3.13.14 \(\int \frac {(c+d x)^{10}}{(a+b x)^9} \, dx\)

Optimal. Leaf size=258 \[ \frac {45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}-\frac {120 d^7 (b c-a d)^3}{b^{11} (a+b x)}-\frac {105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac {84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac {24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac {15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}-\frac {10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac {(b c-a d)^{10}}{8 b^{11} (a+b x)^8}+\frac {d^9 x (10 b c-9 a d)}{b^{10}}+\frac {d^{10} x^2}{2 b^9} \]

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Rubi [A]  time = 0.34, antiderivative size = 258, 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 {d^9 x (10 b c-9 a d)}{b^{10}}-\frac {120 d^7 (b c-a d)^3}{b^{11} (a+b x)}-\frac {105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac {84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac {24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac {15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}+\frac {45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}-\frac {10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac {(b c-a d)^{10}}{8 b^{11} (a+b x)^8}+\frac {d^{10} x^2}{2 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^9,x]

[Out]

(d^9*(10*b*c - 9*a*d)*x)/b^10 + (d^10*x^2)/(2*b^9) - (b*c - a*d)^10/(8*b^11*(a + b*x)^8) - (10*d*(b*c - a*d)^9
)/(7*b^11*(a + b*x)^7) - (15*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^6) - (24*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^
5) - (105*d^4*(b*c - a*d)^6)/(2*b^11*(a + b*x)^4) - (84*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^3) - (105*d^6*(b*c
- a*d)^4)/(b^11*(a + b*x)^2) - (120*d^7*(b*c - a*d)^3)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*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)^9} \, dx &=\int \left (\frac {d^9 (10 b c-9 a d)}{b^{10}}+\frac {d^{10} x}{b^9}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^9}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^8}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^7}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^6}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^5}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^4}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^3}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^2}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)}\right ) \, dx\\ &=\frac {d^9 (10 b c-9 a d) x}{b^{10}}+\frac {d^{10} x^2}{2 b^9}-\frac {(b c-a d)^{10}}{8 b^{11} (a+b x)^8}-\frac {10 d (b c-a d)^9}{7 b^{11} (a+b x)^7}-\frac {15 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^6}-\frac {24 d^3 (b c-a d)^7}{b^{11} (a+b x)^5}-\frac {105 d^4 (b c-a d)^6}{2 b^{11} (a+b x)^4}-\frac {84 d^5 (b c-a d)^5}{b^{11} (a+b x)^3}-\frac {105 d^6 (b c-a d)^4}{b^{11} (a+b x)^2}-\frac {120 d^7 (b c-a d)^3}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [B]  time = 0.32, size = 712, normalized size = 2.76 \begin {gather*} \frac {3601 a^{10} d^{10}+2 a^9 b d^9 (13144 d x-4609 c)+a^8 b^2 d^8 \left (6849 c^2-68704 c d x+81928 d^2 x^2\right )+8 a^7 b^3 d^7 \left (-105 c^3+6534 c^2 d x-27538 c d^2 x^2+17542 d^3 x^3\right )+14 a^6 b^4 d^6 \left (-15 c^4-480 c^3 d x+12348 c^2 d^2 x^2-28112 c d^3 x^3+10010 d^4 x^4\right )-28 a^5 b^5 d^5 \left (3 c^5+60 c^4 d x+840 c^3 d^2 x^2-11508 c^2 d^3 x^3+15050 c d^4 x^4-2744 d^5 x^5\right )-14 a^4 b^6 d^4 \left (3 c^6+48 c^5 d x+420 c^4 d^2 x^2+3360 c^3 d^3 x^3-26250 c^2 d^4 x^4+19040 c d^5 x^5-1064 d^6 x^6\right )-8 a^3 b^7 d^3 \left (3 c^7+42 c^6 d x+294 c^5 d^2 x^2+1470 c^4 d^3 x^3+7350 c^3 d^4 x^4-32340 c^2 d^5 x^5+10780 c d^6 x^6+728 d^7 x^7\right )-a^2 b^8 d^2 \left (15 c^8+192 c^7 d x+1176 c^6 d^2 x^2+4704 c^5 d^3 x^3+14700 c^4 d^4 x^4+47040 c^3 d^5 x^5-105840 c^2 d^6 x^6+4480 c d^7 x^7+3248 d^8 x^8\right )-2 a b^9 d \left (5 c^9+60 c^8 d x+336 c^7 d^2 x^2+1176 c^6 d^3 x^3+2940 c^5 d^4 x^4+5880 c^4 d^5 x^5+11760 c^3 d^6 x^6-10080 c^2 d^7 x^7-2240 c d^8 x^8+140 d^9 x^9\right )+2520 d^8 (a+b x)^8 (b c-a d)^2 \log (a+b x)-\left (b^{10} \left (7 c^{10}+80 c^9 d x+420 c^8 d^2 x^2+1344 c^7 d^3 x^3+2940 c^6 d^4 x^4+4704 c^5 d^5 x^5+5880 c^4 d^6 x^6+6720 c^3 d^7 x^7-560 c d^9 x^9-28 d^{10} x^{10}\right )\right )}{56 b^{11} (a+b x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^9,x]

