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

Optimal. Leaf size=257 \[ \frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac {d^{10} x}{b^{10}} \]

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Rubi [A]  time = 0.31, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} -\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}+\frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac {d^{10} x}{b^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^10,x]

[Out]

(d^10*x)/b^10 - (b*c - a*d)^10/(9*b^11*(a + b*x)^9) - (5*d*(b*c - a*d)^9)/(4*b^11*(a + b*x)^8) - (45*d^2*(b*c
- a*d)^8)/(7*b^11*(a + b*x)^7) - (20*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^6) - (42*d^4*(b*c - a*d)^6)/(b^11*(a +
b*x)^5) - (63*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^4) - (70*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^3) - (60*d^7*(b*
c - a*d)^3)/(b^11*(a + b*x)^2) - (45*d^8*(b*c - a*d)^2)/(b^11*(a + b*x)) + (10*d^9*(b*c - a*d)*Log[a + b*x])/b
^11

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [B]  time = 0.42, size = 708, normalized size = 2.75 \begin {gather*} -\frac {4861 a^{10} d^{10}+a^9 b d^9 (41229 d x-7129 c)+9 a^8 b^2 d^8 \left (140 c^2-6849 c d x+17064 d^2 x^2\right )+12 a^7 b^3 d^7 \left (35 c^3+945 c^2 d x-19602 c d^2 x^2+27342 d^3 x^3\right )+42 a^6 b^4 d^6 \left (5 c^4+90 c^3 d x+1080 c^2 d^2 x^2-12348 c d^3 x^3+10458 d^4 x^4\right )+126 a^5 b^5 d^5 \left (c^5+15 c^4 d x+120 c^3 d^2 x^2+840 c^2 d^3 x^3-5754 c d^4 x^4+2982 d^5 x^5\right )+42 a^4 b^6 d^4 \left (2 c^6+27 c^5 d x+180 c^4 d^2 x^2+840 c^3 d^3 x^3+3780 c^2 d^4 x^4-15750 c d^5 x^5+4704 d^6 x^6\right )+12 a^3 b^7 d^3 \left (5 c^7+63 c^6 d x+378 c^5 d^2 x^2+1470 c^4 d^3 x^3+4410 c^3 d^4 x^4+13230 c^2 d^5 x^5-32340 c d^6 x^6+4536 d^7 x^7\right )+9 a^2 b^8 d^2 \left (5 c^8+60 c^7 d x+336 c^6 d^2 x^2+1176 c^5 d^3 x^3+2940 c^4 d^4 x^4+5880 c^3 d^5 x^5+11760 c^2 d^6 x^6-15120 c d^7 x^7+252 d^8 x^8\right )+a b^9 d \left (35 c^9+405 c^8 d x+2160 c^7 d^2 x^2+7056 c^6 d^3 x^3+15876 c^5 d^4 x^4+26460 c^4 d^5 x^5+35280 c^3 d^6 x^6+45360 c^2 d^7 x^7-22680 c d^8 x^8-2268 d^9 x^9\right )+2520 d^9 (a+b x)^9 (a d-b c) \log (a+b x)+b^{10} \left (28 c^{10}+315 c^9 d x+1620 c^8 d^2 x^2+5040 c^7 d^3 x^3+10584 c^6 d^4 x^4+15876 c^5 d^5 x^5+17640 c^4 d^6 x^6+15120 c^3 d^7 x^7+11340 c^2 d^8 x^8-252 d^{10} x^{10}\right )}{252 b^{11} (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/252*(4861*a^10*d^10 + a^9*b*d^9*(-7129*c + 41229*d*x) + 9*a^8*b^2*d^8*(140*c^2 - 6849*c*d*x + 17064*d^2*x^2
) + 12*a^7*b^3*d^7*(35*c^3 + 945*c^2*d*x - 19602*c*d^2*x^2 + 27342*d^3*x^3) + 42*a^6*b^4*d^6*(5*c^4 + 90*c^3*d
*x + 1080*c^2*d^2*x^2 - 12348*c*d^3*x^3 + 10458*d^4*x^4) + 126*a^5*b^5*d^5*(c^5 + 15*c^4*d*x + 120*c^3*d^2*x^2
+ 840*c^2*d^3*x^3 - 5754*c*d^4*x^4 + 2982*d^5*x^5) + 42*a^4*b^6*d^4*(2*c^6 + 27*c^5*d*x + 180*c^4*d^2*x^2 + 8
40*c^3*d^3*x^3 + 3780*c^2*d^4*x^4 - 15750*c*d^5*x^5 + 4704*d^6*x^6) + 12*a^3*b^7*d^3*(5*c^7 + 63*c^6*d*x + 378
*c^5*d^2*x^2 + 1470*c^4*d^3*x^3 + 4410*c^3*d^4*x^4 + 13230*c^2*d^5*x^5 - 32340*c*d^6*x^6 + 4536*d^7*x^7) + 9*a
^2*b^8*d^2*(5*c^8 + 60*c^7*d*x + 336*c^6*d^2*x^2 + 1176*c^5*d^3*x^3 + 2940*c^4*d^4*x^4 + 5880*c^3*d^5*x^5 + 11
760*c^2*d^6*x^6 - 15120*c*d^7*x^7 + 252*d^8*x^8) + a*b^9*d*(35*c^9 + 405*c^8*d*x + 2160*c^7*d^2*x^2 + 7056*c^6
*d^3*x^3 + 15876*c^5*d^4*x^4 + 26460*c^4*d^5*x^5 + 35280*c^3*d^6*x^6 + 45360*c^2*d^7*x^7 - 22680*c*d^8*x^8 - 2
268*d^9*x^9) + b^10*(28*c^10 + 315*c^9*d*x + 1620*c^8*d^2*x^2 + 5040*c^7*d^3*x^3 + 10584*c^6*d^4*x^4 + 15876*c
^5*d^5*x^5 + 17640*c^4*d^6*x^6 + 15120*c^3*d^7*x^7 + 11340*c^2*d^8*x^8 - 252*d^10*x^10) + 2520*d^9*(-(b*c) + a
*d)*(a + b*x)^9*Log[a + b*x])/(b^11*(a + b*x)^9)

