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

Optimal. Leaf size=30 \[ \frac{a}{9 b^2 (a+b x)^9}-\frac{1}{8 b^2 (a+b x)^8} \]

[Out]

a/(9*b^2*(a + b*x)^9) - 1/(8*b^2*(a + b*x)^8)

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Rubi [A]  time = 0.0135037, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a}{9 b^2 (a+b x)^9}-\frac{1}{8 b^2 (a+b x)^8} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^10,x]

[Out]

a/(9*b^2*(a + b*x)^9) - 1/(8*b^2*(a + b*x)^8)

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}{(a+b x)^{10}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{10}}+\frac{1}{b (a+b x)^9}\right ) \, dx\\ &=\frac{a}{9 b^2 (a+b x)^9}-\frac{1}{8 b^2 (a+b x)^8}\\ \end{align*}

Mathematica [A]  time = 0.0091609, size = 20, normalized size = 0.67 \[ -\frac{a+9 b x}{72 b^2 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^10,x]

[Out]

-(a + 9*b*x)/(72*b^2*(a + b*x)^9)

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Maple [A]  time = 0.005, size = 27, normalized size = 0.9 \begin{align*}{\frac{a}{9\,{b}^{2} \left ( bx+a \right ) ^{9}}}-{\frac{1}{8\,{b}^{2} \left ( bx+a \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x+a)^10,x)

[Out]

1/9*a/b^2/(b*x+a)^9-1/8/b^2/(b*x+a)^8

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Maxima [B]  time = 1.17228, size = 147, normalized size = 4.9 \begin{align*} -\frac{9 \, b x + a}{72 \,{\left (b^{11} x^{9} + 9 \, a b^{10} x^{8} + 36 \, a^{2} b^{9} x^{7} + 84 \, a^{3} b^{8} x^{6} + 126 \, a^{4} b^{7} x^{5} + 126 \, a^{5} b^{6} x^{4} + 84 \, a^{6} b^{5} x^{3} + 36 \, a^{7} b^{4} x^{2} + 9 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(9*b*x + a)/(b^11*x^9 + 9*a*b^10*x^8 + 36*a^2*b^9*x^7 + 84*a^3*b^8*x^6 + 126*a^4*b^7*x^5 + 126*a^5*b^6*x
^4 + 84*a^6*b^5*x^3 + 36*a^7*b^4*x^2 + 9*a^8*b^3*x + a^9*b^2)

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Fricas [B]  time = 1.38064, size = 234, normalized size = 7.8 \begin{align*} -\frac{9 \, b x + a}{72 \,{\left (b^{11} x^{9} + 9 \, a b^{10} x^{8} + 36 \, a^{2} b^{9} x^{7} + 84 \, a^{3} b^{8} x^{6} + 126 \, a^{4} b^{7} x^{5} + 126 \, a^{5} b^{6} x^{4} + 84 \, a^{6} b^{5} x^{3} + 36 \, a^{7} b^{4} x^{2} + 9 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(9*b*x + a)/(b^11*x^9 + 9*a*b^10*x^8 + 36*a^2*b^9*x^7 + 84*a^3*b^8*x^6 + 126*a^4*b^7*x^5 + 126*a^5*b^6*x
^4 + 84*a^6*b^5*x^3 + 36*a^7*b^4*x^2 + 9*a^8*b^3*x + a^9*b^2)

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Sympy [B]  time = 1.07181, size = 116, normalized size = 3.87 \begin{align*} - \frac{a + 9 b x}{72 a^{9} b^{2} + 648 a^{8} b^{3} x + 2592 a^{7} b^{4} x^{2} + 6048 a^{6} b^{5} x^{3} + 9072 a^{5} b^{6} x^{4} + 9072 a^{4} b^{7} x^{5} + 6048 a^{3} b^{8} x^{6} + 2592 a^{2} b^{9} x^{7} + 648 a b^{10} x^{8} + 72 b^{11} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + 9*b*x)/(72*a**9*b**2 + 648*a**8*b**3*x + 2592*a**7*b**4*x**2 + 6048*a**6*b**5*x**3 + 9072*a**5*b**6*x**4
+ 9072*a**4*b**7*x**5 + 6048*a**3*b**8*x**6 + 2592*a**2*b**9*x**7 + 648*a*b**10*x**8 + 72*b**11*x**9)

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Giac [A]  time = 1.20915, size = 24, normalized size = 0.8 \begin{align*} -\frac{9 \, b x + a}{72 \,{\left (b x + a\right )}^{9} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(9*b*x + a)/((b*x + a)^9*b^2)