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

Optimal. Leaf size=205 \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{x}{288 a b \left (a+b x^2\right )^8}-\frac{x}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x/(18*b*(a + b*x^2)^9) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x^2)^7) + (65*x)/(16128*a^3*b*(
a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6
*b*(a + b*x^2)^3) + (715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*ArcTan[(Sqr
t[b]*x)/Sqrt[a]])/(65536*a^(17/2)*b^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.105971, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 199, 205} \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{x}{288 a b \left (a+b x^2\right )^8}-\frac{x}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2)^10,x]

[Out]

-x/(18*b*(a + b*x^2)^9) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x^2)^7) + (65*x)/(16128*a^3*b*(
a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6
*b*(a + b*x^2)^3) + (715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*ArcTan[(Sqr
t[b]*x)/Sqrt[a]])/(65536*a^(17/2)*b^(3/2))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{\int \frac{1}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 \int \frac{1}{\left (a+b x^2\right )^8} \, dx}{96 a b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 \int \frac{1}{\left (a+b x^2\right )^7} \, dx}{1344 a^2 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{715 \int \frac{1}{\left (a+b x^2\right )^6} \, dx}{16128 a^3 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 \int \frac{1}{\left (a+b x^2\right )^5} \, dx}{3584 a^4 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 \int \frac{1}{\left (a+b x^2\right )^4} \, dx}{4096 a^5 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{715 \int \frac{1}{\left (a+b x^2\right )^3} \, dx}{24576 a^6 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{715 \int \frac{1}{\left (a+b x^2\right )^2} \, dx}{32768 a^7 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 \int \frac{1}{a+b x^2} \, dx}{65536 a^8 b}\\ &=-\frac{x}{18 b \left (a+b x^2\right )^9}+\frac{x}{288 a b \left (a+b x^2\right )^8}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{17/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0547511, size = 138, normalized size = 0.67 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (1495494 a^2 b^6 x^{12}+3317886 a^3 b^5 x^{10}+4685824 a^4 b^4 x^8+4349826 a^5 b^3 x^6+2633274 a^6 b^2 x^4+985866 a^7 b x^2-45045 a^8+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{17/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 + 985866*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 4349826*a^5*b^3*x^6 + 4685824*a^4*b
^4*x^8 + 3317886*a^3*b^5*x^10 + 1495494*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45
045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(17/2)*b^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 124, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{715\,x}{65536\,b}}+{\frac{23473\,{x}^{3}}{98304\,a}}+{\frac{20899\,b{x}^{5}}{32768\,{a}^{2}}}+{\frac{241657\,{b}^{2}{x}^{7}}{229376\,{a}^{3}}}+{\frac{143\,{b}^{3}{x}^{9}}{126\,{a}^{4}}}+{\frac{184327\,{b}^{4}{x}^{11}}{229376\,{a}^{5}}}+{\frac{11869\,{b}^{5}{x}^{13}}{32768\,{a}^{6}}}+{\frac{9295\,{b}^{6}{x}^{15}}{98304\,{a}^{7}}}+{\frac{715\,{b}^{7}{x}^{17}}{65536\,{a}^{8}}} \right ) }+{\frac{715}{65536\,{a}^{8}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-715/65536*x/b+23473/98304/a*x^3+20899/32768*b/a^2*x^5+241657/229376*b^2/a^3*x^7+143/126*b^3/a^4*x^9+184327/2
29376*b^4/a^5*x^11+11869/32768/a^6*b^5*x^13+9295/98304/a^7*b^6*x^15+715/65536/a^8*b^7*x^17)/(b*x^2+a)^9+715/65
536/a^8/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29457, size = 1580, normalized size = 7.71 \begin{align*} \left [\frac{90090 \, a b^{9} x^{17} + 780780 \, a^{2} b^{8} x^{15} + 2990988 \, a^{3} b^{7} x^{13} + 6635772 \, a^{4} b^{6} x^{11} + 9371648 \, a^{5} b^{5} x^{9} + 8699652 \, a^{6} b^{4} x^{7} + 5266548 \, a^{7} b^{3} x^{5} + 1971732 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{8257536 \,{\left (a^{9} b^{11} x^{18} + 9 \, a^{10} b^{10} x^{16} + 36 \, a^{11} b^{9} x^{14} + 84 \, a^{12} b^{8} x^{12} + 126 \, a^{13} b^{7} x^{10} + 126 \, a^{14} b^{6} x^{8} + 84 \, a^{15} b^{5} x^{6} + 36 \, a^{16} b^{4} x^{4} + 9 \, a^{17} b^{3} x^{2} + a^{18} b^{2}\right )}}, \frac{45045 \, a b^{9} x^{17} + 390390 \, a^{2} b^{8} x^{15} + 1495494 \, a^{3} b^{7} x^{13} + 3317886 \, a^{4} b^{6} x^{11} + 4685824 \, a^{5} b^{5} x^{9} + 4349826 \, a^{6} b^{4} x^{7} + 2633274 \, a^{7} b^{3} x^{5} + 985866 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{4128768 \,{\left (a^{9} b^{11} x^{18} + 9 \, a^{10} b^{10} x^{16} + 36 \, a^{11} b^{9} x^{14} + 84 \, a^{12} b^{8} x^{12} + 126 \, a^{13} b^{7} x^{10} + 126 \, a^{14} b^{6} x^{8} + 84 \, a^{15} b^{5} x^{6} + 36 \, a^{16} b^{4} x^{4} + 9 \, a^{17} b^{3} x^{2} + a^{18} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8257536*(90090*a*b^9*x^17 + 780780*a^2*b^8*x^15 + 2990988*a^3*b^7*x^13 + 6635772*a^4*b^6*x^11 + 9371648*a^5
*b^5*x^9 + 8699652*a^6*b^4*x^7 + 5266548*a^7*b^3*x^5 + 1971732*a^8*b^2*x^3 - 90090*a^9*b*x - 45045*(b^9*x^18 +
9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a
^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^9*b^11*x^18 + 9*a
^10*b^10*x^16 + 36*a^11*b^9*x^14 + 84*a^12*b^8*x^12 + 126*a^13*b^7*x^10 + 126*a^14*b^6*x^8 + 84*a^15*b^5*x^6 +
36*a^16*b^4*x^4 + 9*a^17*b^3*x^2 + a^18*b^2), 1/4128768*(45045*a*b^9*x^17 + 390390*a^2*b^8*x^15 + 1495494*a^3
*b^7*x^13 + 3317886*a^4*b^6*x^11 + 4685824*a^5*b^5*x^9 + 4349826*a^6*b^4*x^7 + 2633274*a^7*b^3*x^5 + 985866*a^
8*b^2*x^3 - 45045*a^9*b*x + 45045*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x
^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/
(a^9*b^11*x^18 + 9*a^10*b^10*x^16 + 36*a^11*b^9*x^14 + 84*a^12*b^8*x^12 + 126*a^13*b^7*x^10 + 126*a^14*b^6*x^8
+ 84*a^15*b^5*x^6 + 36*a^16*b^4*x^4 + 9*a^17*b^3*x^2 + a^18*b^2)]

