^7\,x^5+346528\,a^2\,b^8\,c^2\,d^8\,x^6+136136\,a^2\,b^8\,c\,d^9\,x^7+24310\,a^2\,b^8\,d^{10}\,x^8+5005\,a\,b^9\,c^9\,d+51051\,a\,b^9\,c^8\,d^2\,x+233376\,a\,b^9\,c^7\,d^3\,x^2+628320\,a\,b^9\,c^6\,d^4\,x^3+1099560\,a\,b^9\,c^5\,d^5\,x^4+1299480\,a\,b^9\,c^4\,d^6\,x^5+1039584\,a\,b^9\,c^3\,d^7\,x^6+544544\,a\,b^9\,c^2\,d^8\,x^7+170170\,a\,b^9\,c\,d^9\,x^8+24310\,a\,b^9\,d^{10}\,x^9+8008\,b^{10}\,c^{10}+85085\,b^{10}\,c^9\,d\,x+408408\,b^{10}\,c^8\,d^2\,x^2+1166880\,b^{10}\,c^7\,d^3\,x^3+2199120\,b^{10}\,c^6\,d^4\,x^4+2858856\,b^{10}\,c^5\,d^5\,x^5+2598960\,b^{10}\,c^4\,d^6\,x^6+1633632\,b^{10}\,c^3\,d^7\,x^7+680680\,b^{10}\,c^2\,d^8\,x^8+170170\,b^{10}\,c\,d^9\,x^9+19448\,b^{10}\,d^{10}\,x^{10}}{136136\,a^{17}\,b^{11}+2314312\,a^{16}\,b^{12}\,x+18514496\,a^{15}\,b^{13}\,x^2+92572480\,a^{14}\,b^{14}\,x^3+324003680\,a^{13}\,b^{15}\,x^4+842409568\,a^{12}\,b^{16}\,x^5+1684819136\,a^{11}\,b^{17}\,x^6+2647572928\,a^{10}\,b^{18}\,x^7+3309466160\,a^9\,b^{19}\,x^8+3309466160\,a^8\,b^{20}\,x^9+2647572928\,a^7\,b^{21}\,x^{10}[phone]\,a^6\,b^{22}\,x^{11}[phone]\,a^5\,b^{23}\,x^{12}[phone]\,a^4\,b^{24}\,x^{13}[phone]\,a^3\,b^{25}\,x^{14}[phone]\,a^2\,b^{26}\,x^{15}+2314312\,a\,b^{27}\,x^{16}+136136\,b^{28}\,x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^18,x)

[Out]

-(a^10*d^10 + 8008*b^10*c^10 + 19448*b^10*d^10*x^10 + 24310*a*b^9*d^10*x^9 + 170170*b^10*c*d^9*x^9 + 3003*a^2*
b^8*c^8*d^2 + 1716*a^3*b^7*c^7*d^3 + 924*a^4*b^6*c^6*d^4 + 462*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 84*a^7*
b^3*c^3*d^7 + 28*a^8*b^2*c^2*d^8 + 136*a^8*b^2*d^10*x^2 + 680*a^7*b^3*d^10*x^3 + 2380*a^6*b^4*d^10*x^4 + 6188*
a^5*b^5*d^10*x^5 + 12376*a^4*b^6*d^10*x^6 + 19448*a^3*b^7*d^10*x^7 + 24310*a^2*b^8*d^10*x^8 + 408408*b^10*c^8*
d^2*x^2 + 1166880*b^10*c^7*d^3*x^3 + 2199120*b^10*c^6*d^4*x^4 + 2858856*b^10*c^5*d^5*x^5 + 2598960*b^10*c^4*d^
6*x^6 + 1633632*b^10*c^3*d^7*x^7 + 680680*b^10*c^2*d^8*x^8 + 5005*a*b^9*c^9*d + 7*a^9*b*c*d^9 + 17*a^9*b*d^10*
x + 85085*b^10*c^9*d*x + 125664*a^2*b^8*c^6*d^4*x^2 + 62832*a^3*b^7*c^5*d^5*x^2 + 28560*a^4*b^6*c^4*d^6*x^2 +
11424*a^5*b^5*c^3*d^7*x^2 + 3808*a^6*b^4*c^2*d^8*x^2 + 314160*a^2*b^8*c^5*d^5*x^3 + 142800*a^3*b^7*c^4*d^6*x^3
+ 57120*a^4*b^6*c^3*d^7*x^3 + 19040*a^5*b^5*c^2*d^8*x^3 + 499800*a^2*b^8*c^4*d^6*x^4 + 199920*a^3*b^7*c^3*d^7
*x^4 + 66640*a^4*b^6*c^2*d^8*x^4 + 519792*a^2*b^8*c^3*d^7*x^5 + 173264*a^3*b^7*c^2*d^8*x^5 + 346528*a^2*b^8*c^
2*d^8*x^6 + 51051*a*b^9*c^8*d^2*x + 119*a^8*b^2*c*d^9*x + 170170*a*b^9*c*d^9*x^8 + 29172*a^2*b^8*c^7*d^3*x + 1
5708*a^3*b^7*c^6*d^4*x + 7854*a^4*b^6*c^5*d^5*x + 3570*a^5*b^5*c^4*d^6*x + 1428*a^6*b^4*c^3*d^7*x + 476*a^7*b^
3*c^2*d^8*x + 233376*a*b^9*c^7*d^3*x^2 + 952*a^7*b^3*c*d^9*x^2 + 628320*a*b^9*c^6*d^4*x^3 + 4760*a^6*b^4*c*d^9
*x^3 + 1099560*a*b^9*c^5*d^5*x^4 + 16660*a^5*b^5*c*d^9*x^4 + 1299480*a*b^9*c^4*d^6*x^5 + 43316*a^4*b^6*c*d^9*x
^5 + 1039584*a*b^9*c^3*d^7*x^6 + 86632*a^3*b^7*c*d^9*x^6 + 544544*a*b^9*c^2*d^8*x^7 + 136136*a^2*b^8*c*d^9*x^7
)/(136136*a^17*b^11 + 136136*b^28*x^17 + 2314312*a^16*b^12*x + 2314312*a*b^27*x^16 + 18514496*a^15*b^13*x^2 +
92572480*a^14*b^14*x^3 + 324003680*a^13*b^15*x^4 + 842409568*a^12*b^16*x^5 + 1684819136*a^11*b^17*x^6 + 264757
2928*a^10*b^18*x^7 + 3309466160*a^9*b^19*x^8 + 3309466160*a^8*b^20*x^9 + 2647572928*a^7*b^21*x^10 + 1684819136
*a^6*b^22*x^11 + 842409568*a^5*b^23*x^12 + 324003680*a^4*b^24*x^13 + 92572480*a^3*b^25*x^14 + 18514496*a^2*b^2
6*x^15)

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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)**18,x)

[Out]

Timed out

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