3.209 \(\int \frac{x^8}{(a+b x)^7} \, dx\)

Optimal. Leaf size=128 \[ -\frac{a^8}{6 b^9 (a+b x)^6}+\frac{8 a^7}{5 b^9 (a+b x)^5}-\frac{7 a^6}{b^9 (a+b x)^4}+\frac{56 a^5}{3 b^9 (a+b x)^3}-\frac{35 a^4}{b^9 (a+b x)^2}+\frac{56 a^3}{b^9 (a+b x)}+\frac{28 a^2 \log (a+b x)}{b^9}-\frac{7 a x}{b^8}+\frac{x^2}{2 b^7} \]

[Out]

(-7*a*x)/b^8 + x^2/(2*b^7) - a^8/(6*b^9*(a + b*x)^6) + (8*a^7)/(5*b^9*(a + b*x)^5) - (7*a^6)/(b^9*(a + b*x)^4)
+ (56*a^5)/(3*b^9*(a + b*x)^3) - (35*a^4)/(b^9*(a + b*x)^2) + (56*a^3)/(b^9*(a + b*x)) + (28*a^2*Log[a + b*x]
)/b^9

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Rubi [A]  time = 0.0889429, antiderivative size = 128, 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} \[ -\frac{a^8}{6 b^9 (a+b x)^6}+\frac{8 a^7}{5 b^9 (a+b x)^5}-\frac{7 a^6}{b^9 (a+b x)^4}+\frac{56 a^5}{3 b^9 (a+b x)^3}-\frac{35 a^4}{b^9 (a+b x)^2}+\frac{56 a^3}{b^9 (a+b x)}+\frac{28 a^2 \log (a+b x)}{b^9}-\frac{7 a x}{b^8}+\frac{x^2}{2 b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x)^7,x]

[Out]

(-7*a*x)/b^8 + x^2/(2*b^7) - a^8/(6*b^9*(a + b*x)^6) + (8*a^7)/(5*b^9*(a + b*x)^5) - (7*a^6)/(b^9*(a + b*x)^4)
+ (56*a^5)/(3*b^9*(a + b*x)^3) - (35*a^4)/(b^9*(a + b*x)^2) + (56*a^3)/(b^9*(a + b*x)) + (28*a^2*Log[a + b*x]
)/b^9

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^8}{(a+b x)^7} \, dx &=\int \left (-\frac{7 a}{b^8}+\frac{x}{b^7}+\frac{a^8}{b^8 (a+b x)^7}-\frac{8 a^7}{b^8 (a+b x)^6}+\frac{28 a^6}{b^8 (a+b x)^5}-\frac{56 a^5}{b^8 (a+b x)^4}+\frac{70 a^4}{b^8 (a+b x)^3}-\frac{56 a^3}{b^8 (a+b x)^2}+\frac{28 a^2}{b^8 (a+b x)}\right ) \, dx\\ &=-\frac{7 a x}{b^8}+\frac{x^2}{2 b^7}-\frac{a^8}{6 b^9 (a+b x)^6}+\frac{8 a^7}{5 b^9 (a+b x)^5}-\frac{7 a^6}{b^9 (a+b x)^4}+\frac{56 a^5}{3 b^9 (a+b x)^3}-\frac{35 a^4}{b^9 (a+b x)^2}+\frac{56 a^3}{b^9 (a+b x)}+\frac{28 a^2 \log (a+b x)}{b^9}\\ \end{align*}

Mathematica [A]  time = 0.0718687, size = 104, normalized size = 0.81 \[ \frac{-\frac{5 a^8}{(a+b x)^6}+\frac{48 a^7}{(a+b x)^5}-\frac{210 a^6}{(a+b x)^4}+\frac{560 a^5}{(a+b x)^3}-\frac{1050 a^4}{(a+b x)^2}+\frac{1680 a^3}{a+b x}+840 a^2 \log (a+b x)-210 a b x+15 b^2 x^2}{30 b^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x)^7,x]

[Out]

(-210*a*b*x + 15*b^2*x^2 - (5*a^8)/(a + b*x)^6 + (48*a^7)/(a + b*x)^5 - (210*a^6)/(a + b*x)^4 + (560*a^5)/(a +
b*x)^3 - (1050*a^4)/(a + b*x)^2 + (1680*a^3)/(a + b*x) + 840*a^2*Log[a + b*x])/(30*b^9)

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Maple [A]  time = 0.01, size = 121, normalized size = 1. \begin{align*} -7\,{\frac{ax}{{b}^{8}}}+{\frac{{x}^{2}}{2\,{b}^{7}}}-{\frac{{a}^{8}}{6\,{b}^{9} \left ( bx+a \right ) ^{6}}}+{\frac{8\,{a}^{7}}{5\,{b}^{9} \left ( bx+a \right ) ^{5}}}-7\,{\frac{{a}^{6}}{{b}^{9} \left ( bx+a \right ) ^{4}}}+{\frac{56\,{a}^{5}}{3\,{b}^{9} \left ( bx+a \right ) ^{3}}}-35\,{\frac{{a}^{4}}{{b}^{9} \left ( bx+a \right ) ^{2}}}+56\,{\frac{{a}^{3}}{{b}^{9} \left ( bx+a \right ) }}+28\,{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x+a)^7,x)

