\(\int \frac {x^9}{(a+b x)^7} \, dx\) [208]

Optimal result

Integrand size = 11, antiderivative size = 139 \[ \int \frac {x^9}{(a+b x)^7} \, dx=\frac {28 a^2 x}{b^9}-\frac {7 a x^2}{2 b^8}+\frac {x^3}{3 b^7}+\frac {a^9}{6 b^{10} (a+b x)^6}-\frac {9 a^8}{5 b^{10} (a+b x)^5}+\frac {9 a^7}{b^{10} (a+b x)^4}-\frac {28 a^6}{b^{10} (a+b x)^3}+\frac {63 a^5}{b^{10} (a+b x)^2}-\frac {126 a^4}{b^{10} (a+b x)}-\frac {84 a^3 \log (a+b x)}{b^{10}} \]

[Out]

28*a^2*x/b^9-7/2*a*x^2/b^8+1/3*x^3/b^7+1/6*a^9/b^10/(b*x+a)^6-9/5*a^8/b^10/(b*x+a)^5+9*a^7/b^10/(b*x+a)^4-28*a
^6/b^10/(b*x+a)^3+63*a^5/b^10/(b*x+a)^2-126*a^4/b^10/(b*x+a)-84*a^3*ln(b*x+a)/b^10

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^9}{(a+b x)^7} \, dx=\frac {a^9}{6 b^{10} (a+b x)^6}-\frac {9 a^8}{5 b^{10} (a+b x)^5}+\frac {9 a^7}{b^{10} (a+b x)^4}-\frac {28 a^6}{b^{10} (a+b x)^3}+\frac {63 a^5}{b^{10} (a+b x)^2}-\frac {126 a^4}{b^{10} (a+b x)}-\frac {84 a^3 \log (a+b x)}{b^{10}}+\frac {28 a^2 x}{b^9}-\frac {7 a x^2}{2 b^8}+\frac {x^3}{3 b^7} \]

[In]

Int[x^9/(a + b*x)^7,x]

[Out]

(28*a^2*x)/b^9 - (7*a*x^2)/(2*b^8) + x^3/(3*b^7) + a^9/(6*b^10*(a + b*x)^6) - (9*a^8)/(5*b^10*(a + b*x)^5) + (
9*a^7)/(b^10*(a + b*x)^4) - (28*a^6)/(b^10*(a + b*x)^3) + (63*a^5)/(b^10*(a + b*x)^2) - (126*a^4)/(b^10*(a + b
*x)) - (84*a^3*Log[a + b*x])/b^10

Rule 45

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*} \text {integral}& = \int \left (\frac {28 a^2}{b^9}-\frac {7 a x}{b^8}+\frac {x^2}{b^7}-\frac {a^9}{b^9 (a+b x)^7}+\frac {9 a^8}{b^9 (a+b x)^6}-\frac {36 a^7}{b^9 (a+b x)^5}+\frac {84 a^6}{b^9 (a+b x)^4}-\frac {126 a^5}{b^9 (a+b x)^3}+\frac {126 a^4}{b^9 (a+b x)^2}-\frac {84 a^3}{b^9 (a+b x)}\right ) \, dx \\ & = \frac {28 a^2 x}{b^9}-\frac {7 a x^2}{2 b^8}+\frac {x^3}{3 b^7}+\frac {a^9}{6 b^{10} (a+b x)^6}-\frac {9 a^8}{5 b^{10} (a+b x)^5}+\frac {9 a^7}{b^{10} (a+b x)^4}-\frac {28 a^6}{b^{10} (a+b x)^3}+\frac {63 a^5}{b^{10} (a+b x)^2}-\frac {126 a^4}{b^{10} (a+b x)}-\frac {84 a^3 \log (a+b x)}{b^{10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92 \[ \int \frac {x^9}{(a+b x)^7} \, dx=-\frac {2509 a^9+12534 a^8 b x+23775 a^7 b^2 x^2+19100 a^6 b^3 x^3+1725 a^5 b^4 x^4-6870 a^4 b^5 x^5-3665 a^3 b^6 x^6-360 a^2 b^7 x^7+45 a b^8 x^8-10 b^9 x^9+2520 a^3 (a+b x)^6 \log (a+b x)}{30 b^{10} (a+b x)^6} \]

