3.225 \(\int \frac{x^9}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=154 \[ \frac{a^9}{9 b^{10} (a+b x)^9}-\frac{9 a^8}{8 b^{10} (a+b x)^8}+\frac{36 a^7}{7 b^{10} (a+b x)^7}-\frac{14 a^6}{b^{10} (a+b x)^6}+\frac{126 a^5}{5 b^{10} (a+b x)^5}-\frac{63 a^4}{2 b^{10} (a+b x)^4}+\frac{28 a^3}{b^{10} (a+b x)^3}-\frac{18 a^2}{b^{10} (a+b x)^2}+\frac{9 a}{b^{10} (a+b x)}+\frac{\log (a+b x)}{b^{10}} \]

[Out]

a^9/(9*b^10*(a + b*x)^9) - (9*a^8)/(8*b^10*(a + b*x)^8) + (36*a^7)/(7*b^10*(a + b*x)^7) - (14*a^6)/(b^10*(a +
b*x)^6) + (126*a^5)/(5*b^10*(a + b*x)^5) - (63*a^4)/(2*b^10*(a + b*x)^4) + (28*a^3)/(b^10*(a + b*x)^3) - (18*a
^2)/(b^10*(a + b*x)^2) + (9*a)/(b^10*(a + b*x)) + Log[a + b*x]/b^10

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Rubi [A]  time = 0.108229, antiderivative size = 154, 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} \[ \frac{a^9}{9 b^{10} (a+b x)^9}-\frac{9 a^8}{8 b^{10} (a+b x)^8}+\frac{36 a^7}{7 b^{10} (a+b x)^7}-\frac{14 a^6}{b^{10} (a+b x)^6}+\frac{126 a^5}{5 b^{10} (a+b x)^5}-\frac{63 a^4}{2 b^{10} (a+b x)^4}+\frac{28 a^3}{b^{10} (a+b x)^3}-\frac{18 a^2}{b^{10} (a+b x)^2}+\frac{9 a}{b^{10} (a+b x)}+\frac{\log (a+b x)}{b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x)^10,x]

[Out]

a^9/(9*b^10*(a + b*x)^9) - (9*a^8)/(8*b^10*(a + b*x)^8) + (36*a^7)/(7*b^10*(a + b*x)^7) - (14*a^6)/(b^10*(a +
b*x)^6) + (126*a^5)/(5*b^10*(a + b*x)^5) - (63*a^4)/(2*b^10*(a + b*x)^4) + (28*a^3)/(b^10*(a + b*x)^3) - (18*a
^2)/(b^10*(a + b*x)^2) + (9*a)/(b^10*(a + b*x)) + 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)^{10}} \, dx &=\int \left (-\frac{a^9}{b^9 (a+b x)^{10}}+\frac{9 a^8}{b^9 (a+b x)^9}-\frac{36 a^7}{b^9 (a+b x)^8}+\frac{84 a^6}{b^9 (a+b x)^7}-\frac{126 a^5}{b^9 (a+b x)^6}+\frac{126 a^4}{b^9 (a+b x)^5}-\frac{84 a^3}{b^9 (a+b x)^4}+\frac{36 a^2}{b^9 (a+b x)^3}-\frac{9 a}{b^9 (a+b x)^2}+\frac{1}{b^9 (a+b x)}\right ) \, dx\\ &=\frac{a^9}{9 b^{10} (a+b x)^9}-\frac{9 a^8}{8 b^{10} (a+b x)^8}+\frac{36 a^7}{7 b^{10} (a+b x)^7}-\frac{14 a^6}{b^{10} (a+b x)^6}+\frac{126 a^5}{5 b^{10} (a+b x)^5}-\frac{63 a^4}{2 b^{10} (a+b x)^4}+\frac{28 a^3}{b^{10} (a+b x)^3}-\frac{18 a^2}{b^{10} (a+b x)^2}+\frac{9 a}{b^{10} (a+b x)}+\frac{\log (a+b x)}{b^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0487159, size = 111, normalized size = 0.72 \[ \frac{a \left (235224 a^6 b^2 x^2+518616 a^5 b^3 x^3+725004 a^4 b^4 x^4+661500 a^3 b^5 x^5+388080 a^2 b^6 x^6+61641 a^7 b x+7129 a^8+136080 a b^7 x^7+22680 b^8 x^8\right )}{2520 b^{10} (a+b x)^9}+\frac{\log (a+b x)}{b^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x)^10,x]

