3.3.32 \(\int \frac {x^2}{(a+b x)^{10}} \, dx\) [232]

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]

-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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Rubi [A]
time = 0.02, antiderivative size = 47, 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^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 {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^10,x]

[Out]

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

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^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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.66 \begin {gather*} -\frac {a^2+9 a b x+36 b^2 x^2}{252 b^3 (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^10,x]

[Out]

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

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(47)=94\).
time = 2.75, size = 119, normalized size = 2.53 \begin {gather*} \frac {-a^2-9 a b x-36 b^2 x^2}{252 b^3 \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^2/(a + b*x)^10,x]')

[Out]

(-a ^ 2 - 9 a b x - 36 b ^ 2 x ^ 2) / (252 b ^ 3 (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 = 0.10, size = 42, normalized size = 0.89

method	result	size
gosper	\(-\frac {36 x^{2} b^{2}+9 a b x +a^{2}}{252 \left (b x +a \right )^{9} b^{3}}\)	\(30\)
norman	\(\frac {-\frac {x^{2}}{7 b}-\frac {a x}{28 b^{2}}-\frac {a^{2}}{252 b^{3}}}{\left (b x +a \right )^{9}}\)	\(33\)
risch	\(\frac {-\frac {x^{2}}{7 b}-\frac {a x}{28 b^{2}}-\frac {a^{2}}{252 b^{3}}}{\left (b x +a \right )^{9}}\)	\(33\)
default	\(-\frac {a^{2}}{9 b^{3} \left (b x +a \right )^{9}}+\frac {a}{4 b^{3} \left (b x +a \right )^{8}}-\frac {1}{7 b^{3} \left (b x +a \right )^{7}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).
time = 0.25, size = 120, normalized size = 2.55 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).
time = 0.31, size = 120, normalized size = 2.55 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (41) = 82\).
time = 0.31, size = 128, normalized size = 2.72 \begin {gather*} \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 {gather*}

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 = 0.00, size = 34, normalized size = 0.72 \begin {gather*} \frac {-36 x^{2} b^{2}-9 x b a-a^{2}}{252 b^{3} \left (x b+a\right )^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.15, size = 31, normalized size = 0.66 \begin {gather*} -\frac {8\,a^2+72\,a\,b\,x+288\,b^2\,x^2}{2016\,b^3\,{\left (a+b\,x\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x)^10,x)

[Out]

-(8*a^2 + 288*b^2*x^2 + 72*a*b*x)/(2016*b^3*(a + b*x)^9)

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