3.131 \(\int x^3 (a+b x)^{10} \, dx\)

Optimal. Leaf size=64 \[ -\frac{a^3 (a+b x)^{11}}{11 b^4}+\frac{a^2 (a+b x)^{12}}{4 b^4}+\frac{(a+b x)^{14}}{14 b^4}-\frac{3 a (a+b x)^{13}}{13 b^4} \]

[Out]

-(a^3*(a + b*x)^11)/(11*b^4) + (a^2*(a + b*x)^12)/(4*b^4) - (3*a*(a + b*x)^13)/(
13*b^4) + (a + b*x)^14/(14*b^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.0920351, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^3 (a+b x)^{11}}{11 b^4}+\frac{a^2 (a+b x)^{12}}{4 b^4}+\frac{(a+b x)^{14}}{14 b^4}-\frac{3 a (a+b x)^{13}}{13 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^10,x]

[Out]

-(a^3*(a + b*x)^11)/(11*b^4) + (a^2*(a + b*x)^12)/(4*b^4) - (3*a*(a + b*x)^13)/(
13*b^4) + (a + b*x)^14/(14*b^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.0495, size = 56, normalized size = 0.88 \[ - \frac{a^{3} \left (a + b x\right )^{11}}{11 b^{4}} + \frac{a^{2} \left (a + b x\right )^{12}}{4 b^{4}} - \frac{3 a \left (a + b x\right )^{13}}{13 b^{4}} + \frac{\left (a + b x\right )^{14}}{14 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**10,x)

[Out]

-a**3*(a + b*x)**11/(11*b**4) + a**2*(a + b*x)**12/(4*b**4) - 3*a*(a + b*x)**13/
(13*b**4) + (a + b*x)**14/(14*b**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00423498, size = 128, normalized size = 2. \[ \frac{a^{10} x^4}{4}+2 a^9 b x^5+\frac{15}{2} a^8 b^2 x^6+\frac{120}{7} a^7 b^3 x^7+\frac{105}{4} a^6 b^4 x^8+28 a^5 b^5 x^9+21 a^4 b^6 x^{10}+\frac{120}{11} a^3 b^7 x^{11}+\frac{15}{4} a^2 b^8 x^{12}+\frac{10}{13} a b^9 x^{13}+\frac{b^{10} x^{14}}{14} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x)^10,x]

[Out]

(a^10*x^4)/4 + 2*a^9*b*x^5 + (15*a^8*b^2*x^6)/2 + (120*a^7*b^3*x^7)/7 + (105*a^6
*b^4*x^8)/4 + 28*a^5*b^5*x^9 + 21*a^4*b^6*x^10 + (120*a^3*b^7*x^11)/11 + (15*a^2
*b^8*x^12)/4 + (10*a*b^9*x^13)/13 + (b^10*x^14)/14

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 113, normalized size = 1.8 \[{\frac{{b}^{10}{x}^{14}}{14}}+{\frac{10\,a{b}^{9}{x}^{13}}{13}}+{\frac{15\,{a}^{2}{b}^{8}{x}^{12}}{4}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{11}}{11}}+21\,{a}^{4}{b}^{6}{x}^{10}+28\,{a}^{5}{b}^{5}{x}^{9}+{\frac{105\,{a}^{6}{b}^{4}{x}^{8}}{4}}+{\frac{120\,{a}^{7}{b}^{3}{x}^{7}}{7}}+{\frac{15\,{a}^{8}{b}^{2}{x}^{6}}{2}}+2\,{a}^{9}b{x}^{5}+{\frac{{a}^{10}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^10,x)

[Out]

1/14*b^10*x^14+10/13*a*b^9*x^13+15/4*a^2*b^8*x^12+120/11*a^3*b^7*x^11+21*a^4*b^6
*x^10+28*a^5*b^5*x^9+105/4*a^6*b^4*x^8+120/7*a^7*b^3*x^7+15/2*a^8*b^2*x^6+2*a^9*
b*x^5+1/4*a^10*x^4

