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

Optimal. Leaf size=139 \[ \frac{a^9}{6 b^{10} (a+b x)^6}-\frac{9 a^8}{5 b^{10} (a+b x)^5}+\frac{9 a^7}{b^{10} (a+b x)^4}-\frac{28 a^6}{b^{10} (a+b x)^3}+\frac{63 a^5}{b^{10} (a+b x)^2}-\frac{126 a^4}{b^{10} (a+b x)}+\frac{28 a^2 x}{b^9}-\frac{84 a^3 \log (a+b x)}{b^{10}}-\frac{7 a x^2}{2 b^8}+\frac{x^3}{3 b^7} \]

[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

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Rubi [A]  time = 0.113718, antiderivative size = 139, 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} \[ \frac{a^9}{6 b^{10} (a+b x)^6}-\frac{9 a^8}{5 b^{10} (a+b x)^5}+\frac{9 a^7}{b^{10} (a+b x)^4}-\frac{28 a^6}{b^{10} (a+b x)^3}+\frac{63 a^5}{b^{10} (a+b x)^2}-\frac{126 a^4}{b^{10} (a+b x)}+\frac{28 a^2 x}{b^9}-\frac{84 a^3 \log (a+b x)}{b^{10}}-\frac{7 a x^2}{2 b^8}+\frac{x^3}{3 b^7} \]

Antiderivative was successfully verified.

[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 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^9}{(a+b x)^7} \, dx &=\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]  time = 0.0488204, size = 128, normalized size = 0.92 \[ -\frac{23775 a^7 b^2 x^2+19100 a^6 b^3 x^3+1725 a^5 b^4 x^4-6870 a^4 b^5 x^5-3665 a^3 b^6 x^6-360 a^2 b^7 x^7+12534 a^8 b x+2520 a^3 (a+b x)^6 \log (a+b x)+2509 a^9+45 a b^8 x^8-10 b^9 x^9}{30 b^{10} (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x)^7,x]

[Out]

-(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 - 366
5*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)

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Maple [A]  time = 0.008, size = 132, normalized size = 1. \begin{align*} 28\,{\frac{{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} \left ( bx+a \right ) ^{6}}}-{\frac{9\,{a}^{8}}{5\,{b}^{10} \left ( bx+a \right ) ^{5}}}+9\,{\frac{{a}^{7}}{{b}^{10} \left ( bx+a \right ) ^{4}}}-28\,{\frac{{a}^{6}}{{b}^{10} \left ( bx+a \right ) ^{3}}}+63\,{\frac{{a}^{5}}{{b}^{10} \left ( bx+a \right ) ^{2}}}-126\,{\frac{{a}^{4}}{{b}^{10} \left ( bx+a \right ) }}-84\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.08471, size = 228, normalized size = 1.64 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.43355, size = 541, normalized size = 3.89 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.40578, size = 178, normalized size = 1.28 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.21574, size = 158, normalized size = 1.14 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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