3.1306 \(\int (a+b x)^5 (c+d x)^{10} \, dx\)

Optimal. Leaf size=146 \[ -\frac{b^4 (c+d x)^{15} (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^{14} (b c-a d)^2}{7 d^6}-\frac{10 b^2 (c+d x)^{13} (b c-a d)^3}{13 d^6}+\frac{5 b (c+d x)^{12} (b c-a d)^4}{12 d^6}-\frac{(c+d x)^{11} (b c-a d)^5}{11 d^6}+\frac{b^5 (c+d x)^{16}}{16 d^6} \]

[Out]

-((b*c - a*d)^5*(c + d*x)^11)/(11*d^6) + (5*b*(b*c - a*d)^4*(c + d*x)^12)/(12*d^
6) - (10*b^2*(b*c - a*d)^3*(c + d*x)^13)/(13*d^6) + (5*b^3*(b*c - a*d)^2*(c + d*
x)^14)/(7*d^6) - (b^4*(b*c - a*d)*(c + d*x)^15)/(3*d^6) + (b^5*(c + d*x)^16)/(16
*d^6)

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Rubi [A]  time = 1.07726, antiderivative size = 146, 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 \[ -\frac{b^4 (c+d x)^{15} (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^{14} (b c-a d)^2}{7 d^6}-\frac{10 b^2 (c+d x)^{13} (b c-a d)^3}{13 d^6}+\frac{5 b (c+d x)^{12} (b c-a d)^4}{12 d^6}-\frac{(c+d x)^{11} (b c-a d)^5}{11 d^6}+\frac{b^5 (c+d x)^{16}}{16 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5*(c + d*x)^10,x]

[Out]

-((b*c - a*d)^5*(c + d*x)^11)/(11*d^6) + (5*b*(b*c - a*d)^4*(c + d*x)^12)/(12*d^
6) - (10*b^2*(b*c - a*d)^3*(c + d*x)^13)/(13*d^6) + (5*b^3*(b*c - a*d)^2*(c + d*
x)^14)/(7*d^6) - (b^4*(b*c - a*d)*(c + d*x)^15)/(3*d^6) + (b^5*(c + d*x)^16)/(16
*d^6)

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Rubi in Sympy [A]  time = 96.4866, size = 131, normalized size = 0.9 \[ \frac{b^{5} \left (c + d x\right )^{16}}{16 d^{6}} + \frac{b^{4} \left (c + d x\right )^{15} \left (a d - b c\right )}{3 d^{6}} + \frac{5 b^{3} \left (c + d x\right )^{14} \left (a d - b c\right )^{2}}{7 d^{6}} + \frac{10 b^{2} \left (c + d x\right )^{13} \left (a d - b c\right )^{3}}{13 d^{6}} + \frac{5 b \left (c + d x\right )^{12} \left (a d - b c\right )^{4}}{12 d^{6}} + \frac{\left (c + d x\right )^{11} \left (a d - b c\right )^{5}}{11 d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**5*(c + d*x)**16/(16*d**6) + b**4*(c + d*x)**15*(a*d - b*c)/(3*d**6) + 5*b**3*
(c + d*x)**14*(a*d - b*c)**2/(7*d**6) + 10*b**2*(c + d*x)**13*(a*d - b*c)**3/(13
*d**6) + 5*b*(c + d*x)**12*(a*d - b*c)**4/(12*d**6) + (c + d*x)**11*(a*d - b*c)*
*5/(11*d**6)