[Out]

(3601*a^10*d^10 + 2*a^9*b*d^9*(-4609*c + 13144*d*x) + a^8*b^2*d^8*(6849*c^2 - 68704*c*d*x + 81928*d^2*x^2) + 8
*a^7*b^3*d^7*(-105*c^3 + 6534*c^2*d*x - 27538*c*d^2*x^2 + 17542*d^3*x^3) + 14*a^6*b^4*d^6*(-15*c^4 - 480*c^3*d
*x + 12348*c^2*d^2*x^2 - 28112*c*d^3*x^3 + 10010*d^4*x^4) - 28*a^5*b^5*d^5*(3*c^5 + 60*c^4*d*x + 840*c^3*d^2*x
^2 - 11508*c^2*d^3*x^3 + 15050*c*d^4*x^4 - 2744*d^5*x^5) - 14*a^4*b^6*d^4*(3*c^6 + 48*c^5*d*x + 420*c^4*d^2*x^
2 + 3360*c^3*d^3*x^3 - 26250*c^2*d^4*x^4 + 19040*c*d^5*x^5 - 1064*d^6*x^6) - 8*a^3*b^7*d^3*(3*c^7 + 42*c^6*d*x
+ 294*c^5*d^2*x^2 + 1470*c^4*d^3*x^3 + 7350*c^3*d^4*x^4 - 32340*c^2*d^5*x^5 + 10780*c*d^6*x^6 + 728*d^7*x^7)
- a^2*b^8*d^2*(15*c^8 + 192*c^7*d*x + 1176*c^6*d^2*x^2 + 4704*c^5*d^3*x^3 + 14700*c^4*d^4*x^4 + 47040*c^3*d^5*
x^5 - 105840*c^2*d^6*x^6 + 4480*c*d^7*x^7 + 3248*d^8*x^8) - 2*a*b^9*d*(5*c^9 + 60*c^8*d*x + 336*c^7*d^2*x^2 +
1176*c^6*d^3*x^3 + 2940*c^5*d^4*x^4 + 5880*c^4*d^5*x^5 + 11760*c^3*d^6*x^6 - 10080*c^2*d^7*x^7 - 2240*c*d^8*x^
8 + 140*d^9*x^9) - b^10*(7*c^10 + 80*c^9*d*x + 420*c^8*d^2*x^2 + 1344*c^7*d^3*x^3 + 2940*c^6*d^4*x^4 + 4704*c^
5*d^5*x^5 + 5880*c^4*d^6*x^6 + 6720*c^3*d^7*x^7 - 560*c*d^9*x^9 - 28*d^10*x^10) + 2520*d^8*(b*c - a*d)^2*(a +
b*x)^8*Log[a + b*x])/(56*b^11*(a + b*x)^8)

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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)^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^9,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^9, x]

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fricas [B]  time = 1.39, size = 1296, normalized size = 5.02

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^9,x, algorithm="fricas")

[Out]