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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)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.23, size = 1216, normalized size = 4.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/252*(252*b^10*d^10*x^10 + 2268*a*b^9*d^10*x^9 - 28*b^10*c^10 - 35*a*b^9*c^9*d - 45*a^2*b^8*c^8*d^2 - 60*a^3*
b^7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 - 126*a^5*b^5*c^5*d^5 - 210*a^6*b^4*c^4*d^6 - 420*a^7*b^3*c^3*d^7 - 1260*a^8*
b^2*c^2*d^8 + 7129*a^9*b*c*d^9 - 4861*a^10*d^10 - 2268*(5*b^10*c^2*d^8 - 10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 -
3024*(5*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 - 45*a^2*b^8*c*d^9 + 18*a^3*b^7*d^10)*x^7 - 3528*(5*b^10*c^4*d^6 + 10*
a*b^9*c^3*d^7 + 30*a^2*b^8*c^2*d^8 - 110*a^3*b^7*c*d^9 + 56*a^4*b^6*d^10)*x^6 - 5292*(3*b^10*c^5*d^5 + 5*a*b^9
*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 71*a^5*b^5*d^10)*x^5 - 5292*(2*b^10*c
^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 8
3*a^6*b^4*d^10)*x^4 - 504*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a
^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 651*a^7*b^3*d^10)*x^3 - 108*(15*b^10*c^8*d^2 + 20*
a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b
^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1422*a^8*b^2*d^10)*x^2 - 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*a^2*b^8*c^
7*d^3 + 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^
2*d^8 - 6849*a^8*b^2*c*d^9 + 4581*a^9*b*d^10)*x + 2520*(a^9*b*c*d^9 - a^10*d^10 + (b^10*c*d^9 - a*b^9*d^10)*x^
9 + 9*(a*b^9*c*d^9 - a^2*b^8*d^10)*x^8 + 36*(a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(a^3*b^7*c*d^9 - a^4*b^6*d
^10)*x^6 + 126*(a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 126*(a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 84*(a^6*b^4*c*d^9
- a^7*b^3*d^10)*x^3 + 36*(a^7*b^3*c*d^9 - a^8*b^2*d^10)*x^2 + 9*(a^8*b^2*c*d^9 - a^9*b*d^10)*x)*log(b*x + a))
/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 + 84*a^6*b
^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11)

