3.1317 \(\int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx\)

Optimal. Leaf size=260 \[ \frac {5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac {15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac {60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {210 d^6 x (b c-a d)^4}{b^{10}} \]

[Out]

210*d^6*(-a*d+b*c)^4*x/b^10-1/5*(-a*d+b*c)^10/b^11/(b*x+a)^5-5/2*d*(-a*d+b*c)^9/b^11/(b*x+a)^4-15*d^2*(-a*d+b*
c)^8/b^11/(b*x+a)^3-60*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^2-210*d^4*(-a*d+b*c)^6/b^11/(b*x+a)+60*d^7*(-a*d+b*c)^3*(
b*x+a)^2/b^11+15*d^8*(-a*d+b*c)^2*(b*x+a)^3/b^11+5/2*d^9*(-a*d+b*c)*(b*x+a)^4/b^11+1/5*d^10*(b*x+a)^5/b^11+252
*d^5*(-a*d+b*c)^5*ln(b*x+a)/b^11

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Rubi [A]  time = 0.42, antiderivative size = 260, 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} \[ \frac {5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac {15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac {60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac {210 d^6 x (b c-a d)^4}{b^{10}}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac {d^{10} (a+b x)^5}{5 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^6,x]

[Out]

(210*d^6*(b*c - a*d)^4*x)/b^10 - (b*c - a*d)^10/(5*b^11*(a + b*x)^5) - (5*d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^4
) - (15*d^2*(b*c - a*d)^8)/(b^11*(a + b*x)^3) - (60*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^2) - (210*d^4*(b*c - a*
d)^6)/(b^11*(a + b*x)) + (60*d^7*(b*c - a*d)^3*(a + b*x)^2)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^3)/b^11 + (
5*d^9*(b*c - a*d)*(a + b*x)^4)/(2*b^11) + (d^10*(a + b*x)^5)/(5*b^11) + (252*d^5*(b*c - a*d)^5*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)^6} \, dx &=\int \left (\frac {210 d^6 (b c-a d)^4}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^6}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^5}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^4}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^3}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^2}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)}+\frac {120 d^7 (b c-a d)^3 (a+b x)}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{b^{10}}+\frac {d^{10} (a+b x)^4}{b^{10}}\right ) \, dx\\ &=\frac {210 d^6 (b c-a d)^4 x}{b^{10}}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}+\frac {60 d^7 (b c-a d)^3 (a+b x)^2}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^3}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^4}{2 b^{11}}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 305, normalized size = 1.17 \[ \frac {10 b^3 d^8 x^3 \left (7 a^2 d^2-20 a b c d+15 b^2 c^2\right )+10 b^2 d^7 x^2 \left (-28 a^3 d^3+105 a^2 b c d^2-135 a b^2 c^2 d+60 b^3 c^3\right )+10 b d^6 x \left (126 a^4 d^4-560 a^3 b c d^3+945 a^2 b^2 c^2 d^2-720 a b^3 c^3 d+210 b^4 c^4\right )+5 b^4 d^9 x^4 (5 b c-3 a d)+2520 d^5 (b c-a d)^5 \log (a+b x)-\frac {2100 d^4 (b c-a d)^6}{a+b x}+\frac {600 d^3 (a d-b c)^7}{(a+b x)^2}-\frac {150 d^2 (b c-a d)^8}{(a+b x)^3}+\frac {25 d (a d-b c)^9}{(a+b x)^4}-\frac {2 (b c-a d)^{10}}{(a+b x)^5}+2 b^5 d^{10} x^5}{10 b^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^6,x]

[Out]

