3.198 \(\int \frac{x^3}{(a+b x)^4} \, dx\)

Optimal. Leaf size=58 \[ \frac{a^3}{3 b^4 (a+b x)^3}-\frac{3 a^2}{2 b^4 (a+b x)^2}+\frac{3 a}{b^4 (a+b x)}+\frac{\log (a+b x)}{b^4} \]

[Out]

a^3/(3*b^4*(a + b*x)^3) - (3*a^2)/(2*b^4*(a + b*x)^2) + (3*a)/(b^4*(a + b*x)) +
Log[a + b*x]/b^4

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Rubi [A]  time = 0.0676924, antiderivative size = 58, 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 \[ \frac{a^3}{3 b^4 (a+b x)^3}-\frac{3 a^2}{2 b^4 (a+b x)^2}+\frac{3 a}{b^4 (a+b x)}+\frac{\log (a+b x)}{b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^4,x]

[Out]

a^3/(3*b^4*(a + b*x)^3) - (3*a^2)/(2*b^4*(a + b*x)^2) + (3*a)/(b^4*(a + b*x)) +
Log[a + b*x]/b^4

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Rubi in Sympy [A]  time = 12.7271, size = 53, normalized size = 0.91 \[ \frac{a^{3}}{3 b^{4} \left (a + b x\right )^{3}} - \frac{3 a^{2}}{2 b^{4} \left (a + b x\right )^{2}} + \frac{3 a}{b^{4} \left (a + b x\right )} + \frac{\log{\left (a + b x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**4,x)

[Out]

a**3/(3*b**4*(a + b*x)**3) - 3*a**2/(2*b**4*(a + b*x)**2) + 3*a/(b**4*(a + b*x))
+ log(a + b*x)/b**4

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Mathematica [A]  time = 0.0237264, size = 44, normalized size = 0.76 \[ \frac{\frac{a \left (11 a^2+27 a b x+18 b^2 x^2\right )}{(a+b x)^3}+6 \log (a+b x)}{6 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x)^4,x]

[Out]

((a*(11*a^2 + 27*a*b*x + 18*b^2*x^2))/(a + b*x)^3 + 6*Log[a + b*x])/(6*b^4)

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Maple [A]  time = 0.009, size = 55, normalized size = 1. \[{\frac{{a}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{a}{{b}^{4} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.33587, size = 95, normalized size = 1.64 \[ \frac{18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3}}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{\log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^4,x, algorithm="maxima")

[Out]

1/6*(18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x +
a^3*b^4) + log(b*x + a)/b^4

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Fricas [A]  time = 0.207187, size = 127, normalized size = 2.19 \[ \frac{18 \, a b^{2} x^{2} + 27 \, a^{2} b x + 11 \, a^{3} + 6 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^4,x, algorithm="fricas")

[Out]

1/6*(18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3 + 6*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x +
a^3)*log(b*x + a))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)

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Sympy [A]  time = 1.67152, size = 70, normalized size = 1.21 \[ \frac{11 a^{3} + 27 a^{2} b x + 18 a b^{2} x^{2}}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{\log{\left (a + b x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**4,x)

[Out]

(11*a**3 + 27*a**2*b*x + 18*a*b**2*x**2)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b*
*6*x**2 + 6*b**7*x**3) + log(a + b*x)/b**4

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GIAC/XCAS [A]  time = 0.205453, size = 62, normalized size = 1.07 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{18 \, a b x^{2} + 27 \, a^{2} x + \frac{11 \, a^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x + a)^4,x, algorithm="giac")

[Out]

ln(abs(b*x + a))/b^4 + 1/6*(18*a*b*x^2 + 27*a^2*x + 11*a^3/b)/((b*x + a)^3*b^3)