3.211 \(\int \frac{x^6}{(a+b x)^7} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^6}{6 b^7 (a+b x)^6}+\frac{6 a^5}{5 b^7 (a+b x)^5}-\frac{15 a^4}{4 b^7 (a+b x)^4}+\frac{20 a^3}{3 b^7 (a+b x)^3}-\frac{15 a^2}{2 b^7 (a+b x)^2}+\frac{6 a}{b^7 (a+b x)}+\frac{\log (a+b x)}{b^7} \]

[Out]

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

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Rubi [A]  time = 0.0642931, antiderivative size = 109, 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} \[ -\frac{a^6}{6 b^7 (a+b x)^6}+\frac{6 a^5}{5 b^7 (a+b x)^5}-\frac{15 a^4}{4 b^7 (a+b x)^4}+\frac{20 a^3}{3 b^7 (a+b x)^3}-\frac{15 a^2}{2 b^7 (a+b x)^2}+\frac{6 a}{b^7 (a+b x)}+\frac{\log (a+b x)}{b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^7,x]

[Out]

-a^6/(6*b^7*(a + b*x)^6) + (6*a^5)/(5*b^7*(a + b*x)^5) - (15*a^4)/(4*b^7*(a + b*x)^4) + (20*a^3)/(3*b^7*(a + b
*x)^3) - (15*a^2)/(2*b^7*(a + b*x)^2) + (6*a)/(b^7*(a + b*x)) + 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)^7} \, dx &=\int \left (\frac{a^6}{b^6 (a+b x)^7}-\frac{6 a^5}{b^6 (a+b x)^6}+\frac{15 a^4}{b^6 (a+b x)^5}-\frac{20 a^3}{b^6 (a+b x)^4}+\frac{15 a^2}{b^6 (a+b x)^3}-\frac{6 a}{b^6 (a+b x)^2}+\frac{1}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac{a^6}{6 b^7 (a+b x)^6}+\frac{6 a^5}{5 b^7 (a+b x)^5}-\frac{15 a^4}{4 b^7 (a+b x)^4}+\frac{20 a^3}{3 b^7 (a+b x)^3}-\frac{15 a^2}{2 b^7 (a+b x)^2}+\frac{6 a}{b^7 (a+b x)}+\frac{\log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0377205, size = 77, normalized size = 0.71 \[ \frac{\frac{a \left (1875 a^3 b^2 x^2+2200 a^2 b^3 x^3+822 a^4 b x+147 a^5+1350 a b^4 x^4+360 b^5 x^5\right )}{(a+b x)^6}+60 \log (a+b x)}{60 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^7,x]

[Out]

((a*(147*a^5 + 822*a^4*b*x + 1875*a^3*b^2*x^2 + 2200*a^2*b^3*x^3 + 1350*a*b^4*x^4 + 360*b^5*x^5))/(a + b*x)^6
+ 60*Log[a + b*x])/(60*b^7)

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Maple [A]  time = 0.007, size = 100, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{6\,{b}^{7} \left ( bx+a \right ) ^{6}}}+{\frac{6\,{a}^{5}}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}-{\frac{15\,{a}^{4}}{4\,{b}^{7} \left ( bx+a \right ) ^{4}}}+{\frac{20\,{a}^{3}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}-{\frac{15\,{a}^{2}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+6\,{\frac{a}{{b}^{7} \left ( bx+a \right ) }}+{\frac{\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)^7,x)

[Out]

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

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Maxima [A]  time = 1.08102, size = 184, normalized size = 1.69 \begin{align*} \frac{360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6}}{60 \,{\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} + \frac{\log \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)^7,x, algorithm="maxima")

[Out]

1/60*(360*a*b^5*x^5 + 1350*a^2*b^4*x^4 + 2200*a^3*b^3*x^3 + 1875*a^4*b^2*x^2 + 822*a^5*b*x + 147*a^6)/(b^13*x^
6 + 6*a*b^12*x^5 + 15*a^2*b^11*x^4 + 20*a^3*b^10*x^3 + 15*a^4*b^9*x^2 + 6*a^5*b^8*x + a^6*b^7) + log(b*x + a)/
b^7

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Fricas [A]  time = 1.54159, size = 428, normalized size = 3.93 \begin{align*} \frac{360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6} + 60 \,{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/60*(360*a*b^5*x^5 + 1350*a^2*b^4*x^4 + 2200*a^3*b^3*x^3 + 1875*a^4*b^2*x^2 + 822*a^5*b*x + 147*a^6 + 60*(b^6
*x^6 + 6*a*b^5*x^5 + 15*a^2*b^4*x^4 + 20*a^3*b^3*x^3 + 15*a^4*b^2*x^2 + 6*a^5*b*x + a^6)*log(b*x + a))/(b^13*x
^6 + 6*a*b^12*x^5 + 15*a^2*b^11*x^4 + 20*a^3*b^10*x^3 + 15*a^4*b^9*x^2 + 6*a^5*b^8*x + a^6*b^7)

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Sympy [A]  time = 1.04979, size = 141, normalized size = 1.29 \begin{align*} \frac{147 a^{6} + 822 a^{5} b x + 1875 a^{4} b^{2} x^{2} + 2200 a^{3} b^{3} x^{3} + 1350 a^{2} b^{4} x^{4} + 360 a b^{5} x^{5}}{60 a^{6} b^{7} + 360 a^{5} b^{8} x + 900 a^{4} b^{9} x^{2} + 1200 a^{3} b^{10} x^{3} + 900 a^{2} b^{11} x^{4} + 360 a b^{12} x^{5} + 60 b^{13} x^{6}} + \frac{\log{\left (a + b x \right )}}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**7,x)

[Out]

(147*a**6 + 822*a**5*b*x + 1875*a**4*b**2*x**2 + 2200*a**3*b**3*x**3 + 1350*a**2*b**4*x**4 + 360*a*b**5*x**5)/
(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**1
2*x**5 + 60*b**13*x**6) + log(a + b*x)/b**7

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Giac [A]  time = 1.20754, size = 107, normalized size = 0.98 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac{360 \, a b^{4} x^{5} + 1350 \, a^{2} b^{3} x^{4} + 2200 \, a^{3} b^{2} x^{3} + 1875 \, a^{4} b x^{2} + 822 \, a^{5} x + \frac{147 \, a^{6}}{b}}{60 \,{\left (b x + a\right )}^{6} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^7,x, algorithm="giac")

[Out]

log(abs(b*x + a))/b^7 + 1/60*(360*a*b^4*x^5 + 1350*a^2*b^3*x^4 + 2200*a^3*b^2*x^3 + 1875*a^4*b*x^2 + 822*a^5*x
+ 147*a^6/b)/((b*x + a)^6*b^6)