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Mathematica [B]  time = 0.172805, size = 811, normalized size = 5.55 \[ \frac{1}{16} b^5 d^{10} x^{16}+\frac{1}{3} b^4 d^9 (2 b c+a d) x^{15}+\frac{5}{14} b^3 d^8 \left (9 b^2 c^2+10 a b d c+2 a^2 d^2\right ) x^{14}+\frac{5}{13} b^2 d^7 \left (24 b^3 c^3+45 a b^2 d c^2+20 a^2 b d^2 c+2 a^3 d^3\right ) x^{13}+\frac{5}{12} b d^6 \left (42 b^4 c^4+120 a b^3 d c^3+90 a^2 b^2 d^2 c^2+20 a^3 b d^3 c+a^4 d^4\right ) x^{12}+\frac{1}{11} d^5 \left (252 b^5 c^5+1050 a b^4 d c^4+1200 a^2 b^3 d^2 c^3+450 a^3 b^2 d^3 c^2+50 a^4 b d^4 c+a^5 d^5\right ) x^{11}+\frac{1}{2} c d^4 \left (42 b^5 c^5+252 a b^4 d c^4+420 a^2 b^3 d^2 c^3+240 a^3 b^2 d^3 c^2+45 a^4 b d^4 c+2 a^5 d^5\right ) x^{10}+\frac{5}{3} c^2 d^3 \left (8 b^5 c^5+70 a b^4 d c^4+168 a^2 b^3 d^2 c^3+140 a^3 b^2 d^3 c^2+40 a^4 b d^4 c+3 a^5 d^5\right ) x^9+\frac{15}{8} c^3 d^2 \left (3 b^5 c^5+40 a b^4 d c^4+140 a^2 b^3 d^2 c^3+168 a^3 b^2 d^3 c^2+70 a^4 b d^4 c+8 a^5 d^5\right ) x^8+\frac{5}{7} c^4 d \left (2 b^5 c^5+45 a b^4 d c^4+240 a^2 b^3 d^2 c^3+420 a^3 b^2 d^3 c^2+252 a^4 b d^4 c+42 a^5 d^5\right ) x^7+\frac{1}{6} c^5 \left (b^5 c^5+50 a b^4 d c^4+450 a^2 b^3 d^2 c^3+1200 a^3 b^2 d^3 c^2+1050 a^4 b d^4 c+252 a^5 d^5\right ) x^6+a c^6 \left (b^4 c^4+20 a b^3 d c^3+90 a^2 b^2 d^2 c^2+120 a^3 b d^3 c+42 a^4 d^4\right ) x^5+\frac{5}{4} a^2 c^7 \left (2 b^3 c^3+20 a b^2 d c^2+45 a^2 b d^2 c+24 a^3 d^3\right ) x^4+\frac{5}{3} a^3 c^8 \left (2 b^2 c^2+10 a b d c+9 a^2 d^2\right ) x^3+\frac{5}{2} a^4 c^9 (b c+2 a d) x^2+a^5 c^{10} x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5*(c + d*x)^10,x]

[Out]

a^5*c^10*x + (5*a^4*c^9*(b*c + 2*a*d)*x^2)/2 + (5*a^3*c^8*(2*b^2*c^2 + 10*a*b*c*
d + 9*a^2*d^2)*x^3)/3 + (5*a^2*c^7*(2*b^3*c^3 + 20*a*b^2*c^2*d + 45*a^2*b*c*d^2
+ 24*a^3*d^3)*x^4)/4 + a*c^6*(b^4*c^4 + 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 12
0*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + (c^5*(b^5*c^5 + 50*a*b^4*c^4*d + 450*a^2*b^3*c
^3*d^2 + 1200*a^3*b^2*c^2*d^3 + 1050*a^4*b*c*d^4 + 252*a^5*d^5)*x^6)/6 + (5*c^4*
d*(2*b^5*c^5 + 45*a*b^4*c^4*d + 240*a^2*b^3*c^3*d^2 + 420*a^3*b^2*c^2*d^3 + 252*
a^4*b*c*d^4 + 42*a^5*d^5)*x^7)/7 + (15*c^3*d^2*(3*b^5*c^5 + 40*a*b^4*c^4*d + 140
*a^2*b^3*c^3*d^2 + 168*a^3*b^2*c^2*d^3 + 70*a^4*b*c*d^4 + 8*a^5*d^5)*x^8)/8 + (5
*c^2*d^3*(8*b^5*c^5 + 70*a*b^4*c^4*d + 168*a^2*b^3*c^3*d^2 + 140*a^3*b^2*c^2*d^3
+ 40*a^4*b*c*d^4 + 3*a^5*d^5)*x^9)/3 + (c*d^4*(42*b^5*c^5 + 252*a*b^4*c^4*d + 4
20*a^2*b^3*c^3*d^2 + 240*a^3*b^2*c^2*d^3 + 45*a^4*b*c*d^4 + 2*a^5*d^5)*x^10)/2 +
(d^5*(252*b^5*c^5 + 1050*a*b^4*c^4*d + 1200*a^2*b^3*c^3*d^2 + 450*a^3*b^2*c^2*d
^3 + 50*a^4*b*c*d^4 + a^5*d^5)*x^11)/11 + (5*b*d^6*(42*b^4*c^4 + 120*a*b^3*c^3*d
+ 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 + a^4*d^4)*x^12)/12 + (5*b^2*d^7*(24*b^3*
c^3 + 45*a*b^2*c^2*d + 20*a^2*b*c*d^2 + 2*a^3*d^3)*x^13)/13 + (5*b^3*d^8*(9*b^2*
c^2 + 10*a*b*c*d + 2*a^2*d^2)*x^14)/14 + (b^4*d^9*(2*b*c + a*d)*x^15)/3 + (b^5*d
^10*x^16)/16