[Out]

(a*(7129*a^8 + 61641*a^7*b*x + 235224*a^6*b^2*x^2 + 518616*a^5*b^3*x^3 + 725004*a^4*b^4*x^4 + 661500*a^3*b^5*x
^5 + 388080*a^2*b^6*x^6 + 136080*a*b^7*x^7 + 22680*b^8*x^8))/(2520*b^10*(a + b*x)^9) + Log[a + b*x]/b^10

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Maple [A]  time = 0.01, size = 145, normalized size = 0.9 \begin{align*}{\frac{{a}^{9}}{9\,{b}^{10} \left ( bx+a \right ) ^{9}}}-{\frac{9\,{a}^{8}}{8\,{b}^{10} \left ( bx+a \right ) ^{8}}}+{\frac{36\,{a}^{7}}{7\,{b}^{10} \left ( bx+a \right ) ^{7}}}-14\,{\frac{{a}^{6}}{{b}^{10} \left ( bx+a \right ) ^{6}}}+{\frac{126\,{a}^{5}}{5\,{b}^{10} \left ( bx+a \right ) ^{5}}}-{\frac{63\,{a}^{4}}{2\,{b}^{10} \left ( bx+a \right ) ^{4}}}+28\,{\frac{{a}^{3}}{{b}^{10} \left ( bx+a \right ) ^{3}}}-18\,{\frac{{a}^{2}}{{b}^{10} \left ( bx+a \right ) ^{2}}}+9\,{\frac{a}{{b}^{10} \left ( bx+a \right ) }}+{\frac{\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)^10,x)

[Out]

1/9*a^9/b^10/(b*x+a)^9-9/8*a^8/b^10/(b*x+a)^8+36/7*a^7/b^10/(b*x+a)^7-14*a^6/b^10/(b*x+a)^6+126/5*a^5/b^10/(b*
x+a)^5-63/2*a^4/b^10/(b*x+a)^4+28*a^3/b^10/(b*x+a)^3-18*a^2/b^10/(b*x+a)^2+9*a/b^10/(b*x+a)+ln(b*x+a)/b^10

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Maxima [A]  time = 1.07296, size = 273, normalized size = 1.77 \begin{align*} \frac{22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9}}{2520 \,{\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} + \frac{\log \left (b x + a\right )}{b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x+a)^10,x, algorithm="maxima")

[Out]

1/2520*(22680*a*b^8*x^8 + 136080*a^2*b^7*x^7 + 388080*a^3*b^6*x^6 + 661500*a^4*b^5*x^5 + 725004*a^5*b^4*x^4 +
518616*a^6*b^3*x^3 + 235224*a^7*b^2*x^2 + 61641*a^8*b*x + 7129*a^9)/(b^19*x^9 + 9*a*b^18*x^8 + 36*a^2*b^17*x^7
+ 84*a^3*b^16*x^6 + 126*a^4*b^15*x^5 + 126*a^5*b^14*x^4 + 84*a^6*b^13*x^3 + 36*a^7*b^12*x^2 + 9*a^8*b^11*x +
a^9*b^10) + log(b*x + a)/b^10

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Fricas [B]  time = 1.56912, size = 684, normalized size = 4.44 \begin{align*} \frac{22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9} + 2520 \,{\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right )}{2520 \,{\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x+a)^10,x, algorithm="fricas")

[Out]

