3.13.10 \(\int \frac {(c+d x)^{10}}{(a+b x)^5} \, dx\)

Optimal. Leaf size=262 \[ \frac {2 d^9 (a+b x)^5 (b c-a d)}{b^{11}}+\frac {45 d^8 (a+b x)^4 (b c-a d)^2}{4 b^{11}}+\frac {40 d^7 (a+b x)^3 (b c-a d)^3}{b^{11}}+\frac {105 d^6 (a+b x)^2 (b c-a d)^4}{b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {252 d^5 x (b c-a d)^5}{b^{10}} \]

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Rubi [A]  time = 0.42, antiderivative size = 262, 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 {2 d^9 (a+b x)^5 (b c-a d)}{b^{11}}+\frac {45 d^8 (a+b x)^4 (b c-a d)^2}{4 b^{11}}+\frac {40 d^7 (a+b x)^3 (b c-a d)^3}{b^{11}}+\frac {105 d^6 (a+b x)^2 (b c-a d)^4}{b^{11}}+\frac {252 d^5 x (b c-a d)^5}{b^{10}}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}+\frac {d^{10} (a+b x)^6}{6 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(252*d^5*(b*c - a*d)^5*x)/b^10 - (b*c - a*d)^10/(4*b^11*(a + b*x)^4) - (10*d*(b*c - a*d)^9)/(3*b^11*(a + b*x)^
3) - (45*d^2*(b*c - a*d)^8)/(2*b^11*(a + b*x)^2) - (120*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)) + (105*d^6*(b*c -
a*d)^4*(a + b*x)^2)/b^11 + (40*d^7*(b*c - a*d)^3*(a + b*x)^3)/b^11 + (45*d^8*(b*c - a*d)^2*(a + b*x)^4)/(4*b^1
1) + (2*d^9*(b*c - a*d)*(a + b*x)^5)/b^11 + (d^10*(a + b*x)^6)/(6*b^11) + (210*d^4*(b*c - a*d)^6*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)^5} \, dx &=\int \left (\frac {252 d^5 (b c-a d)^5}{b^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^5}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^4}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^3}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^2}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)}+\frac {210 d^6 (b c-a d)^4 (a+b x)}{b^{10}}+\frac {120 d^7 (b c-a d)^3 (a+b x)^2}{b^{10}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^3}{b^{10}}+\frac {10 d^9 (b c-a d) (a+b x)^4}{b^{10}}+\frac {d^{10} (a+b x)^5}{b^{10}}\right ) \, dx\\ &=\frac {252 d^5 (b c-a d)^5 x}{b^{10}}-\frac {(b c-a d)^{10}}{4 b^{11} (a+b x)^4}-\frac {10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac {120 d^3 (b c-a d)^7}{b^{11} (a+b x)}+\frac {105 d^6 (b c-a d)^4 (a+b x)^2}{b^{11}}+\frac {40 d^7 (b c-a d)^3 (a+b x)^3}{b^{11}}+\frac {45 d^8 (b c-a d)^2 (a+b x)^4}{4 b^{11}}+\frac {2 d^9 (b c-a d) (a+b x)^5}{b^{11}}+\frac {d^{10} (a+b x)^6}{6 b^{11}}+\frac {210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 359, normalized size = 1.37 \begin {gather*} \frac {15 b^4 d^8 x^4 \left (3 a^2 d^2-10 a b c d+9 b^2 c^2\right )+20 b^3 d^7 x^3 \left (-7 a^3 d^3+30 a^2 b c d^2-45 a b^2 c^2 d+24 b^3 c^3\right )+30 b^2 d^6 x^2 \left (14 a^4 d^4-70 a^3 b c d^3+135 a^2 b^2 c^2 d^2-120 a b^3 c^3 d+42 b^4 c^4\right )+12 b d^5 x \left (-126 a^5 d^5+700 a^4 b c d^4-1575 a^3 b^2 c^2 d^3+1800 a^2 b^3 c^3 d^2-1050 a b^4 c^4 d+252 b^5 c^5\right )+12 b^5 d^9 x^5 (2 b c-a d)+2520 d^4 (b c-a d)^6 \log (a+b x)+\frac {1440 d^3 (a d-b c)^7}{a+b x}-\frac {270 d^2 (b c-a d)^8}{(a+b x)^2}+\frac {40 d (a d-b c)^9}{(a+b x)^3}-\frac {3 (b c-a d)^{10}}{(a+b x)^4}+2 b^6 d^{10} x^6}{12 b^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^10/(a + b*x)^5,x]