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Maple [B]  time = 0.003, size = 841, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5*(d*x+c)^10,x)

[Out]

1/16*b^5*d^10*x^16+1/15*(5*a*b^4*d^10+10*b^5*c*d^9)*x^15+1/14*(10*a^2*b^3*d^10+5
0*a*b^4*c*d^9+45*b^5*c^2*d^8)*x^14+1/13*(10*a^3*b^2*d^10+100*a^2*b^3*c*d^9+225*a
*b^4*c^2*d^8+120*b^5*c^3*d^7)*x^13+1/12*(5*a^4*b*d^10+100*a^3*b^2*c*d^9+450*a^2*
b^3*c^2*d^8+600*a*b^4*c^3*d^7+210*b^5*c^4*d^6)*x^12+1/11*(a^5*d^10+50*a^4*b*c*d^
9+450*a^3*b^2*c^2*d^8+1200*a^2*b^3*c^3*d^7+1050*a*b^4*c^4*d^6+252*b^5*c^5*d^5)*x
^11+1/10*(10*a^5*c*d^9+225*a^4*b*c^2*d^8+1200*a^3*b^2*c^3*d^7+2100*a^2*b^3*c^4*d
^6+1260*a*b^4*c^5*d^5+210*b^5*c^6*d^4)*x^10+1/9*(45*a^5*c^2*d^8+600*a^4*b*c^3*d^
7+2100*a^3*b^2*c^4*d^6+2520*a^2*b^3*c^5*d^5+1050*a*b^4*c^6*d^4+120*b^5*c^7*d^3)*
x^9+1/8*(120*a^5*c^3*d^7+1050*a^4*b*c^4*d^6+2520*a^3*b^2*c^5*d^5+2100*a^2*b^3*c^
6*d^4+600*a*b^4*c^7*d^3+45*b^5*c^8*d^2)*x^8+1/7*(210*a^5*c^4*d^6+1260*a^4*b*c^5*
d^5+2100*a^3*b^2*c^6*d^4+1200*a^2*b^3*c^7*d^3+225*a*b^4*c^8*d^2+10*b^5*c^9*d)*x^
7+1/6*(252*a^5*c^5*d^5+1050*a^4*b*c^6*d^4+1200*a^3*b^2*c^7*d^3+450*a^2*b^3*c^8*d
^2+50*a*b^4*c^9*d+b^5*c^10)*x^6+1/5*(210*a^5*c^6*d^4+600*a^4*b*c^7*d^3+450*a^3*b
^2*c^8*d^2+100*a^2*b^3*c^9*d+5*a*b^4*c^10)*x^5+1/4*(120*a^5*c^7*d^3+225*a^4*b*c^
8*d^2+100*a^3*b^2*c^9*d+10*a^2*b^3*c^10)*x^4+1/3*(45*a^5*c^8*d^2+50*a^4*b*c^9*d+
10*a^3*b^2*c^10)*x^3+1/2*(10*a^5*c^9*d+5*a^4*b*c^10)*x^2+a^5*c^10*x

