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

Optimal. Leaf size=30 \[ \frac{(a+b x)^7}{7 b^2}-\frac{a (a+b x)^6}{6 b^2} \]

[Out]

-(a*(a + b*x)^6)/(6*b^2) + (a + b*x)^7/(7*b^2)

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Rubi [A]  time = 0.0276165, 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 \[ \frac{(a+b x)^7}{7 b^2}-\frac{a (a+b x)^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^5,x]

[Out]

-(a*(a + b*x)^6)/(6*b^2) + (a + b*x)^7/(7*b^2)

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Rubi in Sympy [A]  time = 7.14924, size = 24, normalized size = 0.8 \[ - \frac{a \left (a + b x\right )^{6}}{6 b^{2}} + \frac{\left (a + b x\right )^{7}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**5,x)

[Out]

-a*(a + b*x)**6/(6*b**2) + (a + b*x)**7/(7*b**2)

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Mathematica [B]  time = 0.00255794, size = 67, normalized size = 2.23 \[ \frac{a^5 x^2}{2}+\frac{5}{3} a^4 b x^3+\frac{5}{2} a^3 b^2 x^4+2 a^2 b^3 x^5+\frac{5}{6} a b^4 x^6+\frac{b^5 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^5,x]

[Out]

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

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Maple [B]  time = 0.001, size = 58, normalized size = 1.9 \[{\frac{{b}^{5}{x}^{7}}{7}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{5\,{a}^{4}b{x}^{3}}{3}}+{\frac{{a}^{5}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 1.31855, size = 77, normalized size = 2.57 \[ \frac{1}{7} \, b^{5} x^{7} + \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{2} \, a^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.187731, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} b^{5} + \frac{5}{6} x^{6} b^{4} a + 2 x^{5} b^{3} a^{2} + \frac{5}{2} x^{4} b^{2} a^{3} + \frac{5}{3} x^{3} b a^{4} + \frac{1}{2} x^{2} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.1374, size = 65, normalized size = 2.17 \[ \frac{a^{5} x^{2}}{2} + \frac{5 a^{4} b x^{3}}{3} + \frac{5 a^{3} b^{2} x^{4}}{2} + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.204708, size = 77, normalized size = 2.57 \[ \frac{1}{7} \, b^{5} x^{7} + \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{3} \, a^{4} b x^{3} + \frac{1}{2} \, a^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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