(10*b*d^6*(210*b^4*c^4 - 720*a*b^3*c^3*d + 945*a^2*b^2*c^2*d^2 - 560*a^3*b*c*d^3 + 126*a^4*d^4)*x + 10*b^2*d^7
*(60*b^3*c^3 - 135*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 28*a^3*d^3)*x^2 + 10*b^3*d^8*(15*b^2*c^2 - 20*a*b*c*d + 7*a
^2*d^2)*x^3 + 5*b^4*d^9*(5*b*c - 3*a*d)*x^4 + 2*b^5*d^10*x^5 - (2*(b*c - a*d)^10)/(a + b*x)^5 + (25*d*(-(b*c)
+ a*d)^9)/(a + b*x)^4 - (150*d^2*(b*c - a*d)^8)/(a + b*x)^3 + (600*d^3*(-(b*c) + a*d)^7)/(a + b*x)^2 - (2100*d
^4*(b*c - a*d)^6)/(a + b*x) + 2520*d^5*(b*c - a*d)^5*Log[a + b*x])/(10*b^11)

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fricas [B]  time = 0.45, size = 1395, normalized size = 5.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="fricas")

[Out]

1/10*(2*b^10*d^10*x^10 - 2*b^10*c^10 - 5*a*b^9*c^9*d - 15*a^2*b^8*c^8*d^2 - 60*a^3*b^7*c^7*d^3 - 420*a^4*b^6*c
^6*d^4 + 5754*a^5*b^5*c^5*d^5 - 18270*a^6*b^4*c^4*d^6 + 27540*a^7*b^3*c^3*d^7 - 22290*a^8*b^2*c^2*d^8 + 9395*a
^9*b*c*d^9 - 1627*a^10*d^10 + 5*(5*b^10*c*d^9 - a*b^9*d^10)*x^9 + 15*(10*b^10*c^2*d^8 - 5*a*b^9*c*d^9 + a^2*b^
8*d^10)*x^8 + 60*(10*b^10*c^3*d^7 - 10*a*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(5*b^10*c^4*d
^6 - 10*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 - 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + (10500*a*b^9*c^4*d^6 - 3000
0*a^2*b^8*c^3*d^7 + 35250*a^3*b^7*c^2*d^8 - 19375*a^4*b^6*c*d^9 + 4127*a^5*b^5*d^10)*x^5 - 5*(420*b^10*c^6*d^4
- 2520*a*b^9*c^5*d^5 + 2100*a^2*b^8*c^4*d^6 + 4800*a^3*b^7*c^3*d^7 - 10050*a^4*b^6*c^2*d^8 + 6775*a^5*b^5*c*d
^9 - 1607*a^6*b^4*d^10)*x^4 - 10*(60*b^10*c^7*d^3 + 420*a*b^9*c^6*d^4 - 3780*a^2*b^8*c^5*d^5 + 8400*a^3*b^7*c^
4*d^6 - 7800*a^4*b^6*c^3*d^7 + 2550*a^5*b^5*c^2*d^8 + 475*a^6*b^4*c*d^9 - 347*a^7*b^3*d^10)*x^3 - 10*(15*b^10*
c^8*d^2 + 60*a*b^9*c^7*d^3 + 420*a^2*b^8*c^6*d^4 - 4620*a^3*b^7*c^5*d^5 + 12600*a^4*b^6*c^4*d^6 - 16200*a^5*b^
5*c^3*d^7 + 10950*a^6*b^4*c^2*d^8 - 3725*a^7*b^3*c*d^9 + 493*a^8*b^2*d^10)*x^2 - 5*(5*b^10*c^9*d + 15*a*b^9*c^
8*d^2 + 60*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 5250*a^4*b^6*c^5*d^5 + 15750*a^5*b^5*c^4*d^6 - 22500*a^6*b^
4*c^3*d^7 + 17250*a^7*b^3*c^2*d^8 - 6875*a^8*b^2*c*d^9 + 1123*a^9*b*d^10)*x + 2520*(a^5*b^5*c^5*d^5 - 5*a^6*b^
4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 - 10*a^8*b^2*c^2*d^8 + 5*a^9*b*c*d^9 - a^10*d^10 + (b^10*c^5*d^5 - 5*a*b^9*c^4*
d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 5*(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 - 10*a^4*b^6*c^2*d^8 + 5*a^5*b^5*c*d^9 - a^6*b^4*d^10)*x^4 + 10*(a^2*b^8*c^5
*d^5 - 5*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 - 10*a^5*b^5*c^2*d^8 + 5*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 10*
(a^3*b^7*c^5*d^5 - 5*a^4*b^6*c^4*d^6 + 10*a^5*b^5*c^3*d^7 - 10*a^6*b^4*c^2*d^8 + 5*a^7*b^3*c*d^9 - a^8*b^2*d^1
0)*x^2 + 5*(a^4*b^6*c^5*d^5 - 5*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c^3*d^7 - 10*a^7*b^3*c^2*d^8 + 5*a^8*b^2*c*d^9 -
a^9*b*d^10)*x)*log(b*x + a))/(b^16*x^5 + 5*a*b^15*x^4 + 10*a^2*b^14*x^3 + 10*a^3*b^13*x^2 + 5*a^4*b^12*x + a^5
*b^11)

