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

Optimal. Leaf size=65 \[ -\frac{a^4}{3 b^5 (a+b x)^3}+\frac{2 a^3}{b^5 (a+b x)^2}-\frac{6 a^2}{b^5 (a+b x)}-\frac{4 a \log (a+b x)}{b^5}+\frac{x}{b^4} \]

[Out]

x/b^4 - a^4/(3*b^5*(a + b*x)^3) + (2*a^3)/(b^5*(a + b*x)^2) - (6*a^2)/(b^5*(a + b*x)) - (4*a*Log[a + b*x])/b^5

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Rubi [A]  time = 0.0363399, antiderivative size = 65, 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} \[ -\frac{a^4}{3 b^5 (a+b x)^3}+\frac{2 a^3}{b^5 (a+b x)^2}-\frac{6 a^2}{b^5 (a+b x)}-\frac{4 a \log (a+b x)}{b^5}+\frac{x}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^4,x]

[Out]

x/b^4 - a^4/(3*b^5*(a + b*x)^3) + (2*a^3)/(b^5*(a + b*x)^2) - (6*a^2)/(b^5*(a + b*x)) - (4*a*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)^4} \, dx &=\int \left (\frac{1}{b^4}+\frac{a^4}{b^4 (a+b x)^4}-\frac{4 a^3}{b^4 (a+b x)^3}+\frac{6 a^2}{b^4 (a+b x)^2}-\frac{4 a}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{x}{b^4}-\frac{a^4}{3 b^5 (a+b x)^3}+\frac{2 a^3}{b^5 (a+b x)^2}-\frac{6 a^2}{b^5 (a+b x)}-\frac{4 a \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0674403, size = 51, normalized size = 0.78 \[ -\frac{\frac{a^2 \left (13 a^2+30 a b x+18 b^2 x^2\right )}{(a+b x)^3}+12 a \log (a+b x)-3 b x}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^4,x]

[Out]

-(-3*b*x + (a^2*(13*a^2 + 30*a*b*x + 18*b^2*x^2))/(a + b*x)^3 + 12*a*Log[a + b*x])/(3*b^5)

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Maple [A]  time = 0.008, size = 64, normalized size = 1. \begin{align*}{\frac{x}{{b}^{4}}}-{\frac{{a}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+2\,{\frac{{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{a\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)^4,x)

[Out]

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

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Maxima [A]  time = 1.08027, size = 107, normalized size = 1.65 \begin{align*} -\frac{18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac{x}{b^{4}} - \frac{4 \, a \log \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)^4,x, algorithm="maxima")

[Out]

-1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5) + x/b^4 - 4*a*log(
b*x + a)/b^5

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Fricas [A]  time = 1.43883, size = 244, normalized size = 3.75 \begin{align*} \frac{3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4} - 12 \,{\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b^4*x^4 + 9*a*b^3*x^3 - 9*a^2*b^2*x^2 - 27*a^3*b*x - 13*a^4 - 12*(a*b^3*x^3 + 3*a^2*b^2*x^2 + 3*a^3*b*x
+ a^4)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)

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Sympy [A]  time = 0.708941, size = 80, normalized size = 1.23 \begin{align*} - \frac{4 a \log{\left (a + b x \right )}}{b^{5}} - \frac{13 a^{4} + 30 a^{3} b x + 18 a^{2} b^{2} x^{2}}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac{x}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a*log(a + b*x)/b**5 - (13*a**4 + 30*a**3*b*x + 18*a**2*b**2*x**2)/(3*a**3*b**5 + 9*a**2*b**6*x + 9*a*b**7*x
**2 + 3*b**8*x**3) + x/b**4

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Giac [A]  time = 1.19794, size = 74, normalized size = 1.14 \begin{align*} \frac{x}{b^{4}} - \frac{4 \, a \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \,{\left (b x + a\right )}^{3} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^4 - 4*a*log(abs(b*x + a))/b^5 - 1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/((b*x + a)^3*b^5)