1/56*(28*b^10*d^10*x^10 - 7*b^10*c^10 - 10*a*b^9*c^9*d - 15*a^2*b^8*c^8*d^2 - 24*a^3*b^7*c^7*d^3 - 42*a^4*b^6*
c^6*d^4 - 84*a^5*b^5*c^5*d^5 - 210*a^6*b^4*c^4*d^6 - 840*a^7*b^3*c^3*d^7 + 6849*a^8*b^2*c^2*d^8 - 9218*a^9*b*c
*d^9 + 3601*a^10*d^10 + 280*(2*b^10*c*d^9 - a*b^9*d^10)*x^9 + 112*(40*a*b^9*c*d^9 - 29*a^2*b^8*d^10)*x^8 - 448
*(15*b^10*c^3*d^7 - 45*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + 13*a^3*b^7*d^10)*x^7 - 392*(15*b^10*c^4*d^6 + 60*a*b
^9*c^3*d^7 - 270*a^2*b^8*c^2*d^8 + 220*a^3*b^7*c*d^9 - 38*a^4*b^6*d^10)*x^6 - 784*(6*b^10*c^5*d^5 + 15*a*b^9*c
^4*d^6 + 60*a^2*b^8*c^3*d^7 - 330*a^3*b^7*c^2*d^8 + 340*a^4*b^6*c*d^9 - 98*a^5*b^5*d^10)*x^5 - 980*(3*b^10*c^6
*d^4 + 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 60*a^3*b^7*c^3*d^7 - 375*a^4*b^6*c^2*d^8 + 430*a^5*b^5*c*d^9 - 1
43*a^6*b^4*d^10)*x^4 - 112*(12*b^10*c^7*d^3 + 21*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 + 42
0*a^4*b^6*c^3*d^7 - 2877*a^5*b^5*c^2*d^8 + 3514*a^6*b^4*c*d^9 - 1253*a^7*b^3*d^10)*x^3 - 28*(15*b^10*c^8*d^2 +
24*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 84*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 840*a^5*b^5*c^3*d^7 - 6174
*a^6*b^4*c^2*d^8 + 7868*a^7*b^3*c*d^9 - 2926*a^8*b^2*d^10)*x^2 - 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2*
b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*b
^3*c^2*d^8 + 8588*a^8*b^2*c*d^9 - 3286*a^9*b*d^10)*x + 2520*(a^8*b^2*c^2*d^8 - 2*a^9*b*c*d^9 + a^10*d^10 + (b^
10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 8*(a*b^9*c^2*d^8 - 2*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 28*(
a^2*b^8*c^2*d^8 - 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 56*(a^3*b^7*c^2*d^8 - 2*a^4*b^6*c*d^9 + a^5*b^5*d^10)*
x^5 + 70*(a^4*b^6*c^2*d^8 - 2*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 56*(a^5*b^5*c^2*d^8 - 2*a^6*b^4*c*d^9 + a^7*
b^3*d^10)*x^3 + 28*(a^6*b^4*c^2*d^8 - 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 8*(a^7*b^3*c^2*d^8 - 2*a^8*b^2*c*d
^9 + a^9*b*d^10)*x)*log(b*x + a))/(b^19*x^8 + 8*a*b^18*x^7 + 28*a^2*b^17*x^6 + 56*a^3*b^16*x^5 + 70*a^4*b^15*x
^4 + 56*a^5*b^14*x^3 + 28*a^6*b^13*x^2 + 8*a^7*b^12*x + a^8*b^11)

