3.200 \(\int \frac{x}{(a+b x)^4} \, dx\)

Optimal. Leaf size=30 \[ \frac{a}{3 b^2 (a+b x)^3}-\frac{1}{2 b^2 (a+b x)^2} \]

[Out]

a/(3*b^2*(a + b*x)^3) - 1/(2*b^2*(a + b*x)^2)

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Rubi [A]  time = 0.0303053, 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 \[ \frac{a}{3 b^2 (a+b x)^3}-\frac{1}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x)^4,x]

[Out]

a/(3*b^2*(a + b*x)^3) - 1/(2*b^2*(a + b*x)^2)

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Rubi in Sympy [A]  time = 5.5859, size = 26, normalized size = 0.87 \[ \frac{a}{3 b^{2} \left (a + b x\right )^{3}} - \frac{1}{2 b^{2} \left (a + b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x+a)**4,x)

[Out]

a/(3*b**2*(a + b*x)**3) - 1/(2*b**2*(a + b*x)**2)

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Mathematica [A]  time = 0.00747, size = 20, normalized size = 0.67 \[ -\frac{a+3 b x}{6 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x)^4,x]

[Out]

-(a + 3*b*x)/(6*b^2*(a + b*x)^3)

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Maple [A]  time = 0.008, size = 27, normalized size = 0.9 \[{\frac{a}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{1}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x+a)^4,x)

[Out]

1/3*a/b^2/(b*x+a)^3-1/2/b^2/(b*x+a)^2

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Maxima [A]  time = 1.3368, size = 58, normalized size = 1.93 \[ -\frac{3 \, b x + a}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*x + a)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

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Fricas [A]  time = 0.205653, size = 58, normalized size = 1.93 \[ -\frac{3 \, b x + a}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*x + a)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

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Sympy [A]  time = 1.49142, size = 44, normalized size = 1.47 \[ - \frac{a + 3 b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x+a)**4,x)

[Out]

-(a + 3*b*x)/(6*a**3*b**2 + 18*a**2*b**3*x + 18*a*b**4*x**2 + 6*b**5*x**3)

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GIAC/XCAS [A]  time = 0.200993, size = 24, normalized size = 0.8 \[ -\frac{3 \, b x + a}{6 \,{\left (b x + a\right )}^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(3*b*x + a)/((b*x + a)^3*b^2)