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

Optimal. Leaf size=118 \[ \frac{a^7}{6 b^8 (a+b x)^6}-\frac{7 a^6}{5 b^8 (a+b x)^5}+\frac{21 a^5}{4 b^8 (a+b x)^4}-\frac{35 a^4}{3 b^8 (a+b x)^3}+\frac{35 a^3}{2 b^8 (a+b x)^2}-\frac{21 a^2}{b^8 (a+b x)}-\frac{7 a \log (a+b x)}{b^8}+\frac{x}{b^7} \]

[Out]

x/b^7 + a^7/(6*b^8*(a + b*x)^6) - (7*a^6)/(5*b^8*(a + b*x)^5) + (21*a^5)/(4*b^8*(a + b*x)^4) - (35*a^4)/(3*b^8
*(a + b*x)^3) + (35*a^3)/(2*b^8*(a + b*x)^2) - (21*a^2)/(b^8*(a + b*x)) - (7*a*Log[a + b*x])/b^8

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Rubi [A]  time = 0.0705354, antiderivative size = 118, 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} \[ \frac{a^7}{6 b^8 (a+b x)^6}-\frac{7 a^6}{5 b^8 (a+b x)^5}+\frac{21 a^5}{4 b^8 (a+b x)^4}-\frac{35 a^4}{3 b^8 (a+b x)^3}+\frac{35 a^3}{2 b^8 (a+b x)^2}-\frac{21 a^2}{b^8 (a+b x)}-\frac{7 a \log (a+b x)}{b^8}+\frac{x}{b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x)^7,x]

[Out]

x/b^7 + a^7/(6*b^8*(a + b*x)^6) - (7*a^6)/(5*b^8*(a + b*x)^5) + (21*a^5)/(4*b^8*(a + b*x)^4) - (35*a^4)/(3*b^8
*(a + b*x)^3) + (35*a^3)/(2*b^8*(a + b*x)^2) - (21*a^2)/(b^8*(a + b*x)) - (7*a*Log[a + b*x])/b^8

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^7}{(a+b x)^7} \, dx &=\int \left (\frac{1}{b^7}-\frac{a^7}{b^7 (a+b x)^7}+\frac{7 a^6}{b^7 (a+b x)^6}-\frac{21 a^5}{b^7 (a+b x)^5}+\frac{35 a^4}{b^7 (a+b x)^4}-\frac{35 a^3}{b^7 (a+b x)^3}+\frac{21 a^2}{b^7 (a+b x)^2}-\frac{7 a}{b^7 (a+b x)}\right ) \, dx\\ &=\frac{x}{b^7}+\frac{a^7}{6 b^8 (a+b x)^6}-\frac{7 a^6}{5 b^8 (a+b x)^5}+\frac{21 a^5}{4 b^8 (a+b x)^4}-\frac{35 a^4}{3 b^8 (a+b x)^3}+\frac{35 a^3}{2 b^8 (a+b x)^2}-\frac{21 a^2}{b^8 (a+b x)}-\frac{7 a \log (a+b x)}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.0485582, size = 104, normalized size = 0.88 \[ -\frac{7725 a^5 b^2 x^2+8200 a^4 b^3 x^3+4050 a^3 b^4 x^4+360 a^2 b^5 x^5+3594 a^6 b x+669 a^7-360 a b^6 x^6+420 a (a+b x)^6 \log (a+b x)-60 b^7 x^7}{60 b^8 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x)^7,x]

[Out]

-(669*a^7 + 3594*a^6*b*x + 7725*a^5*b^2*x^2 + 8200*a^4*b^3*x^3 + 4050*a^3*b^4*x^4 + 360*a^2*b^5*x^5 - 360*a*b^
6*x^6 - 60*b^7*x^7 + 420*a*(a + b*x)^6*Log[a + b*x])/(60*b^8*(a + b*x)^6)