[Out]

(12*b*d^5*(252*b^5*c^5 - 1050*a*b^4*c^4*d + 1800*a^2*b^3*c^3*d^2 - 1575*a^3*b^2*c^2*d^3 + 700*a^4*b*c*d^4 - 12
6*a^5*d^5)*x + 30*b^2*d^6*(42*b^4*c^4 - 120*a*b^3*c^3*d + 135*a^2*b^2*c^2*d^2 - 70*a^3*b*c*d^3 + 14*a^4*d^4)*x
^2 + 20*b^3*d^7*(24*b^3*c^3 - 45*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 7*a^3*d^3)*x^3 + 15*b^4*d^8*(9*b^2*c^2 - 10*a*
b*c*d + 3*a^2*d^2)*x^4 + 12*b^5*d^9*(2*b*c - a*d)*x^5 + 2*b^6*d^10*x^6 - (3*(b*c - a*d)^10)/(a + b*x)^4 + (40*
d*(-(b*c) + a*d)^9)/(a + b*x)^3 - (270*d^2*(b*c - a*d)^8)/(a + b*x)^2 + (1440*d^3*(-(b*c) + a*d)^7)/(a + b*x)
+ 2520*d^4*(b*c - a*d)^6*Log[a + b*x])/(12*b^11)

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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)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^5,x]

[Out]

IntegrateAlgebraic[(c + d*x)^10/(a + b*x)^5, x]

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fricas [B]  time = 1.25, size = 1365, normalized size = 5.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^5,x, algorithm="fricas")

[Out]

1/12*(2*b^10*d^10*x^10 - 3*b^10*c^10 - 10*a*b^9*c^9*d - 45*a^2*b^8*c^8*d^2 - 360*a^3*b^7*c^7*d^3 + 5250*a^4*b^
6*c^6*d^4 - 19404*a^5*b^5*c^5*d^5 + 35910*a^6*b^4*c^4*d^6 - 38280*a^7*b^3*c^3*d^7 + 23985*a^8*b^2*c^2*d^8 - 82
50*a^9*b*c*d^9 + 1207*a^10*d^10 + 4*(6*b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(15*b^10*c^2*d^8 - 6*a*b^9*c*d^9 + a^2
*b^8*d^10)*x^8 + 24*(20*b^10*c^3*d^7 - 15*a*b^9*c^2*d^8 + 6*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 84*(15*b^10*c^
4*d^6 - 20*a*b^9*c^3*d^7 + 15*a^2*b^8*c^2*d^8 - 6*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 504*(6*b^10*c^5*d^5 - 15
*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 - 15*a^3*b^7*c^2*d^8 + 6*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + (12096*a*b^9*
c^5*d^5 - 42840*a^2*b^8*c^4*d^6 + 66720*a^3*b^7*c^3*d^7 - 54765*a^4*b^6*c^2*d^8 + 23250*a^5*b^5*c*d^9 - 4043*a
^6*b^4*d^10)*x^4 - 4*(360*b^10*c^7*d^3 - 2520*a*b^9*c^6*d^4 + 3024*a^2*b^8*c^5*d^5 + 5040*a^3*b^7*c^4*d^6 - 16
320*a^4*b^6*c^3*d^7 + 16965*a^5*b^5*c^2*d^8 - 8130*a^6*b^4*c*d^9 + 1523*a^7*b^3*d^10)*x^3 - 6*(45*b^10*c^8*d^2
+ 360*a*b^9*c^7*d^3 - 3780*a^2*b^8*c^6*d^4 + 10584*a^3*b^7*c^5*d^5 - 13860*a^4*b^6*c^4*d^6 + 8880*a^5*b^5*c^3
*d^7 - 1935*a^6*b^4*c^2*d^8 - 570*a^7*b^3*c*d^9 + 263*a^8*b^2*d^10)*x^2 - 4*(10*b^10*c^9*d + 45*a*b^9*c^8*d^2
+ 360*a^2*b^8*c^7*d^3 - 4620*a^3*b^7*c^6*d^4 + 15624*a^4*b^6*c^5*d^5 - 26460*a^5*b^5*c^4*d^6 + 25680*a^6*b^4*c
^3*d^7 - 14535*a^7*b^3*c^2*d^8 + 4470*a^8*b^2*c*d^9 - 577*a^9*b*d^10)*x + 2520*(a^4*b^6*c^6*d^4 - 6*a^5*b^5*c^
5*d^5 + 15*a^6*b^4*c^4*d^6 - 20*a^7*b^3*c^3*d^7 + 15*a^8*b^2*c^2*d^8 - 6*a^9*b*c*d^9 + a^10*d^10 + (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 + 4*(a*b^9*c^6*d^4 - 6*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 - 20*a^4*b^6*c^3*d^7 + 15*a^5*b^5*c^2*
d^8 - 6*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 6*(a^2*b^8*c^6*d^4 - 6*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^4*d^6 - 20*a
^5*b^5*c^3*d^7 + 15*a^6*b^4*c^2*d^8 - 6*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 4*(a^3*b^7*c^6*d^4 - 6*a^4*b^6*c^5
*d^5 + 15*a^5*b^5*c^4*d^6 - 20*a^6*b^4*c^3*d^7 + 15*a^7*b^3*c^2*d^8 - 6*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x
+ a))/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*b^11)

