3.212 \(\int \frac{x^{18}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=197 \[ -\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}+\frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 \sqrt{a} b^{19/2}}-\frac{x^{17}}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^17/(18*b*(a + b*x^2)^9) - (17*x^15)/(288*b^2*(a + b*x^2)^8) - (85*x^13)/(1344*b^3*(a + b*x^2)^7) - (1105*x^
11)/(16128*b^4*(a + b*x^2)^6) - (2431*x^9)/(32256*b^5*(a + b*x^2)^5) - (2431*x^7)/(28672*b^6*(a + b*x^2)^4) -
(2431*x^5)/(24576*b^7*(a + b*x^2)^3) - (12155*x^3)/(98304*b^8*(a + b*x^2)^2) - (12155*x)/(65536*b^9*(a + b*x^2
)) + (12155*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*Sqrt[a]*b^(19/2))

________________________________________________________________________________________

Rubi [A]  time = 0.113243, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {288, 205} \[ -\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}+\frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 \sqrt{a} b^{19/2}}-\frac{x^{17}}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^18/(a + b*x^2)^10,x]

[Out]

-x^17/(18*b*(a + b*x^2)^9) - (17*x^15)/(288*b^2*(a + b*x^2)^8) - (85*x^13)/(1344*b^3*(a + b*x^2)^7) - (1105*x^
11)/(16128*b^4*(a + b*x^2)^6) - (2431*x^9)/(32256*b^5*(a + b*x^2)^5) - (2431*x^7)/(28672*b^6*(a + b*x^2)^4) -
(2431*x^5)/(24576*b^7*(a + b*x^2)^3) - (12155*x^3)/(98304*b^8*(a + b*x^2)^2) - (12155*x)/(65536*b^9*(a + b*x^2
)) + (12155*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*Sqrt[a]*b^(19/2))

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 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^{18}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}+\frac{17 \int \frac{x^{16}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}+\frac{85 \int \frac{x^{14}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}+\frac{1105 \int \frac{x^{12}}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}+\frac{12155 \int \frac{x^{10}}{\left (a+b x^2\right )^6} \, dx}{16128 b^4}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}+\frac{2431 \int \frac{x^8}{\left (a+b x^2\right )^5} \, dx}{3584 b^5}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}+\frac{