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giac [B]  time = 1.34, size = 883, normalized size = 3.40 \[ \frac {252 \, {\left (b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {2 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} - 5754 \, a^{5} b^{5} c^{5} d^{5} + 18270 \, a^{6} b^{4} c^{4} d^{6} - 27540 \, a^{7} b^{3} c^{3} d^{7} + 22290 \, a^{8} b^{2} c^{2} d^{8} - 9395 \, a^{9} b c d^{9} + 1627 \, a^{10} d^{10} + 2100 \, {\left (b^{10} c^{6} d^{4} - 6 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 20 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} - 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 600 \, {\left (b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} - 63 \, a^{2} b^{8} c^{5} d^{5} + 175 \, a^{3} b^{7} c^{4} d^{6} - 245 \, a^{4} b^{6} c^{3} d^{7} + 189 \, a^{5} b^{5} c^{2} d^{8} - 77 \, a^{6} b^{4} c d^{9} + 13 \, a^{7} b^{3} d^{10}\right )} x^{3} + 150 \, {\left (b^{10} c^{8} d^{2} + 4 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 308 \, a^{3} b^{7} c^{5} d^{5} + 910 \, a^{4} b^{6} c^{4} d^{6} - 1316 \, a^{5} b^{5} c^{3} d^{7} + 1036 \, a^{6} b^{4} c^{2} d^{8} - 428 \, a^{7} b^{3} c d^{9} + 73 \, a^{8} b^{2} d^{10}\right )} x^{2} + 25 \, {\left (b^{10} c^{9} d + 3 \, a b^{9} c^{8} d^{2} + 12 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} - 1050 \, a^{4} b^{6} c^{5} d^{5} + 3234 \, a^{5} b^{5} c^{4} d^{6} - 4788 \, a^{6} b^{4} c^{3} d^{7} + 3828 \, a^{7} b^{3} c^{2} d^{8} - 1599 \, a^{8} b^{2} c d^{9} + 275 \, a^{9} b d^{10}\right )} x}{10 \, {\left (b x + a\right )}^{5} b^{11}} + \frac {2 \, b^{24} d^{10} x^{5} + 25 \, b^{24} c d^{9} x^{4} - 15 \, a b^{23} d^{10} x^{4} + 150 \, b^{24} c^{2} d^{8} x^{3} - 200 \, a b^{23} c d^{9} x^{3} + 70 \, a^{2} b^{22} d^{10} x^{3} + 600 \, b^{24} c^{3} d^{7} x^{2} - 1350 \, a b^{23} c^{2} d^{8} x^{2} + 1050 \, a^{2} b^{22} c d^{9} x^{2} - 280 \, a^{3} b^{21} d^{10} x^{2} + 2100 \, b^{24} c^{4} d^{6} x - 7200 \, a b^{23} c^{3} d^{7} x + 9450 \, a^{2} b^{22} c^{2} d^{8} x - 5600 \, a^{3} b^{21} c d^{9} x + 1260 \, a^{4} b^{20} d^{10} x}{10 \, b^{30}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="giac")

[Out]