[In]

Integrate[x^9/(a + b*x)^7,x]

[Out]

-1/30*(2509*a^9 + 12534*a^8*b*x + 23775*a^7*b^2*x^2 + 19100*a^6*b^3*x^3 + 1725*a^5*b^4*x^4 - 6870*a^4*b^5*x^5
- 3665*a^3*b^6*x^6 - 360*a^2*b^7*x^7 + 45*a*b^8*x^8 - 10*b^9*x^9 + 2520*a^3*(a + b*x)^6*Log[a + b*x])/(b^10*(a
+ b*x)^6)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {x^{3}}{3 b^{7}}-\frac {7 a \,x^{2}}{2 b^{8}}+\frac {28 a^{2} x}{b^{9}}+\frac {-126 a^{4} b^{4} x^{5}-567 a^{5} b^{3} x^{4}-1036 a^{6} b^{2} x^{3}-957 a^{7} b \,x^{2}-\frac {2229 a^{8} x}{5}-\frac {2509 a^{9}}{30 b}}{b^{9} \left (b x +a \right )^{6}}-\frac {84 a^{3} \ln \left (b x +a \right )}{b^{10}}\)	\(110\)
norman	\(\frac {\frac {x^{9}}{3 b}-\frac {3 a \,x^{8}}{2 b^{2}}+\frac {12 a^{2} x^{7}}{b^{3}}-\frac {1029 a^{9}}{5 b^{10}}-\frac {504 a^{4} x^{5}}{b^{5}}-\frac {1890 a^{5} x^{4}}{b^{6}}-\frac {3080 a^{6} x^{3}}{b^{7}}-\frac {2625 a^{7} x^{2}}{b^{8}}-\frac {5754 a^{8} x}{5 b^{9}}}{\left (b x +a \right )^{6}}-\frac {84 a^{3} \ln \left (b x +a \right )}{b^{10}}\)	\(114\)
default	\(\frac {\frac {1}{3} b^{2} x^{3}-\frac {7}{2} a b \,x^{2}+28 a^{2} x}{b^{9}}-\frac {9 a^{8}}{5 b^{10} \left (b x +a \right )^{5}}-\frac {84 a^{3} \ln \left (b x +a \right )}{b^{10}}+\frac {a^{9}}{6 b^{10} \left (b x +a \right )^{6}}+\frac {9 a^{7}}{b^{10} \left (b x +a \right )^{4}}-\frac {28 a^{6}}{b^{10} \left (b x +a \right )^{3}}+\frac {63 a^{5}}{b^{10} \left (b x +a \right )^{2}}-\frac {126 a^{4}}{b^{10} \left (b x +a \right )}\)	\(132\)
parallelrisch	\(-\frac {-10 b^{9} x^{9}+45 a \,x^{8} b^{8}+2520 \ln \left (b x +a \right ) x^{6} a^{3} b^{6}-360 a^{2} x^{7} b^{7}+15120 \ln \left (b x +a \right ) x^{5} a^{4} b^{5}+37800 \ln \left (b x +a \right ) x^{4} a^{5} b^{4}+15120 a^{4} x^{5} b^{5}+50400 \ln \left (b x +a \right ) x^{3} a^{6} b^{3}+56700 a^{5} b^{4} x^{4}+37800 \ln \left (b x +a \right ) x^{2} a^{7} b^{2}+92400 a^{6} b^{3} x^{3}+15120 \ln \left (b x +a \right ) x \,a^{8} b +78750 a^{7} b^{2} x^{2}+2520 \ln \left (b x +a \right ) a^{9}+34524 a^{8} b x +6174 a^{9}}{30 b^{10} \left (b x +a \right )^{6}}\)	\(207\)

