3.216 \(\int \frac{x}{(a+b x)^7} \, dx\)

Optimal. Leaf size=30 \[ \frac{a}{6 b^2 (a+b x)^6}-\frac{1}{5 b^2 (a+b x)^5} \]

[Out]

a/(6*b^2*(a + b*x)^6) - 1/(5*b^2*(a + b*x)^5)

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Rubi [A]  time = 0.0318969, 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 \[ \frac{a}{6 b^2 (a+b x)^6}-\frac{1}{5 b^2 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x)^7,x]

[Out]

a/(6*b^2*(a + b*x)^6) - 1/(5*b^2*(a + b*x)^5)

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Rubi in Sympy [A]  time = 5.95895, size = 26, normalized size = 0.87 \[ \frac{a}{6 b^{2} \left (a + b x\right )^{6}} - \frac{1}{5 b^{2} \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x+a)**7,x)

[Out]

a/(6*b**2*(a + b*x)**6) - 1/(5*b**2*(a + b*x)**5)

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Mathematica [A]  time = 0.00867442, size = 20, normalized size = 0.67 \[ -\frac{a+6 b x}{30 b^2 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x)^7,x]

[Out]

-(a + 6*b*x)/(30*b^2*(a + b*x)^6)

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Maple [A]  time = 0.007, size = 27, normalized size = 0.9 \[{\frac{a}{6\,{b}^{2} \left ( bx+a \right ) ^{6}}}-{\frac{1}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)^7,x)

[Out]

1/6*a/b^2/(b*x+a)^6-1/5/b^2/(b*x+a)^5

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Maxima [A]  time = 1.34092, size = 103, normalized size = 3.43 \[ -\frac{6 \, b x + a}{30 \,{\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(6*b*x + a)/(b^8*x^6 + 6*a*b^7*x^5 + 15*a^2*b^6*x^4 + 20*a^3*b^5*x^3 + 15*
a^4*b^4*x^2 + 6*a^5*b^3*x + a^6*b^2)

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Fricas [A]  time = 0.203212, size = 103, normalized size = 3.43 \[ -\frac{6 \, b x + a}{30 \,{\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(6*b*x + a)/(b^8*x^6 + 6*a*b^7*x^5 + 15*a^2*b^6*x^4 + 20*a^3*b^5*x^3 + 15*
a^4*b^4*x^2 + 6*a^5*b^3*x + a^6*b^2)

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Sympy [A]  time = 2.21987, size = 80, normalized size = 2.67 \[ - \frac{a + 6 b x}{30 a^{6} b^{2} + 180 a^{5} b^{3} x + 450 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{3} + 450 a^{2} b^{6} x^{4} + 180 a b^{7} x^{5} + 30 b^{8} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a)**7,x)

[Out]

-(a + 6*b*x)/(30*a**6*b**2 + 180*a**5*b**3*x + 450*a**4*b**4*x**2 + 600*a**3*b**
5*x**3 + 450*a**2*b**6*x**4 + 180*a*b**7*x**5 + 30*b**8*x**6)

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GIAC/XCAS [A]  time = 0.214932, size = 24, normalized size = 0.8 \[ -\frac{6 \, b x + a}{30 \,{\left (b x + a\right )}^{6} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(6*b*x + a)/((b*x + a)^6*b^2)