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giac [B]  time = 1.38, size = 1168, normalized size = 4.46

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^10/(b*x+a)^5,x, algorithm="giac")

[Out]

1/12*(2*d^10 + 24*(b^2*c*d^9 - a*b*d^10)/((b*x + a)*b) + 135*(b^4*c^2*d^8 - 2*a*b^3*c*d^9 + a^2*b^2*d^10)/((b*
x + a)^2*b^2) + 480*(b^6*c^3*d^7 - 3*a*b^5*c^2*d^8 + 3*a^2*b^4*c*d^9 - a^3*b^3*d^10)/((b*x + a)^3*b^3) + 1260*
(b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)/((b*x + a)^4*b^4) + 3024*
(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)/((
b*x + a)^5*b^5))*(b*x + a)^6/b^11 - 210*(b^6*c^6*d^4 - 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d
^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^10)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^11 - 1/12*(3*b^67
*c^10/(b*x + a)^4 + 40*b^66*c^9*d/(b*x + a)^3 - 30*a*b^66*c^9*d/(b*x + a)^4 + 270*b^65*c^8*d^2/(b*x + a)^2 - 3
60*a*b^65*c^8*d^2/(b*x + a)^3 + 135*a^2*b^65*c^8*d^2/(b*x + a)^4 + 1440*b^64*c^7*d^3/(b*x + a) - 2160*a*b^64*c
^7*d^3/(b*x + a)^2 + 1440*a^2*b^64*c^7*d^3/(b*x + a)^3 - 360*a^3*b^64*c^7*d^3/(b*x + a)^4 - 10080*a*b^63*c^6*d
^4/(b*x + a) + 7560*a^2*b^63*c^6*d^4/(b*x + a)^2 - 3360*a^3*b^63*c^6*d^4/(b*x + a)^3 + 630*a^4*b^63*c^6*d^4/(b
*x + a)^4 + 30240*a^2*b^62*c^5*d^5/(b*x + a) - 15120*a^3*b^62*c^5*d^5/(b*x + a)^2 + 5040*a^4*b^62*c^5*d^5/(b*x
+ a)^3 - 756*a^5*b^62*c^5*d^5/(b*x + a)^4 - 50400*a^3*b^61*c^4*d^6/(b*x + a) + 18900*a^4*b^61*c^4*d^6/(b*x +
a)^2 - 5040*a^5*b^61*c^4*d^6/(b*x + a)^3 + 630*a^6*b^61*c^4*d^6/(b*x + a)^4 + 50400*a^4*b^60*c^3*d^7/(b*x + a)
- 15120*a^5*b^60*c^3*d^7/(b*x + a)^2 + 3360*a^6*b^60*c^3*d^7/(b*x + a)^3 - 360*a^7*b^60*c^3*d^7/(b*x + a)^4 -
30240*a^5*b^59*c^2*d^8/(b*x + a) + 7560*a^6*b^59*c^2*d^8/(b*x + a)^2 - 1440*a^7*b^59*c^2*d^8/(b*x + a)^3 + 13
5*a^8*b^59*c^2*d^8/(b*x + a)^4 + 10080*a^6*b^58*c*d^9/(b*x + a) - 2160*a^7*b^58*c*d^9/(b*x + a)^2 + 360*a^8*b^
58*c*d^9/(b*x + a)^3 - 30*a^9*b^58*c*d^9/(b*x + a)^4 - 1440*a^7*b^57*d^10/(b*x + a) + 270*a^8*b^57*d^10/(b*x +
a)^2 - 40*a^9*b^57*d^10/(b*x + a)^3 + 3*a^10*b^57*d^10/(b*x + a)^4)/b^68