1/2520*(22680*a*b^8*x^8 + 136080*a^2*b^7*x^7 + 388080*a^3*b^6*x^6 + 661500*a^4*b^5*x^5 + 725004*a^5*b^4*x^4 +
518616*a^6*b^3*x^3 + 235224*a^7*b^2*x^2 + 61641*a^8*b*x + 7129*a^9 + 2520*(b^9*x^9 + 9*a*b^8*x^8 + 36*a^2*b^7*
x^7 + 84*a^3*b^6*x^6 + 126*a^4*b^5*x^5 + 126*a^5*b^4*x^4 + 84*a^6*b^3*x^3 + 36*a^7*b^2*x^2 + 9*a^8*b*x + a^9)*
log(b*x + a))/(b^19*x^9 + 9*a*b^18*x^8 + 36*a^2*b^17*x^7 + 84*a^3*b^16*x^6 + 126*a^4*b^15*x^5 + 126*a^5*b^14*x
^4 + 84*a^6*b^13*x^3 + 36*a^7*b^12*x^2 + 9*a^8*b^11*x + a^9*b^10)

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Sympy [A]  time = 1.63164, size = 212, normalized size = 1.38 \begin{align*} \frac{7129 a^{9} + 61641 a^{8} b x + 235224 a^{7} b^{2} x^{2} + 518616 a^{6} b^{3} x^{3} + 725004 a^{5} b^{4} x^{4} + 661500 a^{4} b^{5} x^{5} + 388080 a^{3} b^{6} x^{6} + 136080 a^{2} b^{7} x^{7} + 22680 a b^{8} x^{8}}{2520 a^{9} b^{10} + 22680 a^{8} b^{11} x + 90720 a^{7} b^{12} x^{2} + 211680 a^{6} b^{13} x^{3} + 317520 a^{5} b^{14} x^{4} + 317520 a^{4} b^{15} x^{5} + 211680 a^{3} b^{16} x^{6} + 90720 a^{2} b^{17} x^{7} + 22680 a b^{18} x^{8} + 2520 b^{19} x^{9}} + \frac{\log{\left (a + b x \right )}}{b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x+a)**10,x)

[Out]

(7129*a**9 + 61641*a**8*b*x + 235224*a**7*b**2*x**2 + 518616*a**6*b**3*x**3 + 725004*a**5*b**4*x**4 + 661500*a
**4*b**5*x**5 + 388080*a**3*b**6*x**6 + 136080*a**2*b**7*x**7 + 22680*a*b**8*x**8)/(2520*a**9*b**10 + 22680*a*
*8*b**11*x + 90720*a**7*b**12*x**2 + 211680*a**6*b**13*x**3 + 317520*a**5*b**14*x**4 + 317520*a**4*b**15*x**5
+ 211680*a**3*b**16*x**6 + 90720*a**2*b**17*x**7 + 22680*a*b**18*x**8 + 2520*b**19*x**9) + log(a + b*x)/b**10

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Giac [A]  time = 1.1597, size = 151, normalized size = 0.98 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{b^{10}} + \frac{22680 \, a b^{7} x^{8} + 136080 \, a^{2} b^{6} x^{7} + 388080 \, a^{3} b^{5} x^{6} + 661500 \, a^{4} b^{4} x^{5} + 725004 \, a^{5} b^{3} x^{4} + 518616 \, a^{6} b^{2} x^{3} + 235224 \, a^{7} b x^{2} + 61641 \, a^{8} x + \frac{7129 \, a^{9}}{b}}{2520 \,{\left (b x + a\right )}^{9} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x+a)^10,x, algorithm="giac")

[Out]

log(abs(b*x + a))/b^10 + 1/2520*(22680*a*b^7*x^8 + 136080*a^2*b^6*x^7 + 388080*a^3*b^5*x^6 + 661500*a^4*b^4*x^
5 + 725004*a^5*b^3*x^4 + 518616*a^6*b^2*x^3 + 235224*a^7*b*x^2 + 61641*a^8*x + 7129*a^9/b)/((b*x + a)^9*b^9)