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giac [B]  time = 1.25, size = 867, normalized size = 3.37 \begin {gather*} \frac {d^{10} x}{b^{10}} + \frac {10 \, {\left (b c d^{9} - a d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {28 \, b^{10} c^{10} + 35 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 420 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} - 7129 \, a^{9} b c d^{9} + 4861 \, a^{10} d^{10} + 11340 \, {\left (b^{10} c^{2} d^{8} - 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 15120 \, {\left (b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} - 9 \, a^{2} b^{8} c d^{9} + 5 \, a^{3} b^{7} d^{10}\right )} x^{7} + 17640 \, {\left (b^{10} c^{4} d^{6} + 2 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 22 \, a^{3} b^{7} c d^{9} + 13 \, a^{4} b^{6} d^{10}\right )} x^{6} + 5292 \, {\left (3 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} - 125 \, a^{4} b^{6} c d^{9} + 77 \, a^{5} b^{5} d^{10}\right )} x^{5} + 5292 \, {\left (2 \, b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} - 137 \, a^{5} b^{5} c d^{9} + 87 \, a^{6} b^{4} d^{10}\right )} x^{4} + 504 \, {\left (10 \, b^{10} c^{7} d^{3} + 14 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 70 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} - 1029 \, a^{6} b^{4} c d^{9} + 669 \, a^{7} b^{3} d^{10}\right )} x^{3} + 108 \, {\left (15 \, b^{10} c^{8} d^{2} + 20 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 140 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} - 2178 \, a^{7} b^{3} c d^{9} + 1443 \, a^{8} b^{2} d^{10}\right )} x^{2} + 9 \, {\left (35 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 60 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 420 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} - 6849 \, a^{8} b^{2} c d^{9} + 4609 \, a^{9} b d^{10}\right )} x}{252 \, {\left (b x + a\right )}^{9} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^10*x/b^10 + 10*(b*c*d^9 - a*d^10)*log(abs(b*x + a))/b^11 - 1/252*(28*b^10*c^10 + 35*a*b^9*c^9*d + 45*a^2*b^8
*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 420*a^7*b^3*c
^3*d^7 + 1260*a^8*b^2*c^2*d^8 - 7129*a^9*b*c*d^9 + 4861*a^10*d^10 + 11340*(b^10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*
b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 - 9*a^2*b^8*c*d^9 + 5*a^3*b^7*d^10)*x^7 + 17640*(b^10*c^
4*d^6 + 2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 22*a^3*b^7*c*d^9 + 13*a^4*b^6*d^10)*x^6 + 5292*(3*b^10*c^5*d^5 +
5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 77*a^5*b^5*d^10)*x^5 + 5292*(
2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c
*d^9 + 87*a^6*b^4*d^10)*x^4 + 504*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^
6 + 70*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 669*a^7*b^3*d^10)*x^3 + 108*(15*b^10*c^8*d
^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 4
20*a^6*b^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1443*a^8*b^2*d^10)*x^2 + 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*a^
2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^
7*b^3*c^2*d^8 - 6849*a^8*b^2*c*d^9 + 4609*a^9*b*d^10)*x)/((b*x + a)^9*b^11)

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maple [B]  time = 0.02, size = 1266, normalized size = 4.93