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maple [B]  time = 0.02, size = 1172, normalized size = 4.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^10/(b*x+a)^5,x)

[Out]

10/3/b^11*d^10/(b*x+a)^3*a^9-10/3/b^2*d/(b*x+a)^3*c^9-45/2/b^11*d^10/(b*x+a)^2*a^8-45/2/b^3*d^2/(b*x+a)^2*c^8+
210/b^11*d^10*ln(b*x+a)*a^6+210/b^5*d^4*ln(b*x+a)*c^6-1/4/b^11/(b*x+a)^4*a^10*d^10+120/b^11*d^10/(b*x+a)*a^7-1
20/b^4*d^3/(b*x+a)*c^7-d^10/b^6*x^5*a+2*d^9/b^5*x^5*c+15/4*d^10/b^7*x^4*a^2+45/4*d^8/b^5*x^4*c^2-35/3*d^10/b^8
*x^3*a^3+40*d^7/b^5*x^3*c^3+35*d^10/b^9*x^2*a^4+105*d^6/b^5*x^2*c^4-126*d^10/b^10*a^5*x+252*d^5/b^5*c^5*x+1/6*
d^10/b^5*x^6-1/4/b/(b*x+a)^4*c^10-25/2*d^9/b^6*x^4*a*c+50*d^9/b^7*x^3*a^2*c-1260/b^10*d^9*ln(b*x+a)*a^5*c+3150
/b^9*d^8*ln(b*x+a)*a^4*c^2-4200/b^8*d^7*ln(b*x+a)*a^3*c^3+3150/b^7*d^6*ln(b*x+a)*a^2*c^4-1260/b^6*d^5*ln(b*x+a
)*a*c^5+5/2/b^10/(b*x+a)^4*a^9*c*d^9-45/4/b^9/(b*x+a)^4*a^8*c^2*d^8+30/b^8/(b*x+a)^4*a^7*c^3*d^7-105/2/b^7/(b*
x+a)^4*a^6*c^4*d^6+63/b^6/(b*x+a)^4*a^5*c^5*d^5-105/2/b^5/(b*x+a)^4*a^4*c^6*d^4+30/b^4/(b*x+a)^4*a^3*c^7*d^3-4
5/4/b^3/(b*x+a)^4*a^2*c^8*d^2+5/2/b^2/(b*x+a)^4*a*c^9*d-840/b^10*d^9/(b*x+a)*a^6*c+2520/b^9*d^8/(b*x+a)*a^5*c^
2-4200/b^8*d^7/(b*x+a)*a^4*c^3+4200/b^7*d^6/(b*x+a)*a^3*c^4-2520/b^6*d^5/(b*x+a)*a^2*c^5+840/b^5*d^4/(b*x+a)*a
*c^6+700*d^9/b^9*a^4*c*x-1575*d^8/b^8*a^3*c^2*x+1800*d^7/b^7*a^2*c^3*x-1050*d^6/b^6*a*c^4*x-30/b^10*d^9/(b*x+a
)^3*a^8*c+120/b^9*d^8/(b*x+a)^3*a^7*c^2-280/b^8*d^7/(b*x+a)^3*a^6*c^3+420/b^7*d^6/(b*x+a)^3*a^5*c^4-420/b^6*d^
5/(b*x+a)^3*a^4*c^5+280/b^5*d^4/(b*x+a)^3*a^3*c^6-120/b^4*d^3/(b*x+a)^3*a^2*c^7+30/b^3*d^2/(b*x+a)^3*a*c^8+180
/b^10*d^9/(b*x+a)^2*a^7*c-630/b^9*d^8/(b*x+a)^2*a^6*c^2+1260/b^8*d^7/(b*x+a)^2*a^5*c^3-1575/b^7*d^6/(b*x+a)^2*
a^4*c^4+1260/b^6*d^5/(b*x+a)^2*a^3*c^5-630/b^5*d^4/(b*x+a)^2*a^2*c^6+180/b^4*d^3/(b*x+a)^2*a*c^7-75*d^8/b^6*x^
3*a*c^2-175*d^9/b^8*x^2*a^3*c+675/2*d^8/b^7*x^2*a^2*c^2-300*d^7/b^6*x^2*a*c^3

