3.170 \(\int \frac{x^4}{(a+b x)^2} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^4}{b^5 (a+b x)}+\frac{3 a^2 x}{b^4}-\frac{4 a^3 \log (a+b x)}{b^5}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]

[Out]

(3*a^2*x)/b^4 - (a*x^2)/b^3 + x^3/(3*b^2) - a^4/(b^5*(a + b*x)) - (4*a^3*Log[a + b*x])/b^5

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Rubi [A]  time = 0.0329027, 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, Rules used = {43} \[ -\frac{a^4}{b^5 (a+b x)}+\frac{3 a^2 x}{b^4}-\frac{4 a^3 \log (a+b x)}{b^5}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^2,x]

[Out]

(3*a^2*x)/b^4 - (a*x^2)/b^3 + x^3/(3*b^2) - a^4/(b^5*(a + b*x)) - (4*a^3*Log[a + b*x])/b^5

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4}{(a+b x)^2} \, dx &=\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{3 a^2 x}{b^4}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2}-\frac{a^4}{b^5 (a+b x)}-\frac{4 a^3 \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0301503, size = 54, normalized size = 0.93 \[ \frac{-\frac{3 a^4}{a+b x}+9 a^2 b x-12 a^3 \log (a+b x)-3 a b^2 x^2+b^3 x^3}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^2,x]

[Out]

(9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 - (3*a^4)/(a + b*x) - 12*a^3*Log[a + b*x])/(3*b^5)

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Maple [A]  time = 0.006, size = 57, normalized size = 1. \begin{align*} 3\,{\frac{{a}^{2}x}{{b}^{4}}}-{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{4}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.25861, size = 80, normalized size = 1.38 \begin{align*} -\frac{a^{4}}{b^{6} x + a b^{5}} - \frac{4 \, a^{3} \log \left (b x + a\right )}{b^{5}} + \frac{b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

-a^4/(b^6*x + a*b^5) - 4*a^3*log(b*x + a)/b^5 + 1/3*(b^2*x^3 - 3*a*b*x^2 + 9*a^2*x)/b^4

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Fricas [A]  time = 1.50341, size = 155, normalized size = 2.67 \begin{align*} \frac{b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \,{\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a^4)*log(b*x + a))/(b^6*x + a*b
^5)

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Sympy [A]  time = 0.487386, size = 54, normalized size = 0.93 \begin{align*} - \frac{a^{4}}{a b^{5} + b^{6} x} - \frac{4 a^{3} \log{\left (a + b x \right )}}{b^{5}} + \frac{3 a^{2} x}{b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{x^{3}}{3 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.21519, size = 107, normalized size = 1.84 \begin{align*} -\frac{{\left (b x + a\right )}^{3}{\left (\frac{6 \, a}{b x + a} - \frac{18 \, a^{2}}{{\left (b x + a\right )}^{2}} - 1\right )}}{3 \, b^{5}} + \frac{4 \, a^{3} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{5}} - \frac{a^{4}}{{\left (b x + a\right )} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/3*(b*x + a)^3*(6*a/(b*x + a) - 18*a^2/(b*x + a)^2 - 1)/b^5 + 4*a^3*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b
^5 - a^4/((b*x + a)*b^5)