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giac [B]  time = 1.29, size = 871, normalized size = 3.38 \begin {gather*} \frac {45 \, {\left (b^{2} c^{2} d^{8} - 2 \, a b c d^{9} + a^{2} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} + \frac {b^{9} d^{10} x^{2} + 20 \, b^{9} c d^{9} x - 18 \, a b^{8} d^{10} x}{2 \, b^{18}} - \frac {7 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 84 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} - 6849 \, a^{8} b^{2} c^{2} d^{8} + 9218 \, a^{9} b c d^{9} - 3601 \, a^{10} d^{10} + 6720 \, {\left (b^{10} c^{3} d^{7} - 3 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} - a^{3} b^{7} d^{10}\right )} x^{7} + 5880 \, {\left (b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} - 18 \, a^{2} b^{8} c^{2} d^{8} + 20 \, a^{3} b^{7} c d^{9} - 7 \, a^{4} b^{6} d^{10}\right )} x^{6} + 2352 \, {\left (2 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} - 110 \, a^{3} b^{7} c^{2} d^{8} + 130 \, a^{4} b^{6} c d^{9} - 47 \, a^{5} b^{5} d^{10}\right )} x^{5} + 2940 \, {\left (b^{10} c^{6} d^{4} + 2 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} - 125 \, a^{4} b^{6} c^{2} d^{8} + 154 \, a^{5} b^{5} c d^{9} - 57 \, a^{6} b^{4} d^{10}\right )} x^{4} + 336 \, {\left (4 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 14 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} - 959 \, a^{5} b^{5} c^{2} d^{8} + 1218 \, a^{6} b^{4} c d^{9} - 459 \, a^{7} b^{3} d^{10}\right )} x^{3} + 84 \, {\left (5 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 28 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} - 2058 \, a^{6} b^{4} c^{2} d^{8} + 2676 \, a^{7} b^{3} c d^{9} - 1023 \, a^{8} b^{2} d^{10}\right )} x^{2} + 8 \, {\left (10 \, b^{10} c^{9} d + 15 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 84 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} - 6534 \, a^{7} b^{3} c^{2} d^{8} + 8658 \, a^{8} b^{2} c d^{9} - 3349 \, a^{9} b d^{10}\right )} x}{56 \, {\left (b x + a\right )}^{8} b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^9,x, algorithm="giac")

[Out]

45*(b^2*c^2*d^8 - 2*a*b*c*d^9 + a^2*d^10)*log(abs(b*x + a))/b^11 + 1/2*(b^9*d^10*x^2 + 20*b^9*c*d^9*x - 18*a*b
^8*d^10*x)/b^18 - 1/56*(7*b^10*c^10 + 10*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^
6*d^4 + 84*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 - 6849*a^8*b^2*c^2*d^8 + 9218*a^9*b*c*d
^9 - 3601*a^10*d^10 + 6720*(b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 5880*(b^10*
c^4*d^6 + 4*a*b^9*c^3*d^7 - 18*a^2*b^8*c^2*d^8 + 20*a^3*b^7*c*d^9 - 7*a^4*b^6*d^10)*x^6 + 2352*(2*b^10*c^5*d^5
+ 5*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 - 110*a^3*b^7*c^2*d^8 + 130*a^4*b^6*c*d^9 - 47*a^5*b^5*d^10)*x^5 + 294
0*(b^10*c^6*d^4 + 2*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 - 125*a^4*b^6*c^2*d^8 + 154*a^5*b^5
*c*d^9 - 57*a^6*b^4*d^10)*x^4 + 336*(4*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 14*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^
6 + 140*a^4*b^6*c^3*d^7 - 959*a^5*b^5*c^2*d^8 + 1218*a^6*b^4*c*d^9 - 459*a^7*b^3*d^10)*x^3 + 84*(5*b^10*c^8*d^
2 + 8*a*b^9*c^7*d^3 + 14*a^2*b^8*c^6*d^4 + 28*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 - 205
8*a^6*b^4*c^2*d^8 + 2676*a^7*b^3*c*d^9 - 1023*a^8*b^2*d^10)*x^2 + 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2
*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*
b^3*c^2*d^8 + 8658*a^8*b^2*c*d^9 - 3349*a^9*b*d^10)*x)/((b*x + a)^8*b^11)

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maple [B]  time = 0.02, size = 1256, normalized size = 4.87

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^9,x)

[Out]