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Maxima [A]  time = 1.34407, size = 1127, normalized size = 7.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b^5*d^10*x^16 + a^5*c^10*x + 1/3*(2*b^5*c*d^9 + a*b^4*d^10)*x^15 + 5/14*(9*
b^5*c^2*d^8 + 10*a*b^4*c*d^9 + 2*a^2*b^3*d^10)*x^14 + 5/13*(24*b^5*c^3*d^7 + 45*
a*b^4*c^2*d^8 + 20*a^2*b^3*c*d^9 + 2*a^3*b^2*d^10)*x^13 + 5/12*(42*b^5*c^4*d^6 +
120*a*b^4*c^3*d^7 + 90*a^2*b^3*c^2*d^8 + 20*a^3*b^2*c*d^9 + a^4*b*d^10)*x^12 +
1/11*(252*b^5*c^5*d^5 + 1050*a*b^4*c^4*d^6 + 1200*a^2*b^3*c^3*d^7 + 450*a^3*b^2*
c^2*d^8 + 50*a^4*b*c*d^9 + a^5*d^10)*x^11 + 1/2*(42*b^5*c^6*d^4 + 252*a*b^4*c^5*
d^5 + 420*a^2*b^3*c^4*d^6 + 240*a^3*b^2*c^3*d^7 + 45*a^4*b*c^2*d^8 + 2*a^5*c*d^9
)*x^10 + 5/3*(8*b^5*c^7*d^3 + 70*a*b^4*c^6*d^4 + 168*a^2*b^3*c^5*d^5 + 140*a^3*b
^2*c^4*d^6 + 40*a^4*b*c^3*d^7 + 3*a^5*c^2*d^8)*x^9 + 15/8*(3*b^5*c^8*d^2 + 40*a*
b^4*c^7*d^3 + 140*a^2*b^3*c^6*d^4 + 168*a^3*b^2*c^5*d^5 + 70*a^4*b*c^4*d^6 + 8*a
^5*c^3*d^7)*x^8 + 5/7*(2*b^5*c^9*d + 45*a*b^4*c^8*d^2 + 240*a^2*b^3*c^7*d^3 + 42
0*a^3*b^2*c^6*d^4 + 252*a^4*b*c^5*d^5 + 42*a^5*c^4*d^6)*x^7 + 1/6*(b^5*c^10 + 50
*a*b^4*c^9*d + 450*a^2*b^3*c^8*d^2 + 1200*a^3*b^2*c^7*d^3 + 1050*a^4*b*c^6*d^4 +
252*a^5*c^5*d^5)*x^6 + (a*b^4*c^10 + 20*a^2*b^3*c^9*d + 90*a^3*b^2*c^8*d^2 + 12
0*a^4*b*c^7*d^3 + 42*a^5*c^6*d^4)*x^5 + 5/4*(2*a^2*b^3*c^10 + 20*a^3*b^2*c^9*d +
45*a^4*b*c^8*d^2 + 24*a^5*c^7*d^3)*x^4 + 5/3*(2*a^3*b^2*c^10 + 10*a^4*b*c^9*d +
9*a^5*c^8*d^2)*x^3 + 5/2*(a^4*b*c^10 + 2*a^5*c^9*d)*x^2

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Fricas [A]  time = 0.213999, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*x^16*d^10*b^5 + 2/3*x^15*d^9*c*b^5 + 1/3*x^15*d^10*b^4*a + 45/14*x^14*d^8*c
^2*b^5 + 25/7*x^14*d^9*c*b^4*a + 5/7*x^14*d^10*b^3*a^2 + 120/13*x^13*d^7*c^3*b^5
+ 225/13*x^13*d^8*c^2*b^4*a + 100/13*x^13*d^9*c*b^3*a^2 + 10/13*x^13*d^10*b^2*a
^3 + 35/2*x^12*d^6*c^4*b^5 + 50*x^12*d^7*c^3*b^4*a + 75/2*x^12*d^8*c^2*b^3*a^2 +
25/3*x^12*d^9*c*b^2*a^3 + 5/12*x^12*d^10*b*a^4 + 252/11*x^11*d^5*c^5*b^5 + 1050
/11*x^11*d^6*c^4*b^4*a + 1200/11*x^11*d^7*c^3*b^3*a^2 + 450/11*x^11*d^8*c^2*b^2*
a^3 + 50/11*x^11*d^9*c*b*a^4 + 1/11*x^11*d^10*a^5 + 21*x^10*d^4*c^6*b^5 + 126*x^
10*d^5*c^5*b^4*a + 210*x^10*d^6*c^4*b^3*a^2 + 120*x^10*d^7*c^3*b^2*a^3 + 45/2*x^
10*d^8*c^2*b*a^4 + x^10*d^9*c*a^5 + 40/3*x^9*d^3*c^7*b^5 + 350/3*x^9*d^4*c^6*b^4
*a + 280*x^9*d^5*c^5*b^3*a^2 + 700/3*x^9*d^6*c^4*b^2*a^3 + 200/3*x^9*d^7*c^3*b*a
^4 + 5*x^9*d^8*c^2*a^5 + 45/8*x^8*d^2*c^8*b^5 + 75*x^8*d^3*c^7*b^4*a + 525/2*x^8
*d^4*c^6*b^3*a^2 + 315*x^8*d^5*c^5*b^2*a^3 + 525/4*x^8*d^6*c^4*b*a^4 + 15*x^8*d^
7*c^3*a^5 + 10/7*x^7*d*c^9*b^5 + 225/7*x^7*d^2*c^8*b^4*a + 1200/7*x^7*d^3*c^7*b^
3*a^2 + 300*x^7*d^4*c^6*b^2*a^3 + 180*x^7*d^5*c^5*b*a^4 + 30*x^7*d^6*c^4*a^5 + 1
/6*x^6*c^10*b^5 + 25/3*x^6*d*c^9*b^4*a + 75*x^6*d^2*c^8*b^3*a^2 + 200*x^6*d^3*c^
7*b^2*a^3 + 175*x^6*d^4*c^6*b*a^4 + 42*x^6*d^5*c^5*a^5 + x^5*c^10*b^4*a + 20*x^5
*d*c^9*b^3*a^2 + 90*x^5*d^2*c^8*b^2*a^3 + 120*x^5*d^3*c^7*b*a^4 + 42*x^5*d^4*c^6
*a^5 + 5/2*x^4*c^10*b^3*a^2 + 25*x^4*d*c^9*b^2*a^3 + 225/4*x^4*d^2*c^8*b*a^4 + 3
0*x^4*d^3*c^7*a^5 + 10/3*x^3*c^10*b^2*a^3 + 50/3*x^3*d*c^9*b*a^4 + 15*x^3*d^2*c^
8*a^5 + 5/2*x^2*c^10*b*a^4 + 5*x^2*d*c^9*a^5 + x*c^10*a^5

