3.181 \(\int \frac{x^6}{(a+b x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{3 a^2 x^2}{b^5}-\frac{a^6}{2 b^7 (a+b x)^2}+\frac{6 a^5}{b^7 (a+b x)}-\frac{10 a^3 x}{b^6}+\frac{15 a^4 \log (a+b x)}{b^7}-\frac{a x^3}{b^4}+\frac{x^4}{4 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0525685, antiderivative size = 86, 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} \[ \frac{3 a^2 x^2}{b^5}-\frac{a^6}{2 b^7 (a+b x)^2}+\frac{6 a^5}{b^7 (a+b x)}-\frac{10 a^3 x}{b^6}+\frac{15 a^4 \log (a+b x)}{b^7}-\frac{a x^3}{b^4}+\frac{x^4}{4 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0398734, size = 77, normalized size = 0.9 \[ \frac{12 a^2 b^2 x^2-\frac{2 a^6}{(a+b x)^2}+\frac{24 a^5}{a+b x}-40 a^3 b x+60 a^4 \log (a+b x)-4 a b^3 x^3+b^4 x^4}{4 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^3,x]

[Out]

(-40*a^3*b*x + 12*a^2*b^2*x^2 - 4*a*b^3*x^3 + b^4*x^4 - (2*a^6)/(a + b*x)^2 + (24*a^5)/(a + b*x) + 60*a^4*Log[
a + b*x])/(4*b^7)

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Maple [A]  time = 0.007, size = 83, normalized size = 1. \begin{align*} -10\,{\frac{{a}^{3}x}{{b}^{6}}}+3\,{\frac{{a}^{2}{x}^{2}}{{b}^{5}}}-{\frac{a{x}^{3}}{{b}^{4}}}+{\frac{{x}^{4}}{4\,{b}^{3}}}-{\frac{{a}^{6}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{a}^{5}}{{b}^{7} \left ( bx+a \right ) }}+15\,{\frac{{a}^{4}\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)^3,x)

[Out]

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

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Maxima [A]  time = 0.994151, size = 123, normalized size = 1.43 \begin{align*} \frac{12 \, a^{5} b x + 11 \, a^{6}}{2 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac{15 \, a^{4} \log \left (b x + a\right )}{b^{7}} + \frac{b^{3} x^{4} - 4 \, a b^{2} x^{3} + 12 \, a^{2} b x^{2} - 40 \, a^{3} x}{4 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*(12*a^5*b*x + 11*a^6)/(b^9*x^2 + 2*a*b^8*x + a^2*b^7) + 15*a^4*log(b*x + a)/b^7 + 1/4*(b^3*x^4 - 4*a*b^2*x
^3 + 12*a^2*b*x^2 - 40*a^3*x)/b^6

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Fricas [A]  time = 1.51429, size = 247, normalized size = 2.87 \begin{align*} \frac{b^{6} x^{6} - 2 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{4} b^{2} x^{2} - 16 \, a^{5} b x + 22 \, a^{6} + 60 \,{\left (a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*x^6 - 2*a*b^5*x^5 + 5*a^2*b^4*x^4 - 20*a^3*b^3*x^3 - 68*a^4*b^2*x^2 - 16*a^5*b*x + 22*a^6 + 60*(a^4*b
^2*x^2 + 2*a^5*b*x + a^6)*log(b*x + a))/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)

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Sympy [A]  time = 0.651687, size = 92, normalized size = 1.07 \begin{align*} \frac{15 a^{4} \log{\left (a + b x \right )}}{b^{7}} - \frac{10 a^{3} x}{b^{6}} + \frac{3 a^{2} x^{2}}{b^{5}} - \frac{a x^{3}}{b^{4}} + \frac{11 a^{6} + 12 a^{5} b x}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac{x^{4}}{4 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**3,x)

[Out]

15*a**4*log(a + b*x)/b**7 - 10*a**3*x/b**6 + 3*a**2*x**2/b**5 - a*x**3/b**4 + (11*a**6 + 12*a**5*b*x)/(2*a**2*
b**7 + 4*a*b**8*x + 2*b**9*x**2) + x**4/(4*b**3)

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Giac [A]  time = 1.20584, size = 112, normalized size = 1.3 \begin{align*} \frac{15 \, a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac{12 \, a^{5} b x + 11 \, a^{6}}{2 \,{\left (b x + a\right )}^{2} b^{7}} + \frac{b^{9} x^{4} - 4 \, a b^{8} x^{3} + 12 \, a^{2} b^{7} x^{2} - 40 \, a^{3} b^{6} x}{4 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^3,x, algorithm="giac")

[Out]

15*a^4*log(abs(b*x + a))/b^7 + 1/2*(12*a^5*b*x + 11*a^6)/((b*x + a)^2*b^7) + 1/4*(b^9*x^4 - 4*a*b^8*x^3 + 12*a
^2*b^7*x^2 - 40*a^3*b^6*x)/b^12