1/2*d^10*x^2/b^9-9*d^10/b^10*a*x+10*d^9/b^9*x*c-15/2/b^11*d^10/(b*x+a)^6*a^8-15/2/b^3*d^2/(b*x+a)^6*c^8-105/b^
7*d^6/(b*x+a)^2*c^4+45/b^11*d^10*ln(b*x+a)*a^2+45/b^9*d^8*ln(b*x+a)*c^2-105/2/b^11*d^10/(b*x+a)^4*a^6-105/2/b^
5*d^4/(b*x+a)^4*c^6+120/b^11*d^10/(b*x+a)*a^3-120/b^8*d^7/(b*x+a)*c^3-1/8/b^11/(b*x+a)^8*a^10*d^10+84/b^11*d^1
0/(b*x+a)^3*a^5-84/b^6*d^5/(b*x+a)^3*c^5+10/7/b^11*d^10/(b*x+a)^7*a^9-10/7/b^2*d/(b*x+a)^7*c^9+24/b^11*d^10/(b
*x+a)^5*a^7-24/b^4*d^3/(b*x+a)^5*c^7-105/b^11*d^10/(b*x+a)^2*a^4-1/8/b/(b*x+a)^8*c^10-360/7/b^4*d^3/(b*x+a)^7*
a^2*c^7+90/7/b^3*d^2/(b*x+a)^7*a*c^8-168/b^10*d^9/(b*x+a)^5*a^6*c+504/b^9*d^8/(b*x+a)^5*a^5*c^2+180/b^7*d^6/(b
*x+a)^7*a^5*c^4-180/b^6*d^5/(b*x+a)^7*a^4*c^5+120/b^5*d^4/(b*x+a)^7*a^3*c^6-840/b^8*d^7/(b*x+a)^5*a^4*c^3+840/
b^7*d^6/(b*x+a)^5*a^3*c^4-504/b^6*d^5/(b*x+a)^5*a^2*c^5-210/b^9*d^8/(b*x+a)^6*a^6*c^2+420/b^8*d^7/(b*x+a)^6*a^
5*c^3-525/b^7*d^6/(b*x+a)^6*a^4*c^4+420/b^6*d^5/(b*x+a)^6*a^3*c^5-210/b^5*d^4/(b*x+a)^6*a^2*c^6+60/b^4*d^3/(b*
x+a)^6*a*c^7+168/b^5*d^4/(b*x+a)^5*a*c^6+315/b^10*d^9/(b*x+a)^4*a^5*c-1575/2/b^9*d^8/(b*x+a)^4*a^4*c^2+1050/b^
8*d^7/(b*x+a)^4*a^3*c^3-1575/2/b^7*d^6/(b*x+a)^4*a^2*c^4+315/b^6*d^5/(b*x+a)^4*a*c^5-360/b^10*d^9/(b*x+a)*a^2*
c+360/b^9*d^8/(b*x+a)*a*c^2+60/b^10*d^9/(b*x+a)^6*a^7*c+420/b^10*d^9/(b*x+a)^2*a^3*c-630/b^9*d^8/(b*x+a)^2*a^2
*c^2+420/b^8*d^7/(b*x+a)^2*a*c^3-90/b^10*d^9*ln(b*x+a)*a*c+5/4/b^10/(b*x+a)^8*a^9*c*d^9-45/8/b^9/(b*x+a)^8*a^8
*c^2*d^8+15/b^8/(b*x+a)^8*a^7*c^3*d^7-105/4/b^7/(b*x+a)^8*a^6*c^4*d^6+63/2/b^6/(b*x+a)^8*a^5*c^5*d^5-105/4/b^5
/(b*x+a)^8*a^4*c^6*d^4+15/b^4/(b*x+a)^8*a^3*c^7*d^3-45/8/b^3/(b*x+a)^8*a^2*c^8*d^2+5/4/b^2/(b*x+a)^8*a*c^9*d-4
20/b^10*d^9/(b*x+a)^3*a^4*c+840/b^9*d^8/(b*x+a)^3*a^3*c^2-840/b^8*d^7/(b*x+a)^3*a^2*c^3+420/b^7*d^6/(b*x+a)^3*
a*c^4-90/7/b^10*d^9/(b*x+a)^7*a^8*c+360/7/b^9*d^8/(b*x+a)^7*a^7*c^2-120/b^8*d^7/(b*x+a)^7*a^6*c^3

