3.168 \(\int \frac{x^6}{(a+b x)^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a^6}{b^7 (a+b x)}+\frac{5 a^4 x}{b^6}-\frac{6 a^5 \log (a+b x)}{b^7}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2} \]

[Out]

(5*a^4*x)/b^6 - (2*a^3*x^2)/b^5 + (a^2*x^3)/b^4 - (a*x^4)/(2*b^3) + x^5/(5*b^2) - a^6/(b^7*(a + b*x)) - (6*a^5
*Log[a + b*x])/b^7

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Rubi [A]  time = 0.0592848, 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} \[ -\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a^6}{b^7 (a+b x)}+\frac{5 a^4 x}{b^6}-\frac{6 a^5 \log (a+b x)}{b^7}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^2,x]

[Out]

(5*a^4*x)/b^6 - (2*a^3*x^2)/b^5 + (a^2*x^3)/b^4 - (a*x^4)/(2*b^3) + x^5/(5*b^2) - a^6/(b^7*(a + b*x)) - (6*a^5
*Log[a + b*x])/b^7

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^6}{(a+b x)^2} \, dx &=\int \left (\frac{5 a^4}{b^6}-\frac{4 a^3 x}{b^5}+\frac{3 a^2 x^2}{b^4}-\frac{2 a x^3}{b^3}+\frac{x^4}{b^2}+\frac{a^6}{b^6 (a+b x)^2}-\frac{6 a^5}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{5 a^4 x}{b^6}-\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2}-\frac{a^6}{b^7 (a+b x)}-\frac{6 a^5 \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0356202, size = 77, normalized size = 0.95 \[ \frac{-20 a^3 b^2 x^2+10 a^2 b^3 x^3-\frac{10 a^6}{a+b x}+50 a^4 b x-60 a^5 \log (a+b x)-5 a b^4 x^4+2 b^5 x^5}{10 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^2,x]

[Out]

(50*a^4*b*x - 20*a^3*b^2*x^2 + 10*a^2*b^3*x^3 - 5*a*b^4*x^4 + 2*b^5*x^5 - (10*a^6)/(a + b*x) - 60*a^5*Log[a +
b*x])/(10*b^7)

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Maple [A]  time = 0.007, size = 78, normalized size = 1. \begin{align*} 5\,{\frac{{a}^{4}x}{{b}^{6}}}-2\,{\frac{{a}^{3}{x}^{2}}{{b}^{5}}}+{\frac{{a}^{2}{x}^{3}}{{b}^{4}}}-{\frac{a{x}^{4}}{2\,{b}^{3}}}+{\frac{{x}^{5}}{5\,{b}^{2}}}-{\frac{{a}^{6}}{{b}^{7} \left ( bx+a \right ) }}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.05872, size = 111, normalized size = 1.37 \begin{align*} -\frac{a^{6}}{b^{8} x + a b^{7}} - \frac{6 \, a^{5} \log \left (b x + a\right )}{b^{7}} + \frac{2 \, b^{4} x^{5} - 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a^{3} b x^{2} + 50 \, a^{4} x}{10 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="maxima")

[Out]

-a^6/(b^8*x + a*b^7) - 6*a^5*log(b*x + a)/b^7 + 1/10*(2*b^4*x^5 - 5*a*b^3*x^4 + 10*a^2*b^2*x^3 - 20*a^3*b*x^2
+ 50*a^4*x)/b^6

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Fricas [A]  time = 1.48875, size = 208, normalized size = 2.57 \begin{align*} \frac{2 \, b^{6} x^{6} - 3 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{4} b^{2} x^{2} + 50 \, a^{5} b x - 10 \, a^{6} - 60 \,{\left (a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{10 \,{\left (b^{8} x + a b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/10*(2*b^6*x^6 - 3*a*b^5*x^5 + 5*a^2*b^4*x^4 - 10*a^3*b^3*x^3 + 30*a^4*b^2*x^2 + 50*a^5*b*x - 10*a^6 - 60*(a^
5*b*x + a^6)*log(b*x + a))/(b^8*x + a*b^7)

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Sympy [A]  time = 0.497526, size = 78, normalized size = 0.96 \begin{align*} - \frac{a^{6}}{a b^{7} + b^{8} x} - \frac{6 a^{5} \log{\left (a + b x \right )}}{b^{7}} + \frac{5 a^{4} x}{b^{6}} - \frac{2 a^{3} x^{2}}{b^{5}} + \frac{a^{2} x^{3}}{b^{4}} - \frac{a x^{4}}{2 b^{3}} + \frac{x^{5}}{5 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.22041, size = 139, normalized size = 1.72 \begin{align*} -\frac{{\left (b x + a\right )}^{5}{\left (\frac{15 \, a}{b x + a} - \frac{50 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac{100 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac{150 \, a^{4}}{{\left (b x + a\right )}^{4}} - 2\right )}}{10 \, b^{7}} + \frac{6 \, a^{5} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} - \frac{a^{6}}{{\left (b x + a\right )} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/10*(b*x + a)^5*(15*a/(b*x + a) - 50*a^2/(b*x + a)^2 + 100*a^3/(b*x + a)^3 - 150*a^4/(b*x + a)^4 - 2)/b^7 +
6*a^5*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 - a^6/((b*x + a)*b^7)