3.213 \(\int \frac{x^{16}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=198 \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{3/2} b^{17/2}}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}+\frac{715 x}{65536 a b^8 \left (a+b x^2\right )}-\frac{715 x}{32768 b^8 \left (a+b x^2\right )^2}-\frac{x^{15}}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^15/(18*b*(a + b*x^2)^9) - (5*x^13)/(96*b^2*(a + b*x^2)^8) - (65*x^11)/(1344*b^3*(a + b*x^2)^7) - (715*x^9)/
(16128*b^4*(a + b*x^2)^6) - (143*x^7)/(3584*b^5*(a + b*x^2)^5) - (143*x^5)/(4096*b^6*(a + b*x^2)^4) - (715*x^3
)/(24576*b^7*(a + b*x^2)^3) - (715*x)/(32768*b^8*(a + b*x^2)^2) + (715*x)/(65536*a*b^8*(a + b*x^2)) + (715*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(3/2)*b^(17/2))

________________________________________________________________________________________

Rubi [A]  time = 0.122302, antiderivative size = 198, 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} \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{3/2} b^{17/2}}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}+\frac{715 x}{65536 a b^8 \left (a+b x^2\right )}-\frac{715 x}{32768 b^8 \left (a+b x^2\right )^2}-\frac{x^{15}}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^16/(a + b*x^2)^10,x]

[Out]

-x^15/(18*b*(a + b*x^2)^9) - (5*x^13)/(96*b^2*(a + b*x^2)^8) - (65*x^11)/(1344*b^3*(a + b*x^2)^7) - (715*x^9)/
(16128*b^4*(a + b*x^2)^6) - (143*x^7)/(3584*b^5*(a + b*x^2)^5) - (143*x^5)/(4096*b^6*(a + b*x^2)^4) - (715*x^3
)/(24576*b^7*(a + b*x^2)^3) - (715*x)/(32768*b^8*(a + b*x^2)^2) + (715*x)/(65536*a*b^8*(a + b*x^2)) + (715*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(3/2)*b^(17/2))

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^{16}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}+\frac{5 \int \frac{x^{14}}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}+\frac{65 \int \frac{x^{12}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}+\frac{715 \int \frac{x^{10}}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}+\frac{715 \int \frac{x^8}{\left (a+b x^2\right )^6} \, dx}{1792 b^4}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}+\frac{143 \int \frac{x^6}{\left (a+b x^2\right )^5} \, dx}{512 b^5}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}+\frac{715 \int \frac{x^4}{\left (a+b x^2\right )^4} \, dx}{4096 b^6}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}+\frac{715 \int \frac{x^2}{\left (a+b x^2\right )^3} \, dx}{8192 b^7}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac{715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac{715 \int \frac{1}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac{715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac{715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac{715 \int \frac{1}{a+b x^2} \, dx}{65536 a b^8}\\ &=-\frac{x^{15}}{18 b \left (a+b x^2\right )^9}-\frac{5 x^{13}}{96 b^2 \left (a+b x^2\right )^8}-\frac{65 x^{11}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{715 x^9}{16128 b^4 \left (a+b x^2\right )^6}-\frac{143 x^7}{3584 b^5 \left (a+b x^2\right )^5}-\frac{143 x^5}{4096 b^6 \left (a+b x^2\right )^4}-\frac{715 x^3}{24576 b^7 \left (a+b x^2\right )^3}-\frac{715 x}{32768 b^8 \left (a+b x^2\right )^2}+\frac{715 x}{65536 a b^8 \left (a+b x^2\right )}+\frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{3/2} b^{17/2}}\\ \end{align*}

Mathematica [A]  time = 0.0685814, size = 138, normalized size = 0.7 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (-2633274 a^2 b^6 x^{12}-4349826 a^3 b^5 x^{10}-4685824 a^4 b^4 x^8-3317886 a^5 b^3 x^6-1495494 a^6 b^2 x^4-390390 a^7 b x^2-45045 a^8-985866 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{3/2} b^{17/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^16/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 - 1495494*a^6*b^2*x^4 - 3317886*a^5*b^3*x^6 - 4685824*a^4*b
^4*x^8 - 4349826*a^3*b^5*x^10 - 2633274*a^2*b^6*x^12 - 985866*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^(3/2)*b^(17/2))

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 124, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{715\,{a}^{7}x}{65536\,{b}^{8}}}-{\frac{9295\,{a}^{6}{x}^{3}}{98304\,{b}^{7}}}-{\frac{11869\,{a}^{5}{x}^{5}}{32768\,{b}^{6}}}-{\frac{184327\,{a}^{4}{x}^{7}}{229376\,{b}^{5}}}-{\frac{143\,{a}^{3}{x}^{9}}{126\,{b}^{4}}}-{\frac{241657\,{a}^{2}{x}^{11}}{229376\,{b}^{3}}}-{\frac{20899\,a{x}^{13}}{32768\,{b}^{2}}}-{\frac{23473\,{x}^{15}}{98304\,b}}+{\frac{715\,{x}^{17}}{65536\,a}} \right ) }+{\frac{715}{65536\,a{b}^{8}}\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^16/(b*x^2+a)^10,x)