252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9 - a^5*d^10)*log(a
bs(b*x + a))/b^11 - 1/10*(2*b^10*c^10 + 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*
c^6*d^4 - 5754*a^5*b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2*d^8 - 9395*
a^9*b*c*d^9 + 1627*a^10*d^10 + 2100*(b^10*c^6*d^4 - 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 20*a^3*b^7*c^3*d^7
+ 15*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 600*(b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*
c^5*d^5 + 175*a^3*b^7*c^4*d^6 - 245*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13*a^7*b^3*d^10
)*x^3 + 150*(b^10*c^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 308*a^3*b^7*c^5*d^5 + 910*a^4*b^6*c^4*d^6 -
1316*a^5*b^5*c^3*d^7 + 1036*a^6*b^4*c^2*d^8 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10*c^9*d + 3*a
*b^9*c^8*d^2 + 12*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a^4*b^6*c^5*d^5 + 3234*a^5*b^5*c^4*d^6 - 4788*a^
6*b^4*c^3*d^7 + 3828*a^7*b^3*c^2*d^8 - 1599*a^8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/((b*x + a)^5*b^11) + 1/10*(2*b^
24*d^10*x^5 + 25*b^24*c*d^9*x^4 - 15*a*b^23*d^10*x^4 + 150*b^24*c^2*d^8*x^3 - 200*a*b^23*c*d^9*x^3 + 70*a^2*b^
22*d^10*x^3 + 600*b^24*c^3*d^7*x^2 - 1350*a*b^23*c^2*d^8*x^2 + 1050*a^2*b^22*c*d^9*x^2 - 280*a^3*b^21*d^10*x^2
+ 2100*b^24*c^4*d^6*x - 7200*a*b^23*c^3*d^7*x + 9450*a^2*b^22*c^2*d^8*x - 5600*a^3*b^21*c*d^9*x + 1260*a^4*b^
20*d^10*x)/b^30

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maple [B]  time = 0.02, size = 1199, normalized size = 4.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^6,x)

[Out]