[In]

int(x^9/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/3*x^3/b^7-7/2*a*x^2/b^8+28*a^2*x/b^9+(-126*a^4*b^4*x^5-567*a^5*b^3*x^4-1036*a^6*b^2*x^3-957*a^7*b*x^2-2229/5
*a^8*x-2509/30*a^9/b)/b^9/(b*x+a)^6-84*a^3*ln(b*x+a)/b^10

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.72 \[ \int \frac {x^9}{(a+b x)^7} \, dx=\frac {10 \, b^{9} x^{9} - 45 \, a b^{8} x^{8} + 360 \, a^{2} b^{7} x^{7} + 3665 \, a^{3} b^{6} x^{6} + 6870 \, a^{4} b^{5} x^{5} - 1725 \, a^{5} b^{4} x^{4} - 19100 \, a^{6} b^{3} x^{3} - 23775 \, a^{7} b^{2} x^{2} - 12534 \, a^{8} b x - 2509 \, a^{9} - 2520 \, {\left (a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{5} + 15 \, a^{5} b^{4} x^{4} + 20 \, a^{6} b^{3} x^{3} + 15 \, a^{7} b^{2} x^{2} + 6 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right )}{30 \, {\left (b^{16} x^{6} + 6 \, a b^{15} x^{5} + 15 \, a^{2} b^{14} x^{4} + 20 \, a^{3} b^{13} x^{3} + 15 \, a^{4} b^{12} x^{2} + 6 \, a^{5} b^{11} x + a^{6} b^{10}\right )}} \]

[In]

integrate(x^9/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/30*(10*b^9*x^9 - 45*a*b^8*x^8 + 360*a^2*b^7*x^7 + 3665*a^3*b^6*x^6 + 6870*a^4*b^5*x^5 - 1725*a^5*b^4*x^4 - 1
9100*a^6*b^3*x^3 - 23775*a^7*b^2*x^2 - 12534*a^8*b*x - 2509*a^9 - 2520*(a^3*b^6*x^6 + 6*a^4*b^5*x^5 + 15*a^5*b
^4*x^4 + 20*a^6*b^3*x^3 + 15*a^7*b^2*x^2 + 6*a^8*b*x + a^9)*log(b*x + a))/(b^16*x^6 + 6*a*b^15*x^5 + 15*a^2*b^
14*x^4 + 20*a^3*b^13*x^3 + 15*a^4*b^12*x^2 + 6*a^5*b^11*x + a^6*b^10)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29 \[ \int \frac {x^9}{(a+b x)^7} \, dx=- \frac {84 a^{3} \log {\left (a + b x \right )}}{b^{10}} + \frac {28 a^{2} x}{b^{9}} - \frac {7 a x^{2}}{2 b^{8}} + \frac {- 2509 a^{9} - 13374 a^{8} b x - 28710 a^{7} b^{2} x^{2} - 31080 a^{6} b^{3} x^{3} - 17010 a^{5} b^{4} x^{4} - 3780 a^{4} b^{5} x^{5}}{30 a^{6} b^{10} + 180 a^{5} b^{11} x + 450 a^{4} b^{12} x^{2} + 600 a^{3} b^{13} x^{3} + 450 a^{2} b^{14} x^{4} + 180 a b^{15} x^{5} + 30 b^{16} x^{6}} + \frac {x^{3}}{3 b^{7}} \]

[In]

integrate(x**9/(b*x+a)**7,x)

[Out]