[Out]

-7*a*x/b^8+1/2*x^2/b^7-1/6*a^8/b^9/(b*x+a)^6+8/5*a^7/b^9/(b*x+a)^5-7*a^6/b^9/(b*x+a)^4+56/3*a^5/b^9/(b*x+a)^3-
35*a^4/b^9/(b*x+a)^2+56*a^3/b^9/(b*x+a)+28*a^2*ln(b*x+a)/b^9

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Maxima [A]  time = 1.08347, size = 212, normalized size = 1.66 \begin{align*} \frac{1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \,{\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} + \frac{28 \, a^{2} \log \left (b x + a\right )}{b^{9}} + \frac{b x^{2} - 14 \, a x}{2 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="maxima")

[Out]

1/30*(1680*a^3*b^5*x^5 + 7350*a^4*b^4*x^4 + 13160*a^5*b^3*x^3 + 11970*a^6*b^2*x^2 + 5508*a^7*b*x + 1023*a^8)/(
b^15*x^6 + 6*a*b^14*x^5 + 15*a^2*b^13*x^4 + 20*a^3*b^12*x^3 + 15*a^4*b^11*x^2 + 6*a^5*b^10*x + a^6*b^9) + 28*a
^2*log(b*x + a)/b^9 + 1/2*(b*x^2 - 14*a*x)/b^8

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Fricas [A]  time = 1.47469, size = 514, normalized size = 4.02 \begin{align*} \frac{15 \, b^{8} x^{8} - 120 \, a b^{7} x^{7} - 1035 \, a^{2} b^{6} x^{6} - 1170 \, a^{3} b^{5} x^{5} + 3375 \, a^{4} b^{4} x^{4} + 10100 \, a^{5} b^{3} x^{3} + 10725 \, a^{6} b^{2} x^{2} + 5298 \, a^{7} b x + 1023 \, a^{8} + 840 \,{\left (a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 20 \, a^{5} b^{3} x^{3} + 15 \, a^{6} b^{2} x^{2} + 6 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{30 \,{\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/30*(15*b^8*x^8 - 120*a*b^7*x^7 - 1035*a^2*b^6*x^6 - 1170*a^3*b^5*x^5 + 3375*a^4*b^4*x^4 + 10100*a^5*b^3*x^3
+ 10725*a^6*b^2*x^2 + 5298*a^7*b*x + 1023*a^8 + 840*(a^2*b^6*x^6 + 6*a^3*b^5*x^5 + 15*a^4*b^4*x^4 + 20*a^5*b^3
*x^3 + 15*a^6*b^2*x^2 + 6*a^7*b*x + a^8)*log(b*x + a))/(b^15*x^6 + 6*a*b^14*x^5 + 15*a^2*b^13*x^4 + 20*a^3*b^1
2*x^3 + 15*a^4*b^11*x^2 + 6*a^5*b^10*x + a^6*b^9)

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Sympy [A]  time = 1.3054, size = 165, normalized size = 1.29 \begin{align*} \frac{28 a^{2} \log{\left (a + b x \right )}}{b^{9}} - \frac{7 a x}{b^{8}} + \frac{1023 a^{8} + 5508 a^{7} b x + 11970 a^{6} b^{2} x^{2} + 13160 a^{5} b^{3} x^{3} + 7350 a^{4} b^{4} x^{4} + 1680 a^{3} b^{5} x^{5}}{30 a^{6} b^{9} + 180 a^{5} b^{10} x + 450 a^{4} b^{11} x^{2} + 600 a^{3} b^{12} x^{3} + 450 a^{2} b^{13} x^{4} + 180 a b^{14} x^{5} + 30 b^{15} x^{6}} + \frac{x^{2}}{2 b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x+a)**7,x)

[Out]

28*a**2*log(a + b*x)/b**9 - 7*a*x/b**8 + (1023*a**8 + 5508*a**7*b*x + 11970*a**6*b**2*x**2 + 13160*a**5*b**3*x
**3 + 7350*a**4*b**4*x**4 + 1680*a**3*b**5*x**5)/(30*a**6*b**9 + 180*a**5*b**10*x + 450*a**4*b**11*x**2 + 600*
a**3*b**12*x**3 + 450*a**2*b**13*x**4 + 180*a*b**14*x**5 + 30*b**15*x**6) + x**2/(2*b**7)

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Giac [A]  time = 1.26158, size = 142, normalized size = 1.11 \begin{align*} \frac{28 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{9}} + \frac{b^{7} x^{2} - 14 \, a b^{6} x}{2 \, b^{14}} + \frac{1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \,{\left (b x + a\right )}^{6} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="giac")

[Out]

28*a^2*log(abs(b*x + a))/b^9 + 1/2*(b^7*x^2 - 14*a*b^6*x)/b^14 + 1/30*(1680*a^3*b^5*x^5 + 7350*a^4*b^4*x^4 + 1
3160*a^5*b^3*x^3 + 11970*a^6*b^2*x^2 + 5508*a^7*b*x + 1023*a^8)/((b*x + a)^6*b^9)