-252/b^11*d^10*ln(b*x+a)*a^5+252/b^6*d^5*ln(b*x+a)*c^5+5/2/b^11*d^10/(b*x+a)^4*a^9-5/2/b^2*d/(b*x+a)^4*c^9-210
/b^11*d^10/(b*x+a)*a^6-210/b^5*d^4/(b*x+a)*c^6-3/2*d^10/b^7*x^4*a+5/2*d^9/b^6*x^4*c+7*d^10/b^8*x^3*a^2+15*d^8/
b^6*x^3*c^2-28*d^10/b^9*x^2*a^3+60*d^7/b^6*x^2*c^3+126*d^10/b^10*a^4*x+210*d^6/b^6*c^4*x-15/b^11*d^10/(b*x+a)^
3*a^8-15/b^3*d^2/(b*x+a)^3*c^8-1/5/b^11/(b*x+a)^5*a^10*d^10+60/b^11*d^10/(b*x+a)^2*a^7-60/b^4*d^3/(b*x+a)^2*c^
7+1260/b^10*d^9*ln(b*x+a)*a^4*c-2520/b^9*d^8*ln(b*x+a)*a^3*c^2-1260/b^6*d^5/(b*x+a)^2*a^2*c^5+420/b^5*d^4/(b*x
+a)^2*a*c^6-135*d^8/b^7*x^2*a*c^2-560*d^9/b^9*a^3*c*x+945*d^8/b^8*a^2*c^2*x-720*d^7/b^7*a*c^3*x+120/b^10*d^9/(
b*x+a)^3*a^7*c-420/b^9*d^8/(b*x+a)^3*a^6*c^2+840/b^8*d^7/(b*x+a)^3*a^5*c^3-1050/b^7*d^6/(b*x+a)^3*a^4*c^4+840/
b^6*d^5/(b*x+a)^3*a^3*c^5-420/b^5*d^4/(b*x+a)^3*a^2*c^6+120/b^4*d^3/(b*x+a)^3*a*c^7+2/b^10/(b*x+a)^5*a^9*c*d^9
-9/b^9/(b*x+a)^5*a^8*c^2*d^8+24/b^8/(b*x+a)^5*a^7*c^3*d^7-42/b^7/(b*x+a)^5*a^6*c^4*d^6+252/5/b^6/(b*x+a)^5*a^5
*c^5*d^5-42/b^5/(b*x+a)^5*a^4*c^6*d^4+24/b^4/(b*x+a)^5*a^3*c^7*d^3-9/b^3/(b*x+a)^5*a^2*c^8*d^2+2520/b^8*d^7*ln
(b*x+a)*a^2*c^3-1260/b^7*d^6*ln(b*x+a)*a*c^4-20*d^9/b^7*x^3*a*c+105*d^9/b^8*x^2*a^2*c-45/2/b^10*d^9/(b*x+a)^4*
a^8*c+90/b^9*d^8/(b*x+a)^4*a^7*c^2-210/b^8*d^7/(b*x+a)^4*a^6*c^3+315/b^7*d^6/(b*x+a)^4*a^5*c^4-315/b^6*d^5/(b*
x+a)^4*a^4*c^5+210/b^5*d^4/(b*x+a)^4*a^3*c^6-90/b^4*d^3/(b*x+a)^4*a^2*c^7+45/2/b^3*d^2/(b*x+a)^4*a*c^8+1260/b^
10*d^9/(b*x+a)*a^5*c-3150/b^9*d^8/(b*x+a)*a^4*c^2+4200/b^8*d^7/(b*x+a)*a^3*c^3-3150/b^7*d^6/(b*x+a)*a^2*c^4+12
60/b^6*d^5/(b*x+a)*a*c^5+2/b^2/(b*x+a)^5*a*c^9*d-420/b^10*d^9/(b*x+a)^2*a^6*c+1260/b^9*d^8/(b*x+a)^2*a^5*c^2-2
100/b^8*d^7/(b*x+a)^2*a^4*c^3+2100/b^7*d^6/(b*x+a)^2*a^3*c^4+1/5*d^10/b^6*x^5-1/5/b/(b*x+a)^5*c^10