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maxima [B]  time = 2.58, size = 945, normalized size = 3.66 \begin {gather*} -\frac {7 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 42 \, a^{4} b^{6} c^{6} d^{4} + 84 \, a^{5} b^{5} c^{5} d^{5} + 210 \, a^{6} b^{4} c^{4} d^{6} + 840 \, a^{7} b^{3} c^{3} d^{7} - 6849 \, a^{8} b^{2} c^{2} d^{8} + 9218 \, a^{9} b c d^{9} - 3601 \, a^{10} d^{10} + 6720 \, {\left (b^{10} c^{3} d^{7} - 3 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} - a^{3} b^{7} d^{10}\right )} x^{7} + 5880 \, {\left (b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} - 18 \, a^{2} b^{8} c^{2} d^{8} + 20 \, a^{3} b^{7} c d^{9} - 7 \, a^{4} b^{6} d^{10}\right )} x^{6} + 2352 \, {\left (2 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 20 \, a^{2} b^{8} c^{3} d^{7} - 110 \, a^{3} b^{7} c^{2} d^{8} + 130 \, a^{4} b^{6} c d^{9} - 47 \, a^{5} b^{5} d^{10}\right )} x^{5} + 2940 \, {\left (b^{10} c^{6} d^{4} + 2 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 20 \, a^{3} b^{7} c^{3} d^{7} - 125 \, a^{4} b^{6} c^{2} d^{8} + 154 \, a^{5} b^{5} c d^{9} - 57 \, a^{6} b^{4} d^{10}\right )} x^{4} + 336 \, {\left (4 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 14 \, a^{2} b^{8} c^{5} d^{5} + 35 \, a^{3} b^{7} c^{4} d^{6} + 140 \, a^{4} b^{6} c^{3} d^{7} - 959 \, a^{5} b^{5} c^{2} d^{8} + 1218 \, a^{6} b^{4} c d^{9} - 459 \, a^{7} b^{3} d^{10}\right )} x^{3} + 84 \, {\left (5 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 14 \, a^{2} b^{8} c^{6} d^{4} + 28 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} + 280 \, a^{5} b^{5} c^{3} d^{7} - 2058 \, a^{6} b^{4} c^{2} d^{8} + 2676 \, a^{7} b^{3} c d^{9} - 1023 \, a^{8} b^{2} d^{10}\right )} x^{2} + 8 \, {\left (10 \, b^{10} c^{9} d + 15 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 42 \, a^{3} b^{7} c^{6} d^{4} + 84 \, a^{4} b^{6} c^{5} d^{5} + 210 \, a^{5} b^{5} c^{4} d^{6} + 840 \, a^{6} b^{4} c^{3} d^{7} - 6534 \, a^{7} b^{3} c^{2} d^{8} + 8658 \, a^{8} b^{2} c d^{9} - 3349 \, a^{9} b d^{10}\right )} x}{56 \, {\left (b^{19} x^{8} + 8 \, a b^{18} x^{7} + 28 \, a^{2} b^{17} x^{6} + 56 \, a^{3} b^{16} x^{5} + 70 \, a^{4} b^{15} x^{4} + 56 \, a^{5} b^{14} x^{3} + 28 \, a^{6} b^{13} x^{2} + 8 \, a^{7} b^{12} x + a^{8} b^{11}\right )}} + \frac {b d^{10} x^{2} + 2 \, {\left (10 \, b c d^{9} - 9 \, a d^{10}\right )} x}{2 \, b^{10}} + \frac {45 \, {\left (b^{2} c^{2} d^{8} - 2 \, a b c d^{9} + a^{2} 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)^9,x, algorithm="maxima")

[Out]

