3.232 \(\int \frac{x^2}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a^2}{9 b^3 (a+b x)^9}+\frac{a}{4 b^3 (a+b x)^8}-\frac{1}{7 b^3 (a+b x)^7} \]

[Out]

-a^2/(9*b^3*(a + b*x)^9) + a/(4*b^3*(a + b*x)^8) - 1/(7*b^3*(a + b*x)^7)

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Rubi [A]  time = 0.0207929, antiderivative size = 47, 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} \[ -\frac{a^2}{9 b^3 (a+b x)^9}+\frac{a}{4 b^3 (a+b x)^8}-\frac{1}{7 b^3 (a+b x)^7} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^10,x]

[Out]

-a^2/(9*b^3*(a + b*x)^9) + a/(4*b^3*(a + b*x)^8) - 1/(7*b^3*(a + b*x)^7)

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^2}{(a+b x)^{10}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{10}}-\frac{2 a}{b^2 (a+b x)^9}+\frac{1}{b^2 (a+b x)^8}\right ) \, dx\\ &=-\frac{a^2}{9 b^3 (a+b x)^9}+\frac{a}{4 b^3 (a+b x)^8}-\frac{1}{7 b^3 (a+b x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0146774, size = 31, normalized size = 0.66 \[ -\frac{a^2+9 a b x+36 b^2 x^2}{252 b^3 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^10,x]

[Out]

-(a^2 + 9*a*b*x + 36*b^2*x^2)/(252*b^3*(a + b*x)^9)

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Maple [A]  time = 0.005, size = 42, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{9\,{b}^{3} \left ( bx+a \right ) ^{9}}}+{\frac{a}{4\,{b}^{3} \left ( bx+a \right ) ^{8}}}-{\frac{1}{7\,{b}^{3} \left ( bx+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^10,x)

[Out]

-1/9*a^2/b^3/(b*x+a)^9+1/4*a/b^3/(b*x+a)^8-1/7/b^3/(b*x+a)^7

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Maxima [B]  time = 1.04763, size = 162, normalized size = 3.45 \begin{align*} -\frac{36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \,{\left (b^{12} x^{9} + 9 \, a b^{11} x^{8} + 36 \, a^{2} b^{10} x^{7} + 84 \, a^{3} b^{9} x^{6} + 126 \, a^{4} b^{8} x^{5} + 126 \, a^{5} b^{7} x^{4} + 84 \, a^{6} b^{6} x^{3} + 36 \, a^{7} b^{5} x^{2} + 9 \, a^{8} b^{4} x + a^{9} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/(b^12*x^9 + 9*a*b^11*x^8 + 36*a^2*b^10*x^7 + 84*a^3*b^9*x^6 + 126*a^4*b^8*
x^5 + 126*a^5*b^7*x^4 + 84*a^6*b^6*x^3 + 36*a^7*b^5*x^2 + 9*a^8*b^4*x + a^9*b^3)

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Fricas [B]  time = 1.47967, size = 259, normalized size = 5.51 \begin{align*} -\frac{36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \,{\left (b^{12} x^{9} + 9 \, a b^{11} x^{8} + 36 \, a^{2} b^{10} x^{7} + 84 \, a^{3} b^{9} x^{6} + 126 \, a^{4} b^{8} x^{5} + 126 \, a^{5} b^{7} x^{4} + 84 \, a^{6} b^{6} x^{3} + 36 \, a^{7} b^{5} x^{2} + 9 \, a^{8} b^{4} x + a^{9} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/(b^12*x^9 + 9*a*b^11*x^8 + 36*a^2*b^10*x^7 + 84*a^3*b^9*x^6 + 126*a^4*b^8*
x^5 + 126*a^5*b^7*x^4 + 84*a^6*b^6*x^3 + 36*a^7*b^5*x^2 + 9*a^8*b^4*x + a^9*b^3)

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Sympy [B]  time = 1.10751, size = 128, normalized size = 2.72 \begin{align*} - \frac{a^{2} + 9 a b x + 36 b^{2} x^{2}}{252 a^{9} b^{3} + 2268 a^{8} b^{4} x + 9072 a^{7} b^{5} x^{2} + 21168 a^{6} b^{6} x^{3} + 31752 a^{5} b^{7} x^{4} + 31752 a^{4} b^{8} x^{5} + 21168 a^{3} b^{9} x^{6} + 9072 a^{2} b^{10} x^{7} + 2268 a b^{11} x^{8} + 252 b^{12} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**10,x)

[Out]

-(a**2 + 9*a*b*x + 36*b**2*x**2)/(252*a**9*b**3 + 2268*a**8*b**4*x + 9072*a**7*b**5*x**2 + 21168*a**6*b**6*x**
3 + 31752*a**5*b**7*x**4 + 31752*a**4*b**8*x**5 + 21168*a**3*b**9*x**6 + 9072*a**2*b**10*x**7 + 2268*a*b**11*x
**8 + 252*b**12*x**9)

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Giac [A]  time = 1.20627, size = 39, normalized size = 0.83 \begin{align*} -\frac{36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \,{\left (b x + a\right )}^{9} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/((b*x + a)^9*b^3)