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maxima [B]  time = 2.25, size = 912, normalized size = 3.51 \[ -\frac {2 \, b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 15 \, a^{2} b^{8} c^{8} d^{2} + 60 \, a^{3} b^{7} c^{7} d^{3} + 420 \, a^{4} b^{6} c^{6} d^{4} - 5754 \, a^{5} b^{5} c^{5} d^{5} + 18270 \, a^{6} b^{4} c^{4} d^{6} - 27540 \, a^{7} b^{3} c^{3} d^{7} + 22290 \, a^{8} b^{2} c^{2} d^{8} - 9395 \, a^{9} b c d^{9} + 1627 \, a^{10} d^{10} + 2100 \, {\left (b^{10} c^{6} d^{4} - 6 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} - 20 \, a^{3} b^{7} c^{3} d^{7} + 15 \, a^{4} b^{6} c^{2} d^{8} - 6 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 600 \, {\left (b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} - 63 \, a^{2} b^{8} c^{5} d^{5} + 175 \, a^{3} b^{7} c^{4} d^{6} - 245 \, a^{4} b^{6} c^{3} d^{7} + 189 \, a^{5} b^{5} c^{2} d^{8} - 77 \, a^{6} b^{4} c d^{9} + 13 \, a^{7} b^{3} d^{10}\right )} x^{3} + 150 \, {\left (b^{10} c^{8} d^{2} + 4 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 308 \, a^{3} b^{7} c^{5} d^{5} + 910 \, a^{4} b^{6} c^{4} d^{6} - 1316 \, a^{5} b^{5} c^{3} d^{7} + 1036 \, a^{6} b^{4} c^{2} d^{8} - 428 \, a^{7} b^{3} c d^{9} + 73 \, a^{8} b^{2} d^{10}\right )} x^{2} + 25 \, {\left (b^{10} c^{9} d + 3 \, a b^{9} c^{8} d^{2} + 12 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} - 1050 \, a^{4} b^{6} c^{5} d^{5} + 3234 \, a^{5} b^{5} c^{4} d^{6} - 4788 \, a^{6} b^{4} c^{3} d^{7} + 3828 \, a^{7} b^{3} c^{2} d^{8} - 1599 \, a^{8} b^{2} c d^{9} + 275 \, a^{9} b d^{10}\right )} x}{10 \, {\left (b^{16} x^{5} + 5 \, a b^{15} x^{4} + 10 \, a^{2} b^{14} x^{3} + 10 \, a^{3} b^{13} x^{2} + 5 \, a^{4} b^{12} x + a^{5} b^{11}\right )}} + \frac {2 \, b^{4} d^{10} x^{5} + 5 \, {\left (5 \, b^{4} c d^{9} - 3 \, a b^{3} d^{10}\right )} x^{4} + 10 \, {\left (15 \, b^{4} c^{2} d^{8} - 20 \, a b^{3} c d^{9} + 7 \, a^{2} b^{2} d^{10}\right )} x^{3} + 10 \, {\left (60 \, b^{4} c^{3} d^{7} - 135 \, a b^{3} c^{2} d^{8} + 105 \, a^{2} b^{2} c d^{9} - 28 \, a^{3} b d^{10}\right )} x^{2} + 10 \, {\left (210 \, b^{4} c^{4} d^{6} - 720 \, a b^{3} c^{3} d^{7} + 945 \, a^{2} b^{2} c^{2} d^{8} - 560 \, a^{3} b c d^{9} + 126 \, a^{4} d^{10}\right )} x}{10 \, b^{10}} + \frac {252 \, {\left (b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/10*(2*b^10*c^10 + 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 - 5754*a^5*
b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2*d^8 - 9395*a^9*b*c*d^9 + 1627*
a^10*d^10 + 2100*(b^10*c^6*d^4 - 6*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 20*a^3*b^7*c^3*d^7 + 15*a^4*b^6*c^2*d^
8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 600*(b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*c^5*d^5 + 175*a^3*b
^7*c^4*d^6 - 245*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13*a^7*b^3*d^10)*x^3 + 150*(b^10*c
^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 308*a^3*b^7*c^5*d^5 + 910*a^4*b^6*c^4*d^6 - 1316*a^5*b^5*c^3*d
^7 + 1036*a^6*b^4*c^2*d^8 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10*c^9*d + 3*a*b^9*c^8*d^2 + 12*a
^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a^4*b^6*c^5*d^5 + 3234*a^5*b^5*c^4*d^6 - 4788*a^6*b^4*c^3*d^7 + 382
8*a^7*b^3*c^2*d^8 - 1599*a^8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/(b^16*x^5 + 5*a*b^15*x^4 + 10*a^2*b^14*x^3 + 10*a^
3*b^13*x^2 + 5*a^4*b^12*x + a^5*b^11) + 1/10*(2*b^4*d^10*x^5 + 5*(5*b^4*c*d^9 - 3*a*b^3*d^10)*x^4 + 10*(15*b^4
*c^2*d^8 - 20*a*b^3*c*d^9 + 7*a^2*b^2*d^10)*x^3 + 10*(60*b^4*c^3*d^7 - 135*a*b^3*c^2*d^8 + 105*a^2*b^2*c*d^9 -
28*a^3*b*d^10)*x^2 + 10*(210*b^4*c^4*d^6 - 720*a*b^3*c^3*d^7 + 945*a^2*b^2*c^2*d^8 - 560*a^3*b*c*d^9 + 126*a^
4*d^10)*x)/b^10 + 252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9
- a^5*d^10)*log(b*x + a)/b^11