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Maple [A]  time = 0.009, size = 109, normalized size = 0.9 \begin{align*}{\frac{x}{{b}^{7}}}+{\frac{{a}^{7}}{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{7\,{a}^{6}}{5\,{b}^{8} \left ( bx+a \right ) ^{5}}}+{\frac{21\,{a}^{5}}{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}-{\frac{35\,{a}^{4}}{3\,{b}^{8} \left ( bx+a \right ) ^{3}}}+{\frac{35\,{a}^{3}}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}-21\,{\frac{{a}^{2}}{{b}^{8} \left ( bx+a \right ) }}-7\,{\frac{a\ln \left ( bx+a \right ) }{{b}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x+a)^7,x)

[Out]

x/b^7+1/6*a^7/b^8/(b*x+a)^6-7/5*a^6/b^8/(b*x+a)^5+21/4*a^5/b^8/(b*x+a)^4-35/3*a^4/b^8/(b*x+a)^3+35/2*a^3/b^8/(
b*x+a)^2-21*a^2/b^8/(b*x+a)-7*a*ln(b*x+a)/b^8

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Maxima [A]  time = 1.08428, size = 196, normalized size = 1.66 \begin{align*} -\frac{1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac{x}{b^{7}} - \frac{7 \, a \log \left (b x + a\right )}{b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4*x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b^2*x^2 + 3654*a^6*b*x + 669*a^7)/(b^
14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 + 6*a^5*b^9*x + a^6*b^8) + x/b^7 -
7*a*log(b*x + a)/b^8

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Fricas [A]  time = 1.55993, size = 479, normalized size = 4.06 \begin{align*} \frac{60 \, b^{7} x^{7} + 360 \, a b^{6} x^{6} - 360 \, a^{2} b^{5} x^{5} - 4050 \, a^{3} b^{4} x^{4} - 8200 \, a^{4} b^{3} x^{3} - 7725 \, a^{5} b^{2} x^{2} - 3594 \, a^{6} b x - 669 \, a^{7} - 420 \,{\left (a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{5} + 15 \, a^{3} b^{4} x^{4} + 20 \, a^{4} b^{3} x^{3} + 15 \, a^{5} b^{2} x^{2} + 6 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^7*x^7 + 360*a*b^6*x^6 - 360*a^2*b^5*x^5 - 4050*a^3*b^4*x^4 - 8200*a^4*b^3*x^3 - 7725*a^5*b^2*x^2 -
3594*a^6*b*x - 669*a^7 - 420*(a*b^6*x^6 + 6*a^2*b^5*x^5 + 15*a^3*b^4*x^4 + 20*a^4*b^3*x^3 + 15*a^5*b^2*x^2 + 6
*a^6*b*x + a^7)*log(b*x + a))/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 +
6*a^5*b^9*x + a^6*b^8)

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Sympy [A]  time = 1.22954, size = 151, normalized size = 1.28 \begin{align*} - \frac{7 a \log{\left (a + b x \right )}}{b^{8}} - \frac{669 a^{7} + 3654 a^{6} b x + 8085 a^{5} b^{2} x^{2} + 9100 a^{4} b^{3} x^{3} + 5250 a^{3} b^{4} x^{4} + 1260 a^{2} b^{5} x^{5}}{60 a^{6} b^{8} + 360 a^{5} b^{9} x + 900 a^{4} b^{10} x^{2} + 1200 a^{3} b^{11} x^{3} + 900 a^{2} b^{12} x^{4} + 360 a b^{13} x^{5} + 60 b^{14} x^{6}} + \frac{x}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*a*log(a + b*x)/b**8 - (669*a**7 + 3654*a**6*b*x + 8085*a**5*b**2*x**2 + 9100*a**4*b**3*x**3 + 5250*a**3*b**
4*x**4 + 1260*a**2*b**5*x**5)/(60*a**6*b**8 + 360*a**5*b**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 + 9
00*a**2*b**12*x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) + x/b**7

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Giac [A]  time = 1.19203, size = 119, normalized size = 1.01 \begin{align*} \frac{x}{b^{7}} - \frac{7 \, a \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \,{\left (b x + a\right )}^{6} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^7 - 7*a*log(abs(b*x + a))/b^8 - 1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4*x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b^
2*x^2 + 3654*a^6*b*x + 669*a^7)/((b*x + a)^6*b^8)