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

60/b^11*d^10/(b*x+a)^2*a^3-60/b^8*d^7/(b*x+a)^2*c^3-10/b^11*d^10*ln(b*x+a)*a+10/b^10*d^9*ln(b*x+a)*c+63/b^11*d
^10/(b*x+a)^4*a^5-63/b^6*d^5/(b*x+a)^4*c^5-45/b^11*d^10/(b*x+a)*a^2-45/b^9*d^8/(b*x+a)*c^2+20/b^11*d^10/(b*x+a
)^6*a^7-20/b^4*d^3/(b*x+a)^6*c^7-45/7/b^11*d^10/(b*x+a)^7*a^8-45/7/b^3*d^2/(b*x+a)^7*c^8-1/9/b^11/(b*x+a)^9*a^
10*d^10+5/4/b^11*d^10/(b*x+a)^8*a^9-5/4/b^2*d/(b*x+a)^8*c^9-70/b^11*d^10/(b*x+a)^3*a^4-70/b^7*d^6/(b*x+a)^3*c^
4-42/b^11*d^10/(b*x+a)^5*a^6-42/b^5*d^4/(b*x+a)^5*c^6+d^10*x/b^10-1/9/b/(b*x+a)^9*c^10-140/b^10*d^9/(b*x+a)^6*
a^6*c+420/b^9*d^8/(b*x+a)^6*a^5*c^2+360/7/b^4*d^3/(b*x+a)^7*a*c^7+10/9/b^10/(b*x+a)^9*a^9*c*d^9-5/b^9/(b*x+a)^
9*a^8*c^2*d^8+40/3/b^8/(b*x+a)^9*a^7*c^3*d^7+700/b^7*d^6/(b*x+a)^6*a^3*c^4-420/b^6*d^5/(b*x+a)^6*a^2*c^5+140/b
^5*d^4/(b*x+a)^6*a*c^6-70/3/b^7/(b*x+a)^9*a^6*c^4*d^6+28/b^6/(b*x+a)^9*a^5*c^5*d^5-70/3/b^5/(b*x+a)^9*a^4*c^6*
d^4+40/3/b^4/(b*x+a)^9*a^3*c^7*d^3-5/b^3/(b*x+a)^9*a^2*c^8*d^2+10/9/b^2/(b*x+a)^9*a*c^9*d+252/b^10*d^9/(b*x+a)
^5*a^5*c-630/b^9*d^8/(b*x+a)^5*a^4*c^2+840/b^8*d^7/(b*x+a)^5*a^3*c^3-630/b^7*d^6/(b*x+a)^5*a^2*c^4+252/b^6*d^5
/(b*x+a)^5*a*c^5-180/b^10*d^9/(b*x+a)^2*a^2*c+180/b^9*d^8/(b*x+a)^2*a*c^2-315/b^10*d^9/(b*x+a)^4*a^4*c+630/b^9
*d^8/(b*x+a)^4*a^3*c^2-630/b^8*d^7/(b*x+a)^4*a^2*c^3+315/b^7*d^6/(b*x+a)^4*a*c^4+90/b^10*d^9/(b*x+a)*a*c+280/b
^8*d^7/(b*x+a)^3*a*c^3+360/7/b^10*d^9/(b*x+a)^7*a^7*c-180/b^9*d^8/(b*x+a)^7*a^6*c^2+360/b^8*d^7/(b*x+a)^7*a^5*
c^3-450/b^7*d^6/(b*x+a)^7*a^4*c^4+360/b^6*d^5/(b*x+a)^7*a^3*c^5-180/b^5*d^4/(b*x+a)^7*a^2*c^6-700/b^8*d^7/(b*x
+a)^6*a^4*c^3-45/4/b^10*d^9/(b*x+a)^8*a^8*c+45/b^9*d^8/(b*x+a)^8*a^7*c^2-105/b^8*d^7/(b*x+a)^8*a^6*c^3+315/2/b
^7*d^6/(b*x+a)^8*a^5*c^4-315/2/b^6*d^5/(b*x+a)^8*a^4*c^5+105/b^5*d^4/(b*x+a)^8*a^3*c^6-45/b^4*d^3/(b*x+a)^8*a^
2*c^7+45/4/b^3*d^2/(b*x+a)^8*a*c^8+280/b^10*d^9/(b*x+a)^3*a^3*c-420/b^9*d^8/(b*x+a)^3*a^2*c^2