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mupad [B]  time = 0.40, size = 1141, normalized size = 4.39 \[ x^3\,\left (\frac {2\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {5\,a^2\,d^{10}}{b^8}+\frac {15\,c^2\,d^8}{b^6}\right )-x^2\,\left (\frac {3\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b}+\frac {10\,a^3\,d^{10}}{b^9}-\frac {60\,c^3\,d^7}{b^6}-\frac {15\,a^2\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{2\,b^2}\right )-x^4\,\left (\frac {3\,a\,d^{10}}{2\,b^7}-\frac {5\,c\,d^9}{2\,b^6}\right )-\frac {x^4\,\left (210\,a^6\,b^3\,d^{10}-1260\,a^5\,b^4\,c\,d^9+3150\,a^4\,b^5\,c^2\,d^8-4200\,a^3\,b^6\,c^3\,d^7+3150\,a^2\,b^7\,c^4\,d^6-1260\,a\,b^8\,c^5\,d^5+210\,b^9\,c^6\,d^4\right )+\frac {1627\,a^{10}\,d^{10}-9395\,a^9\,b\,c\,d^9+22290\,a^8\,b^2\,c^2\,d^8-27540\,a^7\,b^3\,c^3\,d^7+18270\,a^6\,b^4\,c^4\,d^6-5754\,a^5\,b^5\,c^5\,d^5+420\,a^4\,b^6\,c^6\,d^4+60\,a^3\,b^7\,c^7\,d^3+15\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+2\,b^{10}\,c^{10}}{10\,b}+x\,\left (\frac {1375\,a^9\,d^{10}}{2}-\frac {7995\,a^8\,b\,c\,d^9}{2}+9570\,a^7\,b^2\,c^2\,d^8-11970\,a^6\,b^3\,c^3\,d^7+8085\,a^5\,b^4\,c^4\,d^6-2625\,a^4\,b^5\,c^5\,d^5+210\,a^3\,b^6\,c^6\,d^4+30\,a^2\,b^7\,c^7\,d^3+\frac {15\,a\,b^8\,c^8\,d^2}{2}+\frac {5\,b^9\,c^9\,d}{2}\right )+x^3\,\left (780\,a^7\,b^2\,d^{10}-4620\,a^6\,b^3\,c\,d^9+11340\,a^5\,b^4\,c^2\,d^8-14700\,a^4\,b^5\,c^3\,d^7+10500\,a^3\,b^6\,c^4\,d^6-3780\,a^2\,b^7\,c^5\,d^5+420\,a\,b^8\,c^6\,d^4+60\,b^9\,c^7\,d^3\right )+x^2\,\left (1095\,a^8\,b\,d^{10}-6420\,a^7\,b^2\,c\,d^9+15540\,a^6\,b^3\,c^2\,d^8-19740\,a^5\,b^4\,c^3\,d^7+13650\,a^4\,b^5\,c^4\,d^6-4620\,a^3\,b^6\,c^5\,d^5+420\,a^2\,b^7\,c^6\,d^4+60\,a\,b^8\,c^7\,d^3+15\,b^9\,c^8\,d^2\right )}{a^5\,b^{10}+5\,a^4\,b^{11}\,x+10\,a^3\,b^{12}\,x^2+10\,a^2\,b^{13}\,x^3+5\,a\,b^{14}\,x^4+b^{15}\,x^5}+x\,\left (\frac {6\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b}+\frac {20\,a^3\,d^{10}}{b^9}-\frac {120\,c^3\,d^7}{b^6}-\frac {15\,a^2\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b^2}\right )}{b}-\frac {15\,a^4\,d^{10}}{b^{10}}+\frac {210\,c^4\,d^6}{b^6}+\frac {20\,a^3\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b^3}-\frac {15\,a^2\,\left (\frac {6\,a\,\left (\frac {6\,a\,d^{10}}{b^7}-\frac {10\,c\,d^9}{b^6}\right )}{b}-\frac {15\,a^2\,d^{10}}{b^8}+\frac {45\,c^2\,d^8}{b^6}\right )}{b^2}\right )+\frac {d^{10}\,x^5}{5\,b^6}-\frac {\ln \left (a+b\,x\right )\,\left (252\,a^5\,d^{10}-1260\,a^4\,b\,c\,d^9+2520\,a^3\,b^2\,c^2\,d^8-2520\,a^2\,b^3\,c^3\,d^7+1260\,a\,b^4\,c^4\,d^6-252\,b^5\,c^5\,d^5\right )}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^6,x)

