3.79 \(\int x^4 (a+b x)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^5}{5}+\frac{5}{6} a^4 b x^6+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^2 b^3 x^8+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.0624312, antiderivative size = 69, 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 \[ \frac{a^5 x^5}{5}+\frac{5}{6} a^4 b x^6+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^2 b^3 x^8+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x)^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6606, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{5}}{5} + \frac{5 a^{4} b x^{6}}{6} + \frac{10 a^{3} b^{2} x^{7}}{7} + \frac{5 a^{2} b^{3} x^{8}}{4} + \frac{5 a b^{4} x^{9}}{9} + \frac{b^{5} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**6/6 + 10*a**3*b**2*x**7/7 + 5*a**2*b**3*x**8/4 + 5*a*b
**4*x**9/9 + b**5*x**10/10

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Mathematica [A]  time = 0.00294384, size = 69, normalized size = 1. \[ \frac{a^5 x^5}{5}+\frac{5}{6} a^4 b x^6+\frac{10}{7} a^3 b^2 x^7+\frac{5}{4} a^2 b^3 x^8+\frac{5}{9} a b^4 x^9+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x)^5,x]

[Out]

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

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Maple [A]  time = 0.001, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{5\,{a}^{4}b{x}^{6}}{6}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{7}}{7}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{5\,a{b}^{4}{x}^{9}}{9}}+{\frac{{b}^{5}{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.32839, size = 77, normalized size = 1.12 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.182332, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} b^{5} + \frac{5}{9} x^{9} b^{4} a + \frac{5}{4} x^{8} b^{3} a^{2} + \frac{10}{7} x^{7} b^{2} a^{3} + \frac{5}{6} x^{6} b a^{4} + \frac{1}{5} x^{5} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.126919, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{5}}{5} + \frac{5 a^{4} b x^{6}}{6} + \frac{10 a^{3} b^{2} x^{7}}{7} + \frac{5 a^{2} b^{3} x^{8}}{4} + \frac{5 a b^{4} x^{9}}{9} + \frac{b^{5} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**6/6 + 10*a**3*b**2*x**7/7 + 5*a**2*b**3*x**8/4 + 5*a*b
**4*x**9/9 + b**5*x**10/10

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GIAC/XCAS [A]  time = 0.210772, size = 77, normalized size = 1.12 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{9} \, a b^{4} x^{9} + \frac{5}{4} \, a^{2} b^{3} x^{8} + \frac{10}{7} \, a^{3} b^{2} x^{7} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^4,x, algorithm="giac")

[Out]

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