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maxima [B]  time = 2.36, size = 957, normalized size = 3.72 \begin {gather*} \frac {d^{10} x}{b^{10}} - \frac {28 \, b^{10} c^{10} + 35 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 126 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 420 \, a^{7} b^{3} c^{3} d^{7} + 1260 \, a^{8} b^{2} c^{2} d^{8} - 7129 \, a^{9} b c d^{9} + 4861 \, a^{10} d^{10} + 11340 \, {\left (b^{10} c^{2} d^{8} - 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 15120 \, {\left (b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} - 9 \, a^{2} b^{8} c d^{9} + 5 \, a^{3} b^{7} d^{10}\right )} x^{7} + 17640 \, {\left (b^{10} c^{4} d^{6} + 2 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} - 22 \, a^{3} b^{7} c d^{9} + 13 \, a^{4} b^{6} d^{10}\right )} x^{6} + 5292 \, {\left (3 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 30 \, a^{3} b^{7} c^{2} d^{8} - 125 \, a^{4} b^{6} c d^{9} + 77 \, a^{5} b^{5} d^{10}\right )} x^{5} + 5292 \, {\left (2 \, b^{10} c^{6} d^{4} + 3 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 30 \, a^{4} b^{6} c^{2} d^{8} - 137 \, a^{5} b^{5} c d^{9} + 87 \, a^{6} b^{4} d^{10}\right )} x^{4} + 504 \, {\left (10 \, b^{10} c^{7} d^{3} + 14 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 70 \, a^{4} b^{6} c^{3} d^{7} + 210 \, a^{5} b^{5} c^{2} d^{8} - 1029 \, a^{6} b^{4} c d^{9} + 669 \, a^{7} b^{3} d^{10}\right )} x^{3} + 108 \, {\left (15 \, b^{10} c^{8} d^{2} + 20 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 42 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 140 \, a^{5} b^{5} c^{3} d^{7} + 420 \, a^{6} b^{4} c^{2} d^{8} - 2178 \, a^{7} b^{3} c d^{9} + 1443 \, a^{8} b^{2} d^{10}\right )} x^{2} + 9 \, {\left (35 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 60 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 126 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 420 \, a^{6} b^{4} c^{3} d^{7} + 1260 \, a^{7} b^{3} c^{2} d^{8} - 6849 \, a^{8} b^{2} c d^{9} + 4609 \, a^{9} b d^{10}\right )} x}{252 \, {\left (b^{20} x^{9} + 9 \, a b^{19} x^{8} + 36 \, a^{2} b^{18} x^{7} + 84 \, a^{3} b^{17} x^{6} + 126 \, a^{4} b^{16} x^{5} + 126 \, a^{5} b^{15} x^{4} + 84 \, a^{6} b^{14} x^{3} + 36 \, a^{7} b^{13} x^{2} + 9 \, a^{8} b^{12} x + a^{9} b^{11}\right )}} + \frac {10 \, {\left (b c d^{9} - a d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^10*x/b^10 - 1/252*(28*b^10*c^10 + 35*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*
d^4 + 126*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 - 7129*a^9*b*c*d^
9 + 4861*a^10*d^10 + 11340*(b^10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 + 3*a*b^9*c
^2*d^8 - 9*a^2*b^8*c*d^9 + 5*a^3*b^7*d^10)*x^7 + 17640*(b^10*c^4*d^6 + 2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 2
2*a^3*b^7*c*d^9 + 13*a^4*b^6*d^10)*x^6 + 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*