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Sympy [A]  time = 7.63239, size = 286, normalized size = 1.4 \begin{align*} - \frac{715 \sqrt{- \frac{1}{a^{17} b^{3}}} \log{\left (- a^{9} b \sqrt{- \frac{1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac{715 \sqrt{- \frac{1}{a^{17} b^{3}}} \log{\left (a^{9} b \sqrt{- \frac{1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac{- 45045 a^{8} x + 985866 a^{7} b x^{3} + 2633274 a^{6} b^{2} x^{5} + 4349826 a^{5} b^{3} x^{7} + 4685824 a^{4} b^{4} x^{9} + 3317886 a^{3} b^{5} x^{11} + 1495494 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{17} b + 37158912 a^{16} b^{2} x^{2} + 148635648 a^{15} b^{3} x^{4} + 346816512 a^{14} b^{4} x^{6} + 520224768 a^{13} b^{5} x^{8} + 520224768 a^{12} b^{6} x^{10} + 346816512 a^{11} b^{7} x^{12} + 148635648 a^{10} b^{8} x^{14} + 37158912 a^{9} b^{9} x^{16} + 4128768 a^{8} b^{10} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-715*sqrt(-1/(a**17*b**3))*log(-a**9*b*sqrt(-1/(a**17*b**3)) + x)/131072 + 715*sqrt(-1/(a**17*b**3))*log(a**9*
b*sqrt(-1/(a**17*b**3)) + x)/131072 + (-45045*a**8*x + 985866*a**7*b*x**3 + 2633274*a**6*b**2*x**5 + 4349826*a
**5*b**3*x**7 + 4685824*a**4*b**4*x**9 + 3317886*a**3*b**5*x**11 + 1495494*a**2*b**6*x**13 + 390390*a*b**7*x**
15 + 45045*b**8*x**17)/(4128768*a**17*b + 37158912*a**16*b**2*x**2 + 148635648*a**15*b**3*x**4 + 346816512*a**
14*b**4*x**6 + 520224768*a**13*b**5*x**8 + 520224768*a**12*b**6*x**10 + 346816512*a**11*b**7*x**12 + 148635648
*a**10*b**8*x**14 + 37158912*a**9*b**9*x**16 + 4128768*a**8*b**10*x**18)

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Giac [A]  time = 2.21387, size = 173, normalized size = 0.84 \begin{align*} \frac{715 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{8} b} + \frac{45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{8} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

715/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8*b) + 1/4128768*(45045*b^8*x^17 + 390390*a*b^7*x^15 + 1495494*a^
2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 4685824*a^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a
^7*b*x^3 - 45045*a^8*x)/((b*x^2 + a)^9*a^8*b)