3.196 \(\int \frac{x^5}{(a+b x)^4} \, dx\)

Optimal. Leaf size=81 \[ \frac{a^5}{3 b^6 (a+b x)^3}-\frac{5 a^4}{2 b^6 (a+b x)^2}+\frac{10 a^3}{b^6 (a+b x)}+\frac{10 a^2 \log (a+b x)}{b^6}-\frac{4 a x}{b^5}+\frac{x^2}{2 b^4} \]

[Out]

(-4*a*x)/b^5 + x^2/(2*b^4) + a^5/(3*b^6*(a + b*x)^3) - (5*a^4)/(2*b^6*(a + b*x)^2) + (10*a^3)/(b^6*(a + b*x))
+ (10*a^2*Log[a + b*x])/b^6

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Rubi [A]  time = 0.0476785, antiderivative size = 81, 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} \[ \frac{a^5}{3 b^6 (a+b x)^3}-\frac{5 a^4}{2 b^6 (a+b x)^2}+\frac{10 a^3}{b^6 (a+b x)}+\frac{10 a^2 \log (a+b x)}{b^6}-\frac{4 a x}{b^5}+\frac{x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x)^4,x]

[Out]

(-4*a*x)/b^5 + x^2/(2*b^4) + a^5/(3*b^6*(a + b*x)^3) - (5*a^4)/(2*b^6*(a + b*x)^2) + (10*a^3)/(b^6*(a + b*x))
+ (10*a^2*Log[a + b*x])/b^6

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^5}{(a+b x)^4} \, dx &=\int \left (-\frac{4 a}{b^5}+\frac{x}{b^4}-\frac{a^5}{b^5 (a+b x)^4}+\frac{5 a^4}{b^5 (a+b x)^3}-\frac{10 a^3}{b^5 (a+b x)^2}+\frac{10 a^2}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{4 a x}{b^5}+\frac{x^2}{2 b^4}+\frac{a^5}{3 b^6 (a+b x)^3}-\frac{5 a^4}{2 b^6 (a+b x)^2}+\frac{10 a^3}{b^6 (a+b x)}+\frac{10 a^2 \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.041163, size = 68, normalized size = 0.84 \[ \frac{\frac{2 a^5}{(a+b x)^3}-\frac{15 a^4}{(a+b x)^2}+\frac{60 a^3}{a+b x}+60 a^2 \log (a+b x)-24 a b x+3 b^2 x^2}{6 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^4,x]

[Out]

(-24*a*b*x + 3*b^2*x^2 + (2*a^5)/(a + b*x)^3 - (15*a^4)/(a + b*x)^2 + (60*a^3)/(a + b*x) + 60*a^2*Log[a + b*x]
)/(6*b^6)

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \begin{align*} -4\,{\frac{ax}{{b}^{5}}}+{\frac{{x}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-{\frac{5\,{a}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{a}^{3}}{{b}^{6} \left ( bx+a \right ) }}+10\,{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^4,x)

[Out]

-4*a*x/b^5+1/2*x^2/b^4+1/3*a^5/b^6/(b*x+a)^3-5/2*a^4/b^6/(b*x+a)^2+10*a^3/b^6/(b*x+a)+10*a^2*ln(b*x+a)/b^6

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Maxima [A]  time = 1.05502, size = 123, normalized size = 1.52 \begin{align*} \frac{60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac{10 \, a^{2} \log \left (b x + a\right )}{b^{6}} + \frac{b x^{2} - 8 \, a x}{2 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/6*(60*a^3*b^2*x^2 + 105*a^4*b*x + 47*a^5)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 10*a^2*log(b*x +
a)/b^6 + 1/2*(b*x^2 - 8*a*x)/b^5

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Fricas [A]  time = 1.43358, size = 271, normalized size = 3.35 \begin{align*} \frac{3 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} - 63 \, a^{2} b^{3} x^{3} - 9 \, a^{3} b^{2} x^{2} + 81 \, a^{4} b x + 47 \, a^{5} + 60 \,{\left (a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/6*(3*b^5*x^5 - 15*a*b^4*x^4 - 63*a^2*b^3*x^3 - 9*a^3*b^2*x^2 + 81*a^4*b*x + 47*a^5 + 60*(a^2*b^3*x^3 + 3*a^3
*b^2*x^2 + 3*a^4*b*x + a^5)*log(b*x + a))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)

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Sympy [A]  time = 0.730283, size = 94, normalized size = 1.16 \begin{align*} \frac{10 a^{2} \log{\left (a + b x \right )}}{b^{6}} - \frac{4 a x}{b^{5}} + \frac{47 a^{5} + 105 a^{4} b x + 60 a^{3} b^{2} x^{2}}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac{x^{2}}{2 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**4,x)

[Out]

10*a**2*log(a + b*x)/b**6 - 4*a*x/b**5 + (47*a**5 + 105*a**4*b*x + 60*a**3*b**2*x**2)/(6*a**3*b**6 + 18*a**2*b
**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + x**2/(2*b**4)

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Giac [A]  time = 1.21348, size = 97, normalized size = 1.2 \begin{align*} \frac{10 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{b^{4} x^{2} - 8 \, a b^{3} x}{2 \, b^{8}} + \frac{60 \, a^{3} b^{2} x^{2} + 105 \, a^{4} b x + 47 \, a^{5}}{6 \,{\left (b x + a\right )}^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^4,x, algorithm="giac")

[Out]

10*a^2*log(abs(b*x + a))/b^6 + 1/2*(b^4*x^2 - 8*a*b^3*x)/b^8 + 1/6*(60*a^3*b^2*x^2 + 105*a^4*b*x + 47*a^5)/((b
*x + a)^3*b^6)