-84*a**3*log(a + b*x)/b**10 + 28*a**2*x/b**9 - 7*a*x**2/(2*b**8) + (-2509*a**9 - 13374*a**8*b*x - 28710*a**7*b
**2*x**2 - 31080*a**6*b**3*x**3 - 17010*a**5*b**4*x**4 - 3780*a**4*b**5*x**5)/(30*a**6*b**10 + 180*a**5*b**11*
x + 450*a**4*b**12*x**2 + 600*a**3*b**13*x**3 + 450*a**2*b**14*x**4 + 180*a*b**15*x**5 + 30*b**16*x**6) + x**3
/(3*b**7)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22 \[ \int \frac {x^9}{(a+b x)^7} \, dx=-\frac {3780 \, a^{4} b^{5} x^{5} + 17010 \, a^{5} b^{4} x^{4} + 31080 \, a^{6} b^{3} x^{3} + 28710 \, a^{7} b^{2} x^{2} + 13374 \, a^{8} b x + 2509 \, a^{9}}{30 \, {\left (b^{16} x^{6} + 6 \, a b^{15} x^{5} + 15 \, a^{2} b^{14} x^{4} + 20 \, a^{3} b^{13} x^{3} + 15 \, a^{4} b^{12} x^{2} + 6 \, a^{5} b^{11} x + a^{6} b^{10}\right )}} - \frac {84 \, a^{3} \log \left (b x + a\right )}{b^{10}} + \frac {2 \, b^{2} x^{3} - 21 \, a b x^{2} + 168 \, a^{2} x}{6 \, b^{9}} \]

[In]

integrate(x^9/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/30*(3780*a^4*b^5*x^5 + 17010*a^5*b^4*x^4 + 31080*a^6*b^3*x^3 + 28710*a^7*b^2*x^2 + 13374*a^8*b*x + 2509*a^9
)/(b^16*x^6 + 6*a*b^15*x^5 + 15*a^2*b^14*x^4 + 20*a^3*b^13*x^3 + 15*a^4*b^12*x^2 + 6*a^5*b^11*x + a^6*b^10) -
84*a^3*log(b*x + a)/b^10 + 1/6*(2*b^2*x^3 - 21*a*b*x^2 + 168*a^2*x)/b^9

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {x^9}{(a+b x)^7} \, dx=-\frac {84 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{10}} - \frac {3780 \, a^{4} b^{5} x^{5} + 17010 \, a^{5} b^{4} x^{4} + 31080 \, a^{6} b^{3} x^{3} + 28710 \, a^{7} b^{2} x^{2} + 13374 \, a^{8} b x + 2509 \, a^{9}}{30 \, {\left (b x + a\right )}^{6} b^{10}} + \frac {2 \, b^{14} x^{3} - 21 \, a b^{13} x^{2} + 168 \, a^{2} b^{12} x}{6 \, b^{21}} \]

[In]

integrate(x^9/(b*x+a)^7,x, algorithm="giac")

[Out]

-84*a^3*log(abs(b*x + a))/b^10 - 1/30*(3780*a^4*b^5*x^5 + 17010*a^5*b^4*x^4 + 31080*a^6*b^3*x^3 + 28710*a^7*b^
2*x^2 + 13374*a^8*b*x + 2509*a^9)/((b*x + a)^6*b^10) + 1/6*(2*b^14*x^3 - 21*a*b^13*x^2 + 168*a^2*b^12*x)/b^21

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83 \[ \int \frac {x^9}{(a+b x)^7} \, dx=-\frac {\frac {9\,a\,{\left (a+b\,x\right )}^2}{2}-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {126\,a^4}{a+b\,x}-\frac {63\,a^5}{{\left (a+b\,x\right )}^2}+\frac {28\,a^6}{{\left (a+b\,x\right )}^3}-\frac {9\,a^7}{{\left (a+b\,x\right )}^4}+\frac {9\,a^8}{5\,{\left (a+b\,x\right )}^5}-\frac {a^9}{6\,{\left (a+b\,x\right )}^6}+84\,a^3\,\ln \left (a+b\,x\right )-36\,a^2\,b\,x}{b^{10}} \]

[In]

int(x^9/(a + b*x)^7,x)

[Out]

-((9*a*(a + b*x)^2)/2 - (a + b*x)^3/3 + (126*a^4)/(a + b*x) - (63*a^5)/(a + b*x)^2 + (28*a^6)/(a + b*x)^3 - (9
*a^7)/(a + b*x)^4 + (9*a^8)/(5*(a + b*x)^5) - a^9/(6*(a + b*x)^6) + 84*a^3*log(a + b*x) - 36*a^2*b*x)/b^10