-1/56*(7*b^10*c^10 + 10*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 84*a^5*b^
5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 - 6849*a^8*b^2*c^2*d^8 + 9218*a^9*b*c*d^9 - 3601*a^10*d^
10 + 6720*(b^10*c^3*d^7 - 3*a*b^9*c^2*d^8 + 3*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 5880*(b^10*c^4*d^6 + 4*a*b^9
*c^3*d^7 - 18*a^2*b^8*c^2*d^8 + 20*a^3*b^7*c*d^9 - 7*a^4*b^6*d^10)*x^6 + 2352*(2*b^10*c^5*d^5 + 5*a*b^9*c^4*d^
6 + 20*a^2*b^8*c^3*d^7 - 110*a^3*b^7*c^2*d^8 + 130*a^4*b^6*c*d^9 - 47*a^5*b^5*d^10)*x^5 + 2940*(b^10*c^6*d^4 +
2*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 - 125*a^4*b^6*c^2*d^8 + 154*a^5*b^5*c*d^9 - 57*a^6*b
^4*d^10)*x^4 + 336*(4*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 14*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 140*a^4*b^6*c
^3*d^7 - 959*a^5*b^5*c^2*d^8 + 1218*a^6*b^4*c*d^9 - 459*a^7*b^3*d^10)*x^3 + 84*(5*b^10*c^8*d^2 + 8*a*b^9*c^7*d
^3 + 14*a^2*b^8*c^6*d^4 + 28*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 - 2058*a^6*b^4*c^2*d^8
+ 2676*a^7*b^3*c*d^9 - 1023*a^8*b^2*d^10)*x^2 + 8*(10*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 42
*a^3*b^7*c^6*d^4 + 84*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 - 6534*a^7*b^3*c^2*d^8 + 865
8*a^8*b^2*c*d^9 - 3349*a^9*b*d^10)*x)/(b^19*x^8 + 8*a*b^18*x^7 + 28*a^2*b^17*x^6 + 56*a^3*b^16*x^5 + 70*a^4*b^
15*x^4 + 56*a^5*b^14*x^3 + 28*a^6*b^13*x^2 + 8*a^7*b^12*x + a^8*b^11) + 1/2*(b*d^10*x^2 + 2*(10*b*c*d^9 - 9*a*
d^10)*x)/b^10 + 45*(b^2*c^2*d^8 - 2*a*b*c*d^9 + a^2*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.26, size = 946, normalized size = 3.67 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (45\,a^2\,d^{10}-90\,a\,b\,c\,d^9+45\,b^2\,c^2\,d^8\right )}{b^{11}}-\frac {x^4\,\left (-\frac {5985\,a^6\,b^3\,d^{10}}{2}+8085\,a^5\,b^4\,c\,d^9-\frac {13125\,a^4\,b^5\,c^2\,d^8}{2}+1050\,a^3\,b^6\,c^3\,d^7+\frac {525\,a^2\,b^7\,c^4\,d^6}{2}+105\,a\,b^8\,c^5\,d^5+\frac {105\,b^9\,c^6\,d^4}{2}\right )+x^6\,\left (-735\,a^4\,b^5\,d^{10}+2100\,a^3\,b^6\,c\,d^9-1890\,a^2\,b^7\,c^2\,d^8+420\,a\,b^8\,c^3\,d^7+105\,b^9\,c^4\,d^6\right )+\frac {-3601\,a^{10}\,d^{10}+9218\,a^9\,b\,c\,d^9-6849\,a^8\,b^2\,c^2\,d^8+840\,a^7\,b^3\,c^3\,d^7+210\,a^6\,b^4\,c^4\,d^6+84\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+24\,a^3\,b^7\,c^7\,d^3+15\,a^2\,b^8\,c^8\,d^2+10\,a\,b^9\,c^9\,d+7\,b^{10}\,c^{10}}{56\,b}+x\,\left (-\frac {3349\,a^9\,d^{10}}{7}+\frac {8658\,a^8\,b\,c\,d^9}{7}-\frac {6534\,a^7\,b^2\,c^2\,d^8}{7}+120\,a^6\,b^3\,c^3\,d^7+30\,a^5\,b^4\,c^4\,d^6+12\,a^4\,b^5\,c^5\,d^5+6\,a^3\,b^6\,c^6\,d^4+\frac {24\,a^2\,b^7\,c^7\,d^3}{7}+\frac {15\,a\,b^8\,c^8\,d^2}{7}+\frac {10\,b^9\,c^9\,d}{7}\right )+x^3\,\left (-2754\,a^7\,b^2\,d^{10}+7308\,a^6\,b^3\,c\,d^9-5754\,a^5\,b^4\,c^2\,d^8+840\,a^4\,b^5\,c^3\,d^7+210\,a^3\,b^6\,c^4\,d^6+84\,a^2\,b^7\,c^5\,d^5+42\,a\,b^8\,c^6\,d^4+24\,b^9\,c^7\,d^3\right )+x^2\,\left (-\frac {3069\,a^8\,b\,d^{10}}{2}+4014\,a^7\,b^2\,c\,d^9-3087\,a^6\,b^3\,c^2\,d^8+420\,a^5\,b^4\,c^3\,d^7+105\,a^4\,b^5\,c^4\,d^6+42\,a^3\,b^6\,c^5\,d^5+21\,a^2\,b^7\,c^6\,d^4+12\,a\,b^8\,c^7\,d^3+\frac {15\,b^9\,c^8\,d^2}{2}\right )+x^5\,\left (-1974\,a^5\,b^4\,d^{10}+5460\,a^4\,b^5\,c\,d^9-4620\,a^3\,b^6\,c^2\,d^8+840\,a^2\,b^7\,c^3\,d^7+210\,a\,b^8\,c^4\,d^6+84\,b^9\,c^5\,d^5\right )-x^7\,\left (120\,a^3\,b^6\,d^{10}-360\,a^2\,b^7\,c\,d^9+360\,a\,b^8\,c^2\,d^8-120\,b^9\,c^3\,d^7\right )}{a^8\,b^{10}+8\,a^7\,b^{11}\,x+28\,a^6\,b^{12}\,x^2+56\,a^5\,b^{13}\,x^3+70\,a^4\,b^{14}\,x^4+56\,a^3\,b^{15}\,x^5+28\,a^2\,b^{16}\,x^6+8\,a\,b^{17}\,x^7+b^{18}\,x^8}-x\,\left (\frac {9\,a\,d^{10}}{b^{10}}-\frac {10\,c\,d^9}{b^9}\right )+\frac {d^{10}\,x^2}{2\,b^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^9,x)