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maxima [B]  time = 1.93, size = 903, normalized size = 3.45 \begin {gather*} -\frac {3 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 360 \, a^{3} b^{7} c^{7} d^{3} - 5250 \, a^{4} b^{6} c^{6} d^{4} + 19404 \, a^{5} b^{5} c^{5} d^{5} - 35910 \, a^{6} b^{4} c^{4} d^{6} + 38280 \, a^{7} b^{3} c^{3} d^{7} - 23985 \, a^{8} b^{2} c^{2} d^{8} + 8250 \, a^{9} b c d^{9} - 1207 \, a^{10} d^{10} + 1440 \, {\left (b^{10} c^{7} d^{3} - 7 \, 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} + 35 \, a^{4} b^{6} c^{3} d^{7} - 21 \, 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} + 270 \, {\left (b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} - 84 \, a^{2} b^{8} c^{6} d^{4} + 280 \, a^{3} b^{7} c^{5} d^{5} - 490 \, a^{4} b^{6} c^{4} d^{6} + 504 \, a^{5} b^{5} c^{3} d^{7} - 308 \, a^{6} b^{4} c^{2} d^{8} + 104 \, a^{7} b^{3} c d^{9} - 15 \, a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (2 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 72 \, a^{2} b^{8} c^{7} d^{3} - 924 \, a^{3} b^{7} c^{6} d^{4} + 3276 \, a^{4} b^{6} c^{5} d^{5} - 5922 \, a^{5} b^{5} c^{4} d^{6} + 6216 \, a^{6} b^{4} c^{3} d^{7} - 3852 \, a^{7} b^{3} c^{2} d^{8} + 1314 \, a^{8} b^{2} c d^{9} - 191 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{15} x^{4} + 4 \, a b^{14} x^{3} + 6 \, a^{2} b^{13} x^{2} + 4 \, a^{3} b^{12} x + a^{4} b^{11}\right )}} + \frac {2 \, b^{5} d^{10} x^{6} + 12 \, {\left (2 \, b^{5} c d^{9} - a b^{4} d^{10}\right )} x^{5} + 15 \, {\left (9 \, b^{5} c^{2} d^{8} - 10 \, a b^{4} c d^{9} + 3 \, a^{2} b^{3} d^{10}\right )} x^{4} + 20 \, {\left (24 \, b^{5} c^{3} d^{7} - 45 \, a b^{4} c^{2} d^{8} + 30 \, a^{2} b^{3} c d^{9} - 7 \, a^{3} b^{2} d^{10}\right )} x^{3} + 30 \, {\left (42 \, b^{5} c^{4} d^{6} - 120 \, a b^{4} c^{3} d^{7} + 135 \, a^{2} b^{3} c^{2} d^{8} - 70 \, a^{3} b^{2} c d^{9} + 14 \, a^{4} b d^{10}\right )} x^{2} + 12 \, {\left (252 \, b^{5} c^{5} d^{5} - 1050 \, a b^{4} c^{4} d^{6} + 1800 \, a^{2} b^{3} c^{3} d^{7} - 1575 \, a^{3} b^{2} c^{2} d^{8} + 700 \, a^{4} b c d^{9} - 126 \, a^{5} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{6} c^{6} d^{4} - 6 \, a b^{5} c^{5} d^{5} + 15 \, a^{2} b^{4} c^{4} d^{6} - 20 \, a^{3} b^{3} c^{3} d^{7} + 15 \, a^{4} b^{2} c^{2} d^{8} - 6 \, a^{5} b c d^{9} + a^{6} 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)^5,x, algorithm="maxima")

