3.3.25 \(\int \frac {x^9}{(a+b x)^{10}} \, dx\) [225]

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]

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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Rubi [A]
time = 0.07, antiderivative size = 154, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \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 {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.02, size = 111, normalized size = 0.72 \begin {gather*} \frac {a \left (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\right )}{2520 b^{10} (a+b x)^9}+\frac {\log (a+b x)}{b^{10}} \end {gather*}

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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Mathics [A]
time = 3.86, size = 288, normalized size = 1.87 \begin {gather*} \frac {\frac {a \left (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\right )}{2520}+\text {Log}\left [a+b x\right ] \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )}{b^{10} \left (a^9+9 a^8 b x+36 a^7 b^2 x^2+84 a^6 b^3 x^3+126 a^5 b^4 x^4+126 a^4 b^5 x^5+84 a^3 b^6 x^6+36 a^2 b^7 x^7+9 a b^8 x^8+b^9 x^9\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('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 +
Log[a + b x] (a ^ 9 + 9 a ^ 8 b x + 36 a ^ 7 b ^ 2 x ^ 2 + 84 a ^ 6 b ^ 3 x ^ 3 + 126 a ^ 5 b ^ 4 x ^ 4 + 126
a ^ 4 b ^ 5 x ^ 5 + 84 a ^ 3 b ^ 6 x ^ 6 + 36 a ^ 2 b ^ 7 x ^ 7 + 9 a b ^ 8 x ^ 8 + b ^ 9 x ^ 9)) / (b ^ 10 (
a ^ 9 + 9 a ^ 8 b x + 36 a ^ 7 b ^ 2 x ^ 2 + 84 a ^ 6 b ^ 3 x ^ 3 + 126 a ^ 5 b ^ 4 x ^ 4 + 126 a ^ 4 b ^ 5 x
^ 5 + 84 a ^ 3 b ^ 6 x ^ 6 + 36 a ^ 2 b ^ 7 x ^ 7 + 9 a b ^ 8 x ^ 8 + b ^ 9 x ^ 9))

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Maple [A]
time = 0.08, size = 145, normalized size = 0.94

method	result	size
norman	\(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\)	\(113\)
risch	\(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\)	\(113\)
default	\(\frac {a^{9}}{9 b^{10} \left (b x +a \right )^{9}}-\frac {9 a^{8}}{8 b^{10} \left (b x +a \right )^{8}}+\frac {36 a^{7}}{7 b^{10} \left (b x +a \right )^{7}}-\frac {14 a^{6}}{b^{10} \left (b x +a \right )^{6}}+\frac {126 a^{5}}{5 b^{10} \left (b x +a \right )^{5}}-\frac {63 a^{4}}{2 b^{10} \left (b x +a \right )^{4}}+\frac {28 a^{3}}{b^{10} \left (b x +a \right )^{3}}-\frac {18 a^{2}}{b^{10} \left (b x +a \right )^{2}}+\frac {9 a}{b^{10} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{10}}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x+a)^10,x,method=_RETURNVERBOSE)

[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 = 0.25, size = 202, normalized size = 1.31 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (144) = 288\).
time = 0.31, size = 292, normalized size = 1.90 \begin {gather*} \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 {gather*}

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 = 0.52, size = 212, normalized size = 1.38 \begin {gather*} \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 {gather*}

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 = 0.00, size = 116, normalized size = 0.75 \begin {gather*} \frac {\frac {1}{2520} \left (22680 b^{7} a x^{8}+136080 b^{6} a^{2} x^{7}+388080 b^{5} a^{3} x^{6}+661500 b^{4} a^{4} x^{5}+725004 b^{3} a^{5} x^{4}+518616 b^{2} a^{6} x^{3}+235224 b a^{7} x^{2}+61641 a^{8} x+\frac {7129 a^{9}}{b}\right )}{b^{9} \left (x b+a\right )^{9}}+\frac {\ln \left |x b+a\right |}{b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Mupad [B]
time = 0.19, size = 117, normalized size = 0.76 \begin {gather*} \frac {\ln \left (a+b\,x\right )+\frac {9\,a}{a+b\,x}-\frac {18\,a^2}{{\left (a+b\,x\right )}^2}+\frac {28\,a^3}{{\left (a+b\,x\right )}^3}-\frac {63\,a^4}{2\,{\left (a+b\,x\right )}^4}+\frac {126\,a^5}{5\,{\left (a+b\,x\right )}^5}-\frac {14\,a^6}{{\left (a+b\,x\right )}^6}+\frac {36\,a^7}{7\,{\left (a+b\,x\right )}^7}-\frac {9\,a^8}{8\,{\left (a+b\,x\right )}^8}+\frac {a^9}{9\,{\left (a+b\,x\right )}^9}}{b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(a + b*x)^10,x)

[Out]

(log(a + b*x) + (9*a)/(a + b*x) - (18*a^2)/(a + b*x)^2 + (28*a^3)/(a + b*x)^3 - (63*a^4)/(2*(a + b*x)^4) + (12
6*a^5)/(5*(a + b*x)^5) - (14*a^6)/(a + b*x)^6 + (36*a^7)/(7*(a + b*x)^7) - (9*a^8)/(8*(a + b*x)^8) + a^9/(9*(a
+ b*x)^9))/b^10

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