b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 77*a^5*b^5*d^10)*x^5 + 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^
4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 87*a^6*b^4*d^10)*x^4 + 504*(10*b^10*c^7*
d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 -
1029*a^6*b^4*c*d^9 + 669*a^7*b^3*d^10)*x^3 + 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42
*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1443*
a^8*b^2*d^10)*x^2 + 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 126*a^4*b^
6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 - 6849*a^8*b^2*c*d^9 + 4609*a^9*b
*d^10)*x)/(b^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16*x^5 + 126*a^5*b^15*x^4 +
84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12*x + a^9*b^11) + 10*(b*c*d^9 - a*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.50, size = 955, normalized size = 3.72 \begin {gather*} \frac {d^{10}\,x}{b^{10}}-\frac {\ln \left (a+b\,x\right )\,\left (10\,a\,d^{10}-10\,b\,c\,d^9\right )}{b^{11}}-\frac {x^4\,\left (1827\,a^6\,b^3\,d^{10}-2877\,a^5\,b^4\,c\,d^9+630\,a^4\,b^5\,c^2\,d^8+210\,a^3\,b^6\,c^3\,d^7+105\,a^2\,b^7\,c^4\,d^6+63\,a\,b^8\,c^5\,d^5+42\,b^9\,c^6\,d^4\right )+x^6\,\left (910\,a^4\,b^5\,d^{10}-1540\,a^3\,b^6\,c\,d^9+420\,a^2\,b^7\,c^2\,d^8+140\,a\,b^8\,c^3\,d^7+70\,b^9\,c^4\,d^6\right )+\frac {4861\,a^{10}\,d^{10}-7129\,a^9\,b\,c\,d^9+1260\,a^8\,b^2\,c^2\,d^8+420\,a^7\,b^3\,c^3\,d^7+210\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+60\,a^3\,b^7\,c^7\,d^3+45\,a^2\,b^8\,c^8\,d^2+35\,a\,b^9\,c^9\,d+28\,b^{10}\,c^{10}}{252\,b}+x\,\left (\frac {4609\,a^9\,d^{10}}{28}-\frac {6849\,a^8\,b\,c\,d^9}{28}+45\,a^7\,b^2\,c^2\,d^8+15\,a^6\,b^3\,c^3\,d^7+\frac {15\,a^5\,b^4\,c^4\,d^6}{2}+\frac {9\,a^4\,b^5\,c^5\,d^5}{2}+3\,a^3\,b^6\,c^6\,d^4+\frac {15\,a^2\,b^7\,c^7\,d^3}{7}+\frac {45\,a\,b^8\,c^8\,d^2}{28}+\frac {5\,b^9\,c^9\,d}{4}\right )+x^8\,\left (45\,a^2\,b^7\,d^{10}-90\,a\,b^8\,c\,d^9+45\,b^9\,c^2\,d^8\right )+x^3\,\left (1338\,a^7\,b^2\,d^{10}-2058\,a^6\,b^3\,c\,d^9+420\,a^5\,b^4\,c^2\,d^8+140\,a^4\,b^5\,c^3\,d^7+70\,a^3\,b^6\,c^4\,d^6+42\,a^2\,b^7\,c^5\,d^5+28\,a\,b^8\,c^6\,d^4+20\,b^9\,c^7\,d^3\right )+x^2\,\left (\frac {4329\,a^8\,b\,d^{10}}{7}-\frac {6534\,a^7\,b^2\,c\,d^9}{7}+180\,a^6\,b^3\,c^2\,d^8+60\,a^5\,b^4\,c^3\,d^7+30\,a^4\,b^5\,c^4\,d^6+18\,a^3\,b^6\,c^5\,d^5+12\,a^2\,b^7\,c^6\,d^4+\frac {60\,a\,b^8\,c^7\,d^3}{7}+\frac {45\,b^9\,c^8\,d^2}{7}\right )+x^5\,\left (1617\,a^5\,b^4\,d^{10}-2625\,a^4\,b^5\,c\,d^9+630\,a^3\,b^6\,c^2\,d^8+210\,a^2\,b^7\,c^3\,d^7+105\,a\,b^8\,c^4\,d^6+63\,b^9\,c^5\,d^5\right )+x^7\,\left (300\,a^3\,b^6\,d^{10}-540\,a^2\,b^7\,c\,d^9+180\,a\,b^8\,c^2\,d^8+60\,b^9\,c^3\,d^7\right )}{a^9\,b^{10}+9\,a^8\,b^{11}\,x+36\,a^7\,b^{12}\,x^2+84\,a^6\,b^{13}\,x^3+126\,a^5\,b^{14}\,x^4+126\,a^4\,b^{15}\,x^5+84\,a^3\,b^{16}\,x^6+36\,a^2\,b^{17}\,x^7+9\,a\,b^{18}\,x^8+b^{19}\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^10,x)