_______________________________________________________________________________________

Maxima [A]  time = 1.33008, size = 151, normalized size = 2.36 \[ \frac{1}{14} \, b^{10} x^{14} + \frac{10}{13} \, a b^{9} x^{13} + \frac{15}{4} \, a^{2} b^{8} x^{12} + \frac{120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac{105}{4} \, a^{6} b^{4} x^{8} + \frac{120}{7} \, a^{7} b^{3} x^{7} + \frac{15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac{1}{4} \, a^{10} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^3,x, algorithm="maxima")

[Out]

1/14*b^10*x^14 + 10/13*a*b^9*x^13 + 15/4*a^2*b^8*x^12 + 120/11*a^3*b^7*x^11 + 21
*a^4*b^6*x^10 + 28*a^5*b^5*x^9 + 105/4*a^6*b^4*x^8 + 120/7*a^7*b^3*x^7 + 15/2*a^
8*b^2*x^6 + 2*a^9*b*x^5 + 1/4*a^10*x^4

_______________________________________________________________________________________

Fricas [A]  time = 0.179613, size = 1, normalized size = 0.02 \[ \frac{1}{14} x^{14} b^{10} + \frac{10}{13} x^{13} b^{9} a + \frac{15}{4} x^{12} b^{8} a^{2} + \frac{120}{11} x^{11} b^{7} a^{3} + 21 x^{10} b^{6} a^{4} + 28 x^{9} b^{5} a^{5} + \frac{105}{4} x^{8} b^{4} a^{6} + \frac{120}{7} x^{7} b^{3} a^{7} + \frac{15}{2} x^{6} b^{2} a^{8} + 2 x^{5} b a^{9} + \frac{1}{4} x^{4} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^3,x, algorithm="fricas")

[Out]

1/14*x^14*b^10 + 10/13*x^13*b^9*a + 15/4*x^12*b^8*a^2 + 120/11*x^11*b^7*a^3 + 21
*x^10*b^6*a^4 + 28*x^9*b^5*a^5 + 105/4*x^8*b^4*a^6 + 120/7*x^7*b^3*a^7 + 15/2*x^
6*b^2*a^8 + 2*x^5*b*a^9 + 1/4*x^4*a^10

_______________________________________________________________________________________

Sympy [A]  time = 0.168623, size = 129, normalized size = 2.02 \[ \frac{a^{10} x^{4}}{4} + 2 a^{9} b x^{5} + \frac{15 a^{8} b^{2} x^{6}}{2} + \frac{120 a^{7} b^{3} x^{7}}{7} + \frac{105 a^{6} b^{4} x^{8}}{4} + 28 a^{5} b^{5} x^{9} + 21 a^{4} b^{6} x^{10} + \frac{120 a^{3} b^{7} x^{11}}{11} + \frac{15 a^{2} b^{8} x^{12}}{4} + \frac{10 a b^{9} x^{13}}{13} + \frac{b^{10} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**10,x)

[Out]

a**10*x**4/4 + 2*a**9*b*x**5 + 15*a**8*b**2*x**6/2 + 120*a**7*b**3*x**7/7 + 105*
a**6*b**4*x**8/4 + 28*a**5*b**5*x**9 + 21*a**4*b**6*x**10 + 120*a**3*b**7*x**11/
11 + 15*a**2*b**8*x**12/4 + 10*a*b**9*x**13/13 + b**10*x**14/14

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201333, size = 151, normalized size = 2.36 \[ \frac{1}{14} \, b^{10} x^{14} + \frac{10}{13} \, a b^{9} x^{13} + \frac{15}{4} \, a^{2} b^{8} x^{12} + \frac{120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac{105}{4} \, a^{6} b^{4} x^{8} + \frac{120}{7} \, a^{7} b^{3} x^{7} + \frac{15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac{1}{4} \, a^{10} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^3,x, algorithm="giac")

[Out]

1/14*b^10*x^14 + 10/13*a*b^9*x^13 + 15/4*a^2*b^8*x^12 + 120/11*a^3*b^7*x^11 + 21
*a^4*b^6*x^10 + 28*a^5*b^5*x^9 + 105/4*a^6*b^4*x^8 + 120/7*a^7*b^3*x^7 + 15/2*a^
8*b^2*x^6 + 2*a^9*b*x^5 + 1/4*a^10*x^4