[Out]

x^3*((2*a*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b - (5*a^2*d^10)/b^8 + (15*c^2*d^8)/b^6) - x^2*((3*a*((6*a*((6*a*
d^10)/b^7 - (10*c*d^9)/b^6))/b - (15*a^2*d^10)/b^8 + (45*c^2*d^8)/b^6))/b + (10*a^3*d^10)/b^9 - (60*c^3*d^7)/b
^6 - (15*a^2*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/(2*b^2)) - x^4*((3*a*d^10)/(2*b^7) - (5*c*d^9)/(2*b^6)) - (x^4
*(210*a^6*b^3*d^10 + 210*b^9*c^6*d^4 - 1260*a*b^8*c^5*d^5 - 1260*a^5*b^4*c*d^9 + 3150*a^2*b^7*c^4*d^6 - 4200*a
^3*b^6*c^3*d^7 + 3150*a^4*b^5*c^2*d^8) + (1627*a^10*d^10 + 2*b^10*c^10 + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d
^3 + 420*a^4*b^6*c^6*d^4 - 5754*a^5*b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^
2*c^2*d^8 + 5*a*b^9*c^9*d - 9395*a^9*b*c*d^9)/(10*b) + x*((1375*a^9*d^10)/2 + (5*b^9*c^9*d)/2 + (15*a*b^8*c^8*
d^2)/2 + 30*a^2*b^7*c^7*d^3 + 210*a^3*b^6*c^6*d^4 - 2625*a^4*b^5*c^5*d^5 + 8085*a^5*b^4*c^4*d^6 - 11970*a^6*b^
3*c^3*d^7 + 9570*a^7*b^2*c^2*d^8 - (7995*a^8*b*c*d^9)/2) + x^3*(780*a^7*b^2*d^10 + 60*b^9*c^7*d^3 + 420*a*b^8*
c^6*d^4 - 4620*a^6*b^3*c*d^9 - 3780*a^2*b^7*c^5*d^5 + 10500*a^3*b^6*c^4*d^6 - 14700*a^4*b^5*c^3*d^7 + 11340*a^
5*b^4*c^2*d^8) + x^2*(1095*a^8*b*d^10 + 15*b^9*c^8*d^2 + 60*a*b^8*c^7*d^3 - 6420*a^7*b^2*c*d^9 + 420*a^2*b^7*c
^6*d^4 - 4620*a^3*b^6*c^5*d^5 + 13650*a^4*b^5*c^4*d^6 - 19740*a^5*b^4*c^3*d^7 + 15540*a^6*b^3*c^2*d^8))/(a^5*b
^10 + b^15*x^5 + 5*a^4*b^11*x + 5*a*b^14*x^4 + 10*a^3*b^12*x^2 + 10*a^2*b^13*x^3) + x*((6*a*((6*a*((6*a*((6*a*
d^10)/b^7 - (10*c*d^9)/b^6))/b - (15*a^2*d^10)/b^8 + (45*c^2*d^8)/b^6))/b + (20*a^3*d^10)/b^9 - (120*c^3*d^7)/
b^6 - (15*a^2*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b^2))/b - (15*a^4*d^10)/b^10 + (210*c^4*d^6)/b^6 + (20*a^3*((
6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b^3 - (15*a^2*((6*a*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b - (15*a^2*d^10)/b^8
+ (45*c^2*d^8)/b^6))/b^2) + (d^10*x^5)/(5*b^6) - (log(a + b*x)*(252*a^5*d^10 - 252*b^5*c^5*d^5 + 1260*a*b^4*c^
4*d^6 - 2520*a^2*b^3*c^3*d^7 + 2520*a^3*b^2*c^2*d^8 - 1260*a^4*b*c*d^9))/b^11

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**10/(b*x+a)**6,x)

[Out]

Timed out

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