3.133 \(\int x (a+b x)^{10} \, dx\)

Optimal. Leaf size=30 \[ \frac{(a+b x)^{12}}{12 b^2}-\frac{a (a+b x)^{11}}{11 b^2} \]

[Out]

-(a*(a + b*x)^11)/(11*b^2) + (a + b*x)^12/(12*b^2)

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Rubi [A]  time = 0.0312031, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^{12}}{12 b^2}-\frac{a (a+b x)^{11}}{11 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^10,x]

[Out]

-(a*(a + b*x)^11)/(11*b^2) + (a + b*x)^12/(12*b^2)

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Rubi in Sympy [A]  time = 12.4348, size = 24, normalized size = 0.8 \[ - \frac{a \left (a + b x\right )^{11}}{11 b^{2}} + \frac{\left (a + b x\right )^{12}}{12 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**10,x)

[Out]

-a*(a + b*x)**11/(11*b**2) + (a + b*x)**12/(12*b**2)

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Mathematica [B]  time = 0.00360685, size = 128, normalized size = 4.27 \[ \frac{a^{10} x^2}{2}+\frac{10}{3} a^9 b x^3+\frac{45}{4} a^8 b^2 x^4+24 a^7 b^3 x^5+35 a^6 b^4 x^6+36 a^5 b^5 x^7+\frac{105}{4} a^4 b^6 x^8+\frac{40}{3} a^3 b^7 x^9+\frac{9}{2} a^2 b^8 x^{10}+\frac{10}{11} a b^9 x^{11}+\frac{b^{10} x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^10,x]

[Out]

(a^10*x^2)/2 + (10*a^9*b*x^3)/3 + (45*a^8*b^2*x^4)/4 + 24*a^7*b^3*x^5 + 35*a^6*b
^4*x^6 + 36*a^5*b^5*x^7 + (105*a^4*b^6*x^8)/4 + (40*a^3*b^7*x^9)/3 + (9*a^2*b^8*
x^10)/2 + (10*a*b^9*x^11)/11 + (b^10*x^12)/12

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Maple [B]  time = 0.003, size = 113, normalized size = 3.8 \[{\frac{{b}^{10}{x}^{12}}{12}}+{\frac{10\,a{b}^{9}{x}^{11}}{11}}+{\frac{9\,{a}^{2}{b}^{8}{x}^{10}}{2}}+{\frac{40\,{a}^{3}{b}^{7}{x}^{9}}{3}}+{\frac{105\,{a}^{4}{b}^{6}{x}^{8}}{4}}+36\,{a}^{5}{b}^{5}{x}^{7}+35\,{a}^{6}{b}^{4}{x}^{6}+24\,{a}^{7}{b}^{3}{x}^{5}+{\frac{45\,{a}^{8}{b}^{2}{x}^{4}}{4}}+{\frac{10\,{a}^{9}b{x}^{3}}{3}}+{\frac{{a}^{10}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^10,x)

[Out]

1/12*b^10*x^12+10/11*a*b^9*x^11+9/2*a^2*b^8*x^10+40/3*a^3*b^7*x^9+105/4*a^4*b^6*
x^8+36*a^5*b^5*x^7+35*a^6*b^4*x^6+24*a^7*b^3*x^5+45/4*a^8*b^2*x^4+10/3*a^9*b*x^3
+1/2*a^10*x^2

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Maxima [A]  time = 1.33495, size = 151, normalized size = 5.03 \[ \frac{1}{12} \, b^{10} x^{12} + \frac{10}{11} \, a b^{9} x^{11} + \frac{9}{2} \, a^{2} b^{8} x^{10} + \frac{40}{3} \, a^{3} b^{7} x^{9} + \frac{105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac{45}{4} \, a^{8} b^{2} x^{4} + \frac{10}{3} \, a^{9} b x^{3} + \frac{1}{2} \, a^{10} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^10*x^12 + 10/11*a*b^9*x^11 + 9/2*a^2*b^8*x^10 + 40/3*a^3*b^7*x^9 + 105/4*
a^4*b^6*x^8 + 36*a^5*b^5*x^7 + 35*a^6*b^4*x^6 + 24*a^7*b^3*x^5 + 45/4*a^8*b^2*x^
4 + 10/3*a^9*b*x^3 + 1/2*a^10*x^2

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Fricas [A]  time = 0.187813, size = 1, normalized size = 0.03 \[ \frac{1}{12} x^{12} b^{10} + \frac{10}{11} x^{11} b^{9} a + \frac{9}{2} x^{10} b^{8} a^{2} + \frac{40}{3} x^{9} b^{7} a^{3} + \frac{105}{4} x^{8} b^{6} a^{4} + 36 x^{7} b^{5} a^{5} + 35 x^{6} b^{4} a^{6} + 24 x^{5} b^{3} a^{7} + \frac{45}{4} x^{4} b^{2} a^{8} + \frac{10}{3} x^{3} b a^{9} + \frac{1}{2} x^{2} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*b^10 + 10/11*x^11*b^9*a + 9/2*x^10*b^8*a^2 + 40/3*x^9*b^7*a^3 + 105/4*
x^8*b^6*a^4 + 36*x^7*b^5*a^5 + 35*x^6*b^4*a^6 + 24*x^5*b^3*a^7 + 45/4*x^4*b^2*a^
8 + 10/3*x^3*b*a^9 + 1/2*x^2*a^10

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Sympy [A]  time = 0.165047, size = 129, normalized size = 4.3 \[ \frac{a^{10} x^{2}}{2} + \frac{10 a^{9} b x^{3}}{3} + \frac{45 a^{8} b^{2} x^{4}}{4} + 24 a^{7} b^{3} x^{5} + 35 a^{6} b^{4} x^{6} + 36 a^{5} b^{5} x^{7} + \frac{105 a^{4} b^{6} x^{8}}{4} + \frac{40 a^{3} b^{7} x^{9}}{3} + \frac{9 a^{2} b^{8} x^{10}}{2} + \frac{10 a b^{9} x^{11}}{11} + \frac{b^{10} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**10,x)

[Out]

a**10*x**2/2 + 10*a**9*b*x**3/3 + 45*a**8*b**2*x**4/4 + 24*a**7*b**3*x**5 + 35*a
**6*b**4*x**6 + 36*a**5*b**5*x**7 + 105*a**4*b**6*x**8/4 + 40*a**3*b**7*x**9/3 +
9*a**2*b**8*x**10/2 + 10*a*b**9*x**11/11 + b**10*x**12/12

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GIAC/XCAS [A]  time = 0.200475, size = 151, normalized size = 5.03 \[ \frac{1}{12} \, b^{10} x^{12} + \frac{10}{11} \, a b^{9} x^{11} + \frac{9}{2} \, a^{2} b^{8} x^{10} + \frac{40}{3} \, a^{3} b^{7} x^{9} + \frac{105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac{45}{4} \, a^{8} b^{2} x^{4} + \frac{10}{3} \, a^{9} b x^{3} + \frac{1}{2} \, a^{10} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*b^10*x^12 + 10/11*a*b^9*x^11 + 9/2*a^2*b^8*x^10 + 40/3*a^3*b^7*x^9 + 105/4*
a^4*b^6*x^8 + 36*a^5*b^5*x^7 + 35*a^6*b^4*x^6 + 24*a^7*b^3*x^5 + 45/4*a^8*b^2*x^
4 + 10/3*a^9*b*x^3 + 1/2*a^10*x^2