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Sympy [A]  time = 0.464677, size = 940, normalized size = 6.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**5*c**10*x + b**5*d**10*x**16/16 + x**15*(a*b**4*d**10/3 + 2*b**5*c*d**9/3) +
x**14*(5*a**2*b**3*d**10/7 + 25*a*b**4*c*d**9/7 + 45*b**5*c**2*d**8/14) + x**13*
(10*a**3*b**2*d**10/13 + 100*a**2*b**3*c*d**9/13 + 225*a*b**4*c**2*d**8/13 + 120
*b**5*c**3*d**7/13) + x**12*(5*a**4*b*d**10/12 + 25*a**3*b**2*c*d**9/3 + 75*a**2
*b**3*c**2*d**8/2 + 50*a*b**4*c**3*d**7 + 35*b**5*c**4*d**6/2) + x**11*(a**5*d**
10/11 + 50*a**4*b*c*d**9/11 + 450*a**3*b**2*c**2*d**8/11 + 1200*a**2*b**3*c**3*d
**7/11 + 1050*a*b**4*c**4*d**6/11 + 252*b**5*c**5*d**5/11) + x**10*(a**5*c*d**9
+ 45*a**4*b*c**2*d**8/2 + 120*a**3*b**2*c**3*d**7 + 210*a**2*b**3*c**4*d**6 + 12
6*a*b**4*c**5*d**5 + 21*b**5*c**6*d**4) + x**9*(5*a**5*c**2*d**8 + 200*a**4*b*c*
*3*d**7/3 + 700*a**3*b**2*c**4*d**6/3 + 280*a**2*b**3*c**5*d**5 + 350*a*b**4*c**
6*d**4/3 + 40*b**5*c**7*d**3/3) + x**8*(15*a**5*c**3*d**7 + 525*a**4*b*c**4*d**6
/4 + 315*a**3*b**2*c**5*d**5 + 525*a**2*b**3*c**6*d**4/2 + 75*a*b**4*c**7*d**3 +
45*b**5*c**8*d**2/8) + x**7*(30*a**5*c**4*d**6 + 180*a**4*b*c**5*d**5 + 300*a**
3*b**2*c**6*d**4 + 1200*a**2*b**3*c**7*d**3/7 + 225*a*b**4*c**8*d**2/7 + 10*b**5
*c**9*d/7) + x**6*(42*a**5*c**5*d**5 + 175*a**4*b*c**6*d**4 + 200*a**3*b**2*c**7
*d**3 + 75*a**2*b**3*c**8*d**2 + 25*a*b**4*c**9*d/3 + b**5*c**10/6) + x**5*(42*a
**5*c**6*d**4 + 120*a**4*b*c**7*d**3 + 90*a**3*b**2*c**8*d**2 + 20*a**2*b**3*c**
9*d + a*b**4*c**10) + x**4*(30*a**5*c**7*d**3 + 225*a**4*b*c**8*d**2/4 + 25*a**3
*b**2*c**9*d + 5*a**2*b**3*c**10/2) + x**3*(15*a**5*c**8*d**2 + 50*a**4*b*c**9*d
/3 + 10*a**3*b**2*c**10/3) + x**2*(5*a**5*c**9*d + 5*a**4*b*c**10/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219422, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done