[Out]

(d^10*x)/b^10 - (log(a + b*x)*(10*a*d^10 - 10*b*c*d^9))/b^11 - (x^4*(1827*a^6*b^3*d^10 + 42*b^9*c^6*d^4 + 63*a
*b^8*c^5*d^5 - 2877*a^5*b^4*c*d^9 + 105*a^2*b^7*c^4*d^6 + 210*a^3*b^6*c^3*d^7 + 630*a^4*b^5*c^2*d^8) + x^6*(91
0*a^4*b^5*d^10 + 70*b^9*c^4*d^6 + 140*a*b^8*c^3*d^7 - 1540*a^3*b^6*c*d^9 + 420*a^2*b^7*c^2*d^8) + (4861*a^10*d
^10 + 28*b^10*c^10 + 45*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 210*
a^6*b^4*c^4*d^6 + 420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 35*a*b^9*c^9*d - 7129*a^9*b*c*d^9)/(252*b) + x*
((4609*a^9*d^10)/28 + (5*b^9*c^9*d)/4 + (45*a*b^8*c^8*d^2)/28 + (15*a^2*b^7*c^7*d^3)/7 + 3*a^3*b^6*c^6*d^4 + (
9*a^4*b^5*c^5*d^5)/2 + (15*a^5*b^4*c^4*d^6)/2 + 15*a^6*b^3*c^3*d^7 + 45*a^7*b^2*c^2*d^8 - (6849*a^8*b*c*d^9)/2
8) + x^8*(45*a^2*b^7*d^10 + 45*b^9*c^2*d^8 - 90*a*b^8*c*d^9) + x^3*(1338*a^7*b^2*d^10 + 20*b^9*c^7*d^3 + 28*a*
b^8*c^6*d^4 - 2058*a^6*b^3*c*d^9 + 42*a^2*b^7*c^5*d^5 + 70*a^3*b^6*c^4*d^6 + 140*a^4*b^5*c^3*d^7 + 420*a^5*b^4
*c^2*d^8) + x^2*((4329*a^8*b*d^10)/7 + (45*b^9*c^8*d^2)/7 + (60*a*b^8*c^7*d^3)/7 - (6534*a^7*b^2*c*d^9)/7 + 12
*a^2*b^7*c^6*d^4 + 18*a^3*b^6*c^5*d^5 + 30*a^4*b^5*c^4*d^6 + 60*a^5*b^4*c^3*d^7 + 180*a^6*b^3*c^2*d^8) + x^5*(
1617*a^5*b^4*d^10 + 63*b^9*c^5*d^5 + 105*a*b^8*c^4*d^6 - 2625*a^4*b^5*c*d^9 + 210*a^2*b^7*c^3*d^7 + 630*a^3*b^
6*c^2*d^8) + x^7*(300*a^3*b^6*d^10 + 60*b^9*c^3*d^7 + 180*a*b^8*c^2*d^8 - 540*a^2*b^7*c*d^9))/(a^9*b^10 + b^19
*x^9 + 9*a^8*b^11*x + 9*a*b^18*x^8 + 36*a^7*b^12*x^2 + 84*a^6*b^13*x^3 + 126*a^5*b^14*x^4 + 126*a^4*b^15*x^5 +
84*a^3*b^16*x^6 + 36*a^2*b^17*x^7)

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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)**10,x)

[Out]

Timed out

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