3.1316 \(\int \frac{(c+d x)^{10}}{(a+b x)^5} \, dx\)

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

[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^11) + (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

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Rubi [A]  time = 0.950403, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 d^9 (a+b x)^5 (b c-a d)}{b^{11}}+\frac{45 d^8 (a+b x)^4 (b c-a d)^2}{4 b^{11}}+\frac{40 d^7 (a+b x)^3 (b c-a d)^3}{b^{11}}+\frac{105 d^6 (a+b x)^2 (b c-a d)^4}{b^{11}}+\frac{210 d^4 (b c-a d)^6 \log (a+b x)}{b^{11}}-\frac{120 d^3 (b c-a d)^7}{b^{11} (a+b x)}-\frac{45 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^2}-\frac{10 d (b c-a d)^9}{3 b^{11} (a+b x)^3}-\frac{(b c-a d)^{10}}{4 b^{11} (a+b x)^4}+\frac{d^{10} (a+b x)^6}{6 b^{11}}+\frac{252 d^5 x (b c-a d)^5}{b^{10}} \]

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^11) + (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

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Rubi in Sympy [A]  time = 135.097, size = 243, normalized size = 0.93 \[ - \frac{252 d^{5} x \left (a d - b c\right )^{5}}{b^{10}} + \frac{d^{10} \left (a + b x\right )^{6}}{6 b^{11}} - \frac{2 d^{9} \left (a + b x\right )^{5} \left (a d - b c\right )}{b^{11}} + \frac{45 d^{8} \left (a + b x\right )^{4} \left (a d - b c\right )^{2}}{4 b^{11}} - \frac{40 d^{7} \left (a + b x\right )^{3} \left (a d - b c\right )^{3}}{b^{11}} + \frac{105 d^{6} \left (a + b x\right )^{2} \left (a d - b c\right )^{4}}{b^{11}} + \frac{210 d^{4} \left (a d - b c\right )^{6} \log{\left (a + b x \right )}}{b^{11}} + \frac{120 d^{3} \left (a d - b c\right )^{7}}{b^{11} \left (a + b x\right )} - \frac{45 d^{2} \left (a d - b c\right )^{8}}{2 b^{11} \left (a + b x\right )^{2}} + \frac{10 d \left (a d - b c\right )^{9}}{3 b^{11} \left (a + b x\right )^{3}} - \frac{\left (a d - b c\right )^{10}}{4 b^{11} \left (a + b x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**5,x)

[Out]

-252*d**5*x*(a*d - b*c)**5/b**10 + d**10*(a + b*x)**6/(6*b**11) - 2*d**9*(a + b*
x)**5*(a*d - b*c)/b**11 + 45*d**8*(a + b*x)**4*(a*d - b*c)**2/(4*b**11) - 40*d**
7*(a + b*x)**3*(a*d - b*c)**3/b**11 + 105*d**6*(a + b*x)**2*(a*d - b*c)**4/b**11
+ 210*d**4*(a*d - b*c)**6*log(a + b*x)/b**11 + 120*d**3*(a*d - b*c)**7/(b**11*(
a + b*x)) - 45*d**2*(a*d - b*c)**8/(2*b**11*(a + b*x)**2) + 10*d*(a*d - b*c)**9/
(3*b**11*(a + b*x)**3) - (a*d - b*c)**10/(4*b**11*(a + b*x)**4)

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Mathematica [A]  time = 0.335869, size = 359, normalized size = 1.37 \[ \frac{15 b^4 d^8 x^4 \left (3 a^2 d^2-10 a b c d+9 b^2 c^2\right )+20 b^3 d^7 x^3 \left (-7 a^3 d^3+30 a^2 b c d^2-45 a b^2 c^2 d+24 b^3 c^3\right )+30 b^2 d^6 x^2 \left (14 a^4 d^4-70 a^3 b c d^3+135 a^2 b^2 c^2 d^2-120 a b^3 c^3 d+42 b^4 c^4\right )+12 b d^5 x \left (-126 a^5 d^5+700 a^4 b c d^4-1575 a^3 b^2 c^2 d^3+1800 a^2 b^3 c^3 d^2-1050 a b^4 c^4 d+252 b^5 c^5\right )+12 b^5 d^9 x^5 (2 b c-a d)+2520 d^4 (b c-a d)^6 \log (a+b x)+\frac{1440 d^3 (a d-b c)^7}{a+b x}-\frac{270 d^2 (b c-a d)^8}{(a+b x)^2}+\frac{40 d (a d-b c)^9}{(a+b x)^3}-\frac{3 (b c-a d)^{10}}{(a+b x)^4}+2 b^6 d^{10} x^6}{12 b^{11}} \]

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 - 126*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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Maple [B]  time = 0.026, size = 1172, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^5,x)

[Out]

-45/2/b^11*d^10/(b*x+a)^2*a^8-45/2/b^3*d^2/(b*x+a)^2*c^8+120/b^11*d^10/(b*x+a)*a
^7-120/b^4*d^3/(b*x+a)*c^7-1/4/b^11/(b*x+a)^4*a^10*d^10+10/3/b^11*d^10/(b*x+a)^3
*a^9-10/3/b^2*d/(b*x+a)^3*c^9+210/b^5*d^4*ln(b*x+a)*c^6-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+210/b^11*d^10*ln(b*x+a)*a^6-25/2*d^9/b^6*x^4*a*c+50*d^9/b^7*x^3*a^2*c+3
0/b^4/(b*x+a)^4*a^3*c^7*d^3+1800*d^7/b^7*a^2*c^3*x-1050*d^6/b^6*a*c^4*x-1260/b^1
0*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^4-45/4/b^3/(b*x+a)^4*a^2*c^8*d^2+5/2/b^2/(b*x+a)^4*a*c^9*d-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^7-840/b^10*d^9/(b*x+a)*a^6*c+2520/b^9*d^8/(b*x+a)*a^5*c^2-4
200/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^6-30/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^3-75*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+700*d^9/b^9*a^4*c*x-1575*d^8/b^8
*a^3*c^2*x+1/6*d^10/b^5*x^6-1/4/b/(b*x+a)^4*c^10

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Maxima [A]  time = 1.45106, size = 1219, normalized size = 4.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^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 - 1207*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 - 49
0*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 - 1
26*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)*log(b*x + a
)/b^11

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Fricas [A]  time = 0.211774, size = 1843, normalized size = 7.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^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 - 8250*a^9*b*c*d^9 + 120
7*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 - 54
765*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 + 1058
4*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^1
0*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**10/(b*x+a)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228917, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done