3.195 \(\int \frac{x^6}{(a+b x)^4} \, dx\)

Optimal. Leaf size=90 \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]

[Out]

(10*a^2*x)/b^6 - (2*a*x^2)/b^5 + x^3/(3*b^4) - a^6/(3*b^7*(a + b*x)^3) + (3*a^5)/(b^7*(a + b*x)^2) - (15*a^4)/
(b^7*(a + b*x)) - (20*a^3*Log[a + b*x])/b^7

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Rubi [A]  time = 0.0565142, antiderivative size = 90, 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} \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^4,x]

[Out]

(10*a^2*x)/b^6 - (2*a*x^2)/b^5 + x^3/(3*b^4) - a^6/(3*b^7*(a + b*x)^3) + (3*a^5)/(b^7*(a + b*x)^2) - (15*a^4)/
(b^7*(a + b*x)) - (20*a^3*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)^4} \, dx &=\int \left (\frac{10 a^2}{b^6}-\frac{4 a x}{b^5}+\frac{x^2}{b^4}+\frac{a^6}{b^6 (a+b x)^4}-\frac{6 a^5}{b^6 (a+b x)^3}+\frac{15 a^4}{b^6 (a+b x)^2}-\frac{20 a^3}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{10 a^2 x}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4}-\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}-\frac{20 a^3 \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0281285, size = 90, normalized size = 1. \[ -\frac{a^6}{3 b^7 (a+b x)^3}+\frac{3 a^5}{b^7 (a+b x)^2}-\frac{15 a^4}{b^7 (a+b x)}+\frac{10 a^2 x}{b^6}-\frac{20 a^3 \log (a+b x)}{b^7}-\frac{2 a x^2}{b^5}+\frac{x^3}{3 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^4,x]

[Out]

(10*a^2*x)/b^6 - (2*a*x^2)/b^5 + x^3/(3*b^4) - a^6/(3*b^7*(a + b*x)^3) + (3*a^5)/(b^7*(a + b*x)^2) - (15*a^4)/
(b^7*(a + b*x)) - (20*a^3*Log[a + b*x])/b^7

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Maple [A]  time = 0.006, size = 87, normalized size = 1. \begin{align*} 10\,{\frac{{a}^{2}x}{{b}^{6}}}-2\,{\frac{a{x}^{2}}{{b}^{5}}}+{\frac{{x}^{3}}{3\,{b}^{4}}}-{\frac{{a}^{6}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}+3\,{\frac{{a}^{5}}{{b}^{7} \left ( bx+a \right ) ^{2}}}-15\,{\frac{{a}^{4}}{{b}^{7} \left ( bx+a \right ) }}-20\,{\frac{{a}^{3}\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)^4,x)

[Out]

10*a^2*x/b^6-2*a*x^2/b^5+1/3*x^3/b^4-1/3*a^6/b^7/(b*x+a)^3+3*a^5/b^7/(b*x+a)^2-15*a^4/b^7/(b*x+a)-20*a^3*ln(b*
x+a)/b^7

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Maxima [A]  time = 1.11261, size = 138, normalized size = 1.53 \begin{align*} -\frac{45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} - \frac{20 \, a^{3} \log \left (b x + a\right )}{b^{7}} + \frac{b^{2} x^{3} - 6 \, a b x^{2} + 30 \, a^{2} x}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*(45*a^4*b^2*x^2 + 81*a^5*b*x + 37*a^6)/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7) - 20*a^3*log(b*x
+ a)/b^7 + 1/3*(b^2*x^3 - 6*a*b*x^2 + 30*a^2*x)/b^6

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Fricas [A]  time = 1.60368, size = 293, normalized size = 3.26 \begin{align*} \frac{b^{6} x^{6} - 3 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 73 \, a^{3} b^{3} x^{3} + 39 \, a^{4} b^{2} x^{2} - 51 \, a^{5} b x - 37 \, a^{6} - 60 \,{\left (a^{3} b^{3} x^{3} + 3 \, a^{4} b^{2} x^{2} + 3 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*(b^6*x^6 - 3*a*b^5*x^5 + 15*a^2*b^4*x^4 + 73*a^3*b^3*x^3 + 39*a^4*b^2*x^2 - 51*a^5*b*x - 37*a^6 - 60*(a^3*
b^3*x^3 + 3*a^4*b^2*x^2 + 3*a^5*b*x + a^6)*log(b*x + a))/(b^10*x^3 + 3*a*b^9*x^2 + 3*a^2*b^8*x + a^3*b^7)

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Sympy [A]  time = 0.758466, size = 105, normalized size = 1.17 \begin{align*} - \frac{20 a^{3} \log{\left (a + b x \right )}}{b^{7}} + \frac{10 a^{2} x}{b^{6}} - \frac{2 a x^{2}}{b^{5}} - \frac{37 a^{6} + 81 a^{5} b x + 45 a^{4} b^{2} x^{2}}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac{x^{3}}{3 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**4,x)

[Out]

-20*a**3*log(a + b*x)/b**7 + 10*a**2*x/b**6 - 2*a*x**2/b**5 - (37*a**6 + 81*a**5*b*x + 45*a**4*b**2*x**2)/(3*a
**3*b**7 + 9*a**2*b**8*x + 9*a*b**9*x**2 + 3*b**10*x**3) + x**3/(3*b**4)

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Giac [A]  time = 1.20424, size = 112, normalized size = 1.24 \begin{align*} -\frac{20 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \,{\left (b x + a\right )}^{3} b^{7}} + \frac{b^{8} x^{3} - 6 \, a b^{7} x^{2} + 30 \, a^{2} b^{6} x}{3 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^4,x, algorithm="giac")

[Out]

-20*a^3*log(abs(b*x + a))/b^7 - 1/3*(45*a^4*b^2*x^2 + 81*a^5*b*x + 37*a^6)/((b*x + a)^3*b^7) + 1/3*(b^8*x^3 -
6*a*b^7*x^2 + 30*a^2*b^6*x)/b^12