3.102 \(\int x^4 (a+b x)^7 \, dx\)

Optimal. Leaf size=81 \[ \frac{a^4 (a+b x)^8}{8 b^5}-\frac{4 a^3 (a+b x)^9}{9 b^5}+\frac{3 a^2 (a+b x)^{10}}{5 b^5}+\frac{(a+b x)^{12}}{12 b^5}-\frac{4 a (a+b x)^{11}}{11 b^5} \]

[Out]

(a^4*(a + b*x)^8)/(8*b^5) - (4*a^3*(a + b*x)^9)/(9*b^5) + (3*a^2*(a + b*x)^10)/(
5*b^5) - (4*a*(a + b*x)^11)/(11*b^5) + (a + b*x)^12/(12*b^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.0830989, antiderivative size = 81, 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 \[ \frac{a^4 (a+b x)^8}{8 b^5}-\frac{4 a^3 (a+b x)^9}{9 b^5}+\frac{3 a^2 (a+b x)^{10}}{5 b^5}+\frac{(a+b x)^{12}}{12 b^5}-\frac{4 a (a+b x)^{11}}{11 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x)^7,x]

[Out]

(a^4*(a + b*x)^8)/(8*b^5) - (4*a^3*(a + b*x)^9)/(9*b^5) + (3*a^2*(a + b*x)^10)/(
5*b^5) - (4*a*(a + b*x)^11)/(11*b^5) + (a + b*x)^12/(12*b^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.218, size = 75, normalized size = 0.93 \[ \frac{a^{4} \left (a + b x\right )^{8}}{8 b^{5}} - \frac{4 a^{3} \left (a + b x\right )^{9}}{9 b^{5}} + \frac{3 a^{2} \left (a + b x\right )^{10}}{5 b^{5}} - \frac{4 a \left (a + b x\right )^{11}}{11 b^{5}} + \frac{\left (a + b x\right )^{12}}{12 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)**7,x)

[Out]

a**4*(a + b*x)**8/(8*b**5) - 4*a**3*(a + b*x)**9/(9*b**5) + 3*a**2*(a + b*x)**10
/(5*b**5) - 4*a*(a + b*x)**11/(11*b**5) + (a + b*x)**12/(12*b**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00329103, size = 93, normalized size = 1.15 \[ \frac{a^7 x^5}{5}+\frac{7}{6} a^6 b x^6+3 a^5 b^2 x^7+\frac{35}{8} a^4 b^3 x^8+\frac{35}{9} a^3 b^4 x^9+\frac{21}{10} a^2 b^5 x^{10}+\frac{7}{11} a b^6 x^{11}+\frac{b^7 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x)^7,x]

[Out]

(a^7*x^5)/5 + (7*a^6*b*x^6)/6 + 3*a^5*b^2*x^7 + (35*a^4*b^3*x^8)/8 + (35*a^3*b^4
*x^9)/9 + (21*a^2*b^5*x^10)/10 + (7*a*b^6*x^11)/11 + (b^7*x^12)/12

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 80, normalized size = 1. \[{\frac{{b}^{7}{x}^{12}}{12}}+{\frac{7\,a{b}^{6}{x}^{11}}{11}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{10}}{10}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{9}}{9}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{8}}{8}}+3\,{a}^{5}{b}^{2}{x}^{7}+{\frac{7\,{a}^{6}b{x}^{6}}{6}}+{\frac{{a}^{7}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)^7,x)

[Out]

1/12*b^7*x^12+7/11*a*b^6*x^11+21/10*a^2*b^5*x^10+35/9*a^3*b^4*x^9+35/8*a^4*b^3*x
^8+3*a^5*b^2*x^7+7/6*a^6*b*x^6+1/5*a^7*x^5

_______________________________________________________________________________________

Maxima [A]  time = 1.34667, size = 107, normalized size = 1.32 \[ \frac{1}{12} \, b^{7} x^{12} + \frac{7}{11} \, a b^{6} x^{11} + \frac{21}{10} \, a^{2} b^{5} x^{10} + \frac{35}{9} \, a^{3} b^{4} x^{9} + \frac{35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac{7}{6} \, a^{6} b x^{6} + \frac{1}{5} \, a^{7} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^4,x, algorithm="maxima")

[Out]

1/12*b^7*x^12 + 7/11*a*b^6*x^11 + 21/10*a^2*b^5*x^10 + 35/9*a^3*b^4*x^9 + 35/8*a
^4*b^3*x^8 + 3*a^5*b^2*x^7 + 7/6*a^6*b*x^6 + 1/5*a^7*x^5

_______________________________________________________________________________________

Fricas [A]  time = 0.176396, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} b^{7} + \frac{7}{11} x^{11} b^{6} a + \frac{21}{10} x^{10} b^{5} a^{2} + \frac{35}{9} x^{9} b^{4} a^{3} + \frac{35}{8} x^{8} b^{3} a^{4} + 3 x^{7} b^{2} a^{5} + \frac{7}{6} x^{6} b a^{6} + \frac{1}{5} x^{5} a^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^4,x, algorithm="fricas")

[Out]

1/12*x^12*b^7 + 7/11*x^11*b^6*a + 21/10*x^10*b^5*a^2 + 35/9*x^9*b^4*a^3 + 35/8*x
^8*b^3*a^4 + 3*x^7*b^2*a^5 + 7/6*x^6*b*a^6 + 1/5*x^5*a^7

_______________________________________________________________________________________

Sympy [A]  time = 0.135242, size = 92, normalized size = 1.14 \[ \frac{a^{7} x^{5}}{5} + \frac{7 a^{6} b x^{6}}{6} + 3 a^{5} b^{2} x^{7} + \frac{35 a^{4} b^{3} x^{8}}{8} + \frac{35 a^{3} b^{4} x^{9}}{9} + \frac{21 a^{2} b^{5} x^{10}}{10} + \frac{7 a b^{6} x^{11}}{11} + \frac{b^{7} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)**7,x)

[Out]

a**7*x**5/5 + 7*a**6*b*x**6/6 + 3*a**5*b**2*x**7 + 35*a**4*b**3*x**8/8 + 35*a**3
*b**4*x**9/9 + 21*a**2*b**5*x**10/10 + 7*a*b**6*x**11/11 + b**7*x**12/12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207633, size = 107, normalized size = 1.32 \[ \frac{1}{12} \, b^{7} x^{12} + \frac{7}{11} \, a b^{6} x^{11} + \frac{21}{10} \, a^{2} b^{5} x^{10} + \frac{35}{9} \, a^{3} b^{4} x^{9} + \frac{35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac{7}{6} \, a^{6} b x^{6} + \frac{1}{5} \, a^{7} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^4,x, algorithm="giac")

[Out]

1/12*b^7*x^12 + 7/11*a*b^6*x^11 + 21/10*a^2*b^5*x^10 + 35/9*a^3*b^4*x^9 + 35/8*a
^4*b^3*x^8 + 3*a^5*b^2*x^7 + 7/6*a^6*b*x^6 + 1/5*a^7*x^5