3.2.100 \(\int \frac {x^9}{(a+b x^2)^{10}} \, dx\) [200]

Optimal. Leaf size=91 \[ -\frac {a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac {a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac {3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac {a}{3 b^5 \left (a+b x^2\right )^6}-\frac {1}{10 b^5 \left (a+b x^2\right )^5} \]

[Out]

-1/18*a^4/b^5/(b*x^2+a)^9+1/4*a^3/b^5/(b*x^2+a)^8-3/7*a^2/b^5/(b*x^2+a)^7+1/3*a/b^5/(b*x^2+a)^6-1/10/b^5/(b*x^
2+a)^5

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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac {a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac {3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac {a}{3 b^5 \left (a+b x^2\right )^6}-\frac {1}{10 b^5 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/(a + b*x^2)^10,x]

[Out]

-1/18*a^4/(b^5*(a + b*x^2)^9) + a^3/(4*b^5*(a + b*x^2)^8) - (3*a^2)/(7*b^5*(a + b*x^2)^7) + a/(3*b^5*(a + b*x^
2)^6) - 1/(10*b^5*(a + b*x^2)^5)

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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^9}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^4}{b^4 (a+b x)^{10}}-\frac {4 a^3}{b^4 (a+b x)^9}+\frac {6 a^2}{b^4 (a+b x)^8}-\frac {4 a}{b^4 (a+b x)^7}+\frac {1}{b^4 (a+b x)^6}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac {a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac {3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac {a}{3 b^5 \left (a+b x^2\right )^6}-\frac {1}{10 b^5 \left (a+b x^2\right )^5}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 57, normalized size = 0.63 \begin {gather*} -\frac {a^4+9 a^3 b x^2+36 a^2 b^2 x^4+84 a b^3 x^6+126 b^4 x^8}{1260 b^5 \left (a+b x^2\right )^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a + b*x^2)^10,x]

[Out]

-1/1260*(a^4 + 9*a^3*b*x^2 + 36*a^2*b^2*x^4 + 84*a*b^3*x^6 + 126*b^4*x^8)/(b^5*(a + b*x^2)^9)

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Maple [A]
time = 0.09, size = 82, normalized size = 0.90

method	result	size
gosper	\(-\frac {126 x^{8} b^{4}+84 a \,x^{6} b^{3}+36 a^{2} x^{4} b^{2}+9 a^{3} x^{2} b +a^{4}}{1260 \left (b \,x^{2}+a \right )^{9} b^{5}}\)	\(56\)
norman	\(\frac {-\frac {a^{4}}{1260 b^{5}}-\frac {a^{3} x^{2}}{140 b^{4}}-\frac {a^{2} x^{4}}{35 b^{3}}-\frac {a \,x^{6}}{15 b^{2}}-\frac {x^{8}}{10 b}}{\left (b \,x^{2}+a \right )^{9}}\)	\(59\)
risch	\(\frac {-\frac {a^{4}}{1260 b^{5}}-\frac {a^{3} x^{2}}{140 b^{4}}-\frac {a^{2} x^{4}}{35 b^{3}}-\frac {a \,x^{6}}{15 b^{2}}-\frac {x^{8}}{10 b}}{\left (b \,x^{2}+a \right )^{9}}\)	\(59\)
default	\(-\frac {a^{4}}{18 b^{5} \left (b \,x^{2}+a \right )^{9}}+\frac {a^{3}}{4 b^{5} \left (b \,x^{2}+a \right )^{8}}-\frac {3 a^{2}}{7 b^{5} \left (b \,x^{2}+a \right )^{7}}+\frac {a}{3 b^{5} \left (b \,x^{2}+a \right )^{6}}-\frac {1}{10 b^{5} \left (b \,x^{2}+a \right )^{5}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b*x^2+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/18*a^4/b^5/(b*x^2+a)^9+1/4*a^3/b^5/(b*x^2+a)^8-3/7*a^2/b^5/(b*x^2+a)^7+1/3*a/b^5/(b*x^2+a)^6-1/10/b^5/(b*x^
2+a)^5

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Maxima [A]
time = 0.30, size = 146, normalized size = 1.60 \begin {gather*} -\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/(b^14*x^18 + 9*a*b^13*x^16 + 36*a^2*
b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4*b^10*x^10 + 126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b
^6*x^2 + a^9*b^5)

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Fricas [A]
time = 1.23, size = 146, normalized size = 1.60 \begin {gather*} -\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/(b^14*x^18 + 9*a*b^13*x^16 + 36*a^2*
b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4*b^10*x^10 + 126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b
^6*x^2 + a^9*b^5)

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Sympy [A]
time = 0.49, size = 155, normalized size = 1.70 \begin {gather*} \frac {- a^{4} - 9 a^{3} b x^{2} - 36 a^{2} b^{2} x^{4} - 84 a b^{3} x^{6} - 126 b^{4} x^{8}}{1260 a^{9} b^{5} + 11340 a^{8} b^{6} x^{2} + 45360 a^{7} b^{7} x^{4} + 105840 a^{6} b^{8} x^{6} + 158760 a^{5} b^{9} x^{8} + 158760 a^{4} b^{10} x^{10} + 105840 a^{3} b^{11} x^{12} + 45360 a^{2} b^{12} x^{14} + 11340 a b^{13} x^{16} + 1260 b^{14} x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b*x**2+a)**10,x)

[Out]

(-a**4 - 9*a**3*b*x**2 - 36*a**2*b**2*x**4 - 84*a*b**3*x**6 - 126*b**4*x**8)/(1260*a**9*b**5 + 11340*a**8*b**6
*x**2 + 45360*a**7*b**7*x**4 + 105840*a**6*b**8*x**6 + 158760*a**5*b**9*x**8 + 158760*a**4*b**10*x**10 + 10584
0*a**3*b**11*x**12 + 45360*a**2*b**12*x**14 + 11340*a*b**13*x**16 + 1260*b**14*x**18)

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Giac [A]
time = 1.33, size = 55, normalized size = 0.60 \begin {gather*} -\frac {126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \, {\left (b x^{2} + a\right )}^{9} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/((b*x^2 + a)^9*b^5)

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Mupad [B]
time = 4.82, size = 148, normalized size = 1.63 \begin {gather*} -\frac {a^4+9\,a^3\,b\,x^2+36\,a^2\,b^2\,x^4+84\,a\,b^3\,x^6+126\,b^4\,x^8}{1260\,a^9\,b^5+11340\,a^8\,b^6\,x^2+45360\,a^7\,b^7\,x^4+105840\,a^6\,b^8\,x^6+158760\,a^5\,b^9\,x^8+158760\,a^4\,b^{10}\,x^{10}+105840\,a^3\,b^{11}\,x^{12}+45360\,a^2\,b^{12}\,x^{14}+11340\,a\,b^{13}\,x^{16}+1260\,b^{14}\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(a + b*x^2)^10,x)

[Out]

-(a^4 + 126*b^4*x^8 + 9*a^3*b*x^2 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4)/(1260*a^9*b^5 + 1260*b^14*x^18 + 11340*a*b^
13*x^16 + 11340*a^8*b^6*x^2 + 45360*a^7*b^7*x^4 + 105840*a^6*b^8*x^6 + 158760*a^5*b^9*x^8 + 158760*a^4*b^10*x^
10 + 105840*a^3*b^11*x^12 + 45360*a^2*b^12*x^14)

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