[Out]

-1/12*(3*b^10*c^10 + 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*
a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 38280*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 8250*a^9*b*c*d^9 - 1
207*a^10*d^10 + 1440*(b^10*c^7*d^3 - 7*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 + 35*a^4*b^6*c^
3*d^7 - 21*a^5*b^5*c^2*d^8 + 7*a^6*b^4*c*d^9 - a^7*b^3*d^10)*x^3 + 270*(b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 - 84*a^
2*b^8*c^6*d^4 + 280*a^3*b^7*c^5*d^5 - 490*a^4*b^6*c^4*d^6 + 504*a^5*b^5*c^3*d^7 - 308*a^6*b^4*c^2*d^8 + 104*a^
7*b^3*c*d^9 - 15*a^8*b^2*d^10)*x^2 + 20*(2*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 72*a^2*b^8*c^7*d^3 - 924*a^3*b^7*c^6
*d^4 + 3276*a^4*b^6*c^5*d^5 - 5922*a^5*b^5*c^4*d^6 + 6216*a^6*b^4*c^3*d^7 - 3852*a^7*b^3*c^2*d^8 + 1314*a^8*b^
2*c*d^9 - 191*a^9*b*d^10)*x)/(b^15*x^4 + 4*a*b^14*x^3 + 6*a^2*b^13*x^2 + 4*a^3*b^12*x + a^4*b^11) + 1/12*(2*b^
5*d^10*x^6 + 12*(2*b^5*c*d^9 - a*b^4*d^10)*x^5 + 15*(9*b^5*c^2*d^8 - 10*a*b^4*c*d^9 + 3*a^2*b^3*d^10)*x^4 + 20
*(24*b^5*c^3*d^7 - 45*a*b^4*c^2*d^8 + 30*a^2*b^3*c*d^9 - 7*a^3*b^2*d^10)*x^3 + 30*(42*b^5*c^4*d^6 - 120*a*b^4*
c^3*d^7 + 135*a^2*b^3*c^2*d^8 - 70*a^3*b^2*c*d^9 + 14*a^4*b*d^10)*x^2 + 12*(252*b^5*c^5*d^5 - 1050*a*b^4*c^4*d
^6 + 1800*a^2*b^3*c^3*d^7 - 1575*a^3*b^2*c^2*d^8 + 700*a^4*b*c*d^9 - 126*a^5*d^10)*x)/b^10 + 210*(b^6*c^6*d^4
- 6*a*b^5*c^5*d^5 + 15*a^2*b^4*c^4*d^6 - 20*a^3*b^3*c^3*d^7 + 15*a^4*b^2*c^2*d^8 - 6*a^5*b*c*d^9 + a^6*d^10)*l
og(b*x + a)/b^11