[Out]

(-715/65536*a^7/b^8*x-9295/98304*a^6/b^7*x^3-11869/32768*a^5/b^6*x^5-184327/229376*a^4/b^5*x^7-143/126*a^3/b^4
*x^9-241657/229376*a^2/b^3*x^11-20899/32768/b^2*a*x^13-23473/98304/b*x^15+715/65536/a*x^17)/(b*x^2+a)^9+715/65
536/a/b^8/(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^16/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32484, size = 1580, normalized size = 7.98 \begin{align*} \left [\frac{90090 \, a b^{9} x^{17} - 1971732 \, a^{2} b^{8} x^{15} - 5266548 \, a^{3} b^{7} x^{13} - 8699652 \, a^{4} b^{6} x^{11} - 9371648 \, a^{5} b^{5} x^{9} - 6635772 \, a^{6} b^{4} x^{7} - 2990988 \, a^{7} b^{3} x^{5} - 780780 \, 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^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}, \frac{45045 \, a b^{9} x^{17} - 985866 \, a^{2} b^{8} x^{15} - 2633274 \, a^{3} b^{7} x^{13} - 4349826 \, a^{4} b^{6} x^{11} - 4685824 \, a^{5} b^{5} x^{9} - 3317886 \, a^{6} b^{4} x^{7} - 1495494 \, a^{7} b^{3} x^{5} - 390390 \, 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^{2} b^{18} x^{18} + 9 \, a^{3} b^{17} x^{16} + 36 \, a^{4} b^{16} x^{14} + 84 \, a^{5} b^{15} x^{12} + 126 \, a^{6} b^{14} x^{10} + 126 \, a^{7} b^{13} x^{8} + 84 \, a^{8} b^{12} x^{6} + 36 \, a^{9} b^{11} x^{4} + 9 \, a^{10} b^{10} x^{2} + a^{11} b^{9}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^16/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.65989, size = 289, normalized size = 1.46 \begin{align*} - \frac{715 \sqrt{- \frac{1}{a^{3} b^{17}}} \log{\left (- a^{2} b^{8} \sqrt{- \frac{1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac{715 \sqrt{- \frac{1}{a^{3} b^{17}}} \log{\left (a^{2} b^{8} \sqrt{- \frac{1}{a^{3} b^{17}}} + x \right )}}{131072} + \frac{- 45045 a^{8} x - 390390 a^{7} b x^{3} - 1495494 a^{6} b^{2} x^{5} - 3317886 a^{5} b^{3} x^{7} - 4685824 a^{4} b^{4} x^{9} - 4349826 a^{3} b^{5} x^{11} - 2633274 a^{2} b^{6} x^{13} - 985866 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{10} b^{8} + 37158912 a^{9} b^{9} x^{2} + 148635648 a^{8} b^{10} x^{4} + 346816512 a^{7} b^{11} x^{6} + 520224768 a^{6} b^{12} x^{8} + 520224768 a^{5} b^{13} x^{10} + 346816512 a^{4} b^{14} x^{12} + 148635648 a^{3} b^{15} x^{14} + 37158912 a^{2} b^{16} x^{16} + 4128768 a b^{17} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**16/(b*x**2+a)**10,x)

[Out]

-715*sqrt(-1/(a**3*b**17))*log(-a**2*b**8*sqrt(-1/(a**3*b**17)) + x)/131072 + 715*sqrt(-1/(a**3*b**17))*log(a*
*2*b**8*sqrt(-1/(a**3*b**17)) + x)/131072 + (-45045*a**8*x - 390390*a**7*b*x**3 - 1495494*a**6*b**2*x**5 - 331
7886*a**5*b**3*x**7 - 4685824*a**4*b**4*x**9 - 4349826*a**3*b**5*x**11 - 2633274*a**2*b**6*x**13 - 985866*a*b*
*7*x**15 + 45045*b**8*x**17)/(4128768*a**10*b**8 + 37158912*a**9*b**9*x**2 + 148635648*a**8*b**10*x**4 + 34681
6512*a**7*b**11*x**6 + 520224768*a**6*b**12*x**8 + 520224768*a**5*b**13*x**10 + 346816512*a**4*b**14*x**12 + 1
48635648*a**3*b**15*x**14 + 37158912*a**2*b**16*x**16 + 4128768*a*b**17*x**18)

________________________________________________________________________________________

Giac [A]  time = 2.9788, size = 173, normalized size = 0.87 \begin{align*} \frac{715 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a b^{8}} + \frac{45045 \, b^{8} x^{17} - 985866 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^16/(b*x^2+a)^10,x, algorithm="giac")

[Out]

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