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

Optimal. Leaf size=64 \[ -\frac{a^4}{2 b^5 (a+b x)^2}+\frac{4 a^3}{b^5 (a+b x)}+\frac{6 a^2 \log (a+b x)}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0361858, antiderivative size = 64, 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} \[ -\frac{a^4}{2 b^5 (a+b x)^2}+\frac{4 a^3}{b^5 (a+b x)}+\frac{6 a^2 \log (a+b x)}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.027626, size = 55, normalized size = 0.86 \[ \frac{-\frac{a^4}{(a+b x)^2}+\frac{8 a^3}{a+b x}+12 a^2 \log (a+b x)-6 a b x+b^2 x^2}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^3,x]

[Out]

(-6*a*b*x + b^2*x^2 - a^4/(a + b*x)^2 + (8*a^3)/(a + b*x) + 12*a^2*Log[a + b*x])/(2*b^5)

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Maple [A]  time = 0.007, size = 61, normalized size = 1. \begin{align*} -3\,{\frac{ax}{{b}^{4}}}+{\frac{{x}^{2}}{2\,{b}^{3}}}-{\frac{{a}^{4}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}+4\,{\frac{{a}^{3}}{{b}^{5} \left ( bx+a \right ) }}+6\,{\frac{{a}^{2}\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)^3,x)

[Out]

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

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Maxima [A]  time = 1.0957, size = 93, normalized size = 1.45 \begin{align*} \frac{8 \, a^{3} b x + 7 \, a^{4}}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac{6 \, a^{2} \log \left (b x + a\right )}{b^{5}} + \frac{b x^{2} - 6 \, a x}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62256, size = 200, normalized size = 3.12 \begin{align*} \frac{b^{4} x^{4} - 4 \, a b^{3} x^{3} - 11 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x + 7 \, a^{4} + 12 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*x^4 - 4*a*b^3*x^3 - 11*a^2*b^2*x^2 + 2*a^3*b*x + 7*a^4 + 12*(a^2*b^2*x^2 + 2*a^3*b*x + a^4)*log(b*x +
a))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)

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Sympy [A]  time = 0.622383, size = 70, normalized size = 1.09 \begin{align*} \frac{6 a^{2} \log{\left (a + b x \right )}}{b^{5}} - \frac{3 a x}{b^{4}} + \frac{7 a^{4} + 8 a^{3} b x}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac{x^{2}}{2 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**3,x)

[Out]

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

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Giac [A]  time = 1.13793, size = 82, normalized size = 1.28 \begin{align*} \frac{6 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{b^{3} x^{2} - 6 \, a b^{2} x}{2 \, b^{6}} + \frac{8 \, a^{3} b x + 7 \, a^{4}}{2 \,{\left (b x + a\right )}^{2} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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