[Out]

(log(a + b*x)*(45*a^2*d^10 + 45*b^2*c^2*d^8 - 90*a*b*c*d^9))/b^11 - (x^4*((105*b^9*c^6*d^4)/2 - (5985*a^6*b^3*
d^10)/2 + 105*a*b^8*c^5*d^5 + 8085*a^5*b^4*c*d^9 + (525*a^2*b^7*c^4*d^6)/2 + 1050*a^3*b^6*c^3*d^7 - (13125*a^4
*b^5*c^2*d^8)/2) + x^6*(105*b^9*c^4*d^6 - 735*a^4*b^5*d^10 + 420*a*b^8*c^3*d^7 + 2100*a^3*b^6*c*d^9 - 1890*a^2
*b^7*c^2*d^8) + (7*b^10*c^10 - 3601*a^10*d^10 + 15*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 +
84*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 840*a^7*b^3*c^3*d^7 - 6849*a^8*b^2*c^2*d^8 + 10*a*b^9*c^9*d + 9218
*a^9*b*c*d^9)/(56*b) + x*((10*b^9*c^9*d)/7 - (3349*a^9*d^10)/7 + (15*a*b^8*c^8*d^2)/7 + (24*a^2*b^7*c^7*d^3)/7
+ 6*a^3*b^6*c^6*d^4 + 12*a^4*b^5*c^5*d^5 + 30*a^5*b^4*c^4*d^6 + 120*a^6*b^3*c^3*d^7 - (6534*a^7*b^2*c^2*d^8)/
7 + (8658*a^8*b*c*d^9)/7) + x^3*(24*b^9*c^7*d^3 - 2754*a^7*b^2*d^10 + 42*a*b^8*c^6*d^4 + 7308*a^6*b^3*c*d^9 +
84*a^2*b^7*c^5*d^5 + 210*a^3*b^6*c^4*d^6 + 840*a^4*b^5*c^3*d^7 - 5754*a^5*b^4*c^2*d^8) + x^2*((15*b^9*c^8*d^2)
/2 - (3069*a^8*b*d^10)/2 + 12*a*b^8*c^7*d^3 + 4014*a^7*b^2*c*d^9 + 21*a^2*b^7*c^6*d^4 + 42*a^3*b^6*c^5*d^5 + 1
05*a^4*b^5*c^4*d^6 + 420*a^5*b^4*c^3*d^7 - 3087*a^6*b^3*c^2*d^8) + x^5*(84*b^9*c^5*d^5 - 1974*a^5*b^4*d^10 + 2
10*a*b^8*c^4*d^6 + 5460*a^4*b^5*c*d^9 + 840*a^2*b^7*c^3*d^7 - 4620*a^3*b^6*c^2*d^8) - x^7*(120*a^3*b^6*d^10 -
120*b^9*c^3*d^7 + 360*a*b^8*c^2*d^8 - 360*a^2*b^7*c*d^9))/(a^8*b^10 + b^18*x^8 + 8*a^7*b^11*x + 8*a*b^17*x^7 +
28*a^6*b^12*x^2 + 56*a^5*b^13*x^3 + 70*a^4*b^14*x^4 + 56*a^3*b^15*x^5 + 28*a^2*b^16*x^6) - x*((9*a*d^10)/b^10
- (10*c*d^9)/b^9) + (d^10*x^2)/(2*b^9)

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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)**9,x)

[Out]

Timed out

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