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mupad [B]  time = 0.38, size = 1494, normalized size = 5.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^10/(a + b*x)^5,x)

[Out]

x^2*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^
3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/(2*b) - (5*a^4*d^10)/(2*b^9
) + (105*c^4*d^6)/b^5 + (5*a^3*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^3 - (5*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d
^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^2) - x^5*((a*d^10)/b^6 - (2*c*d^9)/b^5) - x^3*((5*a*((5
*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/(3*b) + (10*a^3*d^10)/(3*b^8)
- (40*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/(3*b^2)) + x^4*((5*a*((5*a*d^10)/b^6 - (10*c*
d^9)/b^5))/(4*b) - (5*a^2*d^10)/(2*b^7) + (45*c^2*d^8)/(4*b^5)) - x*((5*a*((5*a*((5*a*((5*a*((5*a*d^10)/b^6 -
(10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b^8 - (120*c^3*d^7)/b^5 - (10*a^
2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b - (5*a^4*d^10)/b^9 + (210*c^4*d^6)/b^5 + (10*a^3*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b^3 - (10*a^2*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)
/b^5))/b^2))/b + (a^5*d^10)/b^10 - (252*c^5*d^5)/b^5 - (5*a^4*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^4 - (10*a^2
*((5*a*((5*a*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b + (10*a^3*d^10)/b
^8 - (120*c^3*d^7)/b^5 - (10*a^2*((5*a*d^10)/b^6 - (10*c*d^9)/b^5))/b^2))/b^2 + (10*a^3*((5*a*((5*a*d^10)/b^6
- (10*c*d^9)/b^5))/b - (10*a^2*d^10)/b^7 + (45*c^2*d^8)/b^5))/b^3) - ((3*b^10*c^10 - 1207*a^10*d^10 + 45*a^2*b
^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 - 5250*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 - 35910*a^6*b^4*c^4*d^6 + 3828
0*a^7*b^3*c^3*d^7 - 23985*a^8*b^2*c^2*d^8 + 10*a*b^9*c^9*d + 8250*a^9*b*c*d^9)/(12*b) + x*((10*b^9*c^9*d)/3 -
(955*a^9*d^10)/3 + 15*a*b^8*c^8*d^2 + 120*a^2*b^7*c^7*d^3 - 1540*a^3*b^6*c^6*d^4 + 5460*a^4*b^5*c^5*d^5 - 9870
*a^5*b^4*c^4*d^6 + 10360*a^6*b^3*c^3*d^7 - 6420*a^7*b^2*c^2*d^8 + 2190*a^8*b*c*d^9) - x^3*(120*a^7*b^2*d^10 -
120*b^9*c^7*d^3 + 840*a*b^8*c^6*d^4 - 840*a^6*b^3*c*d^9 - 2520*a^2*b^7*c^5*d^5 + 4200*a^3*b^6*c^4*d^6 - 4200*a
^4*b^5*c^3*d^7 + 2520*a^5*b^4*c^2*d^8) + x^2*((45*b^9*c^8*d^2)/2 - (675*a^8*b*d^10)/2 + 180*a*b^8*c^7*d^3 + 23
40*a^7*b^2*c*d^9 - 1890*a^2*b^7*c^6*d^4 + 6300*a^3*b^6*c^5*d^5 - 11025*a^4*b^5*c^4*d^6 + 11340*a^5*b^4*c^3*d^7
- 6930*a^6*b^3*c^2*d^8))/(a^4*b^10 + b^14*x^4 + 4*a^3*b^11*x + 4*a*b^13*x^3 + 6*a^2*b^12*x^2) + (log(a + b*x)
*(210*a^6*d^10 + 210*b^6*c^6*d^4 - 1260*a*b^5*c^5*d^5 + 3150*a^2*b^4*c^4*d^6 - 4200*a^3*b^3*c^3*d^7 + 3150*a^4
*b^2*c^2*d^8 - 1260*a^5*b*c*d^9))/b^11 + (d^10*x^6)/(6*b^5)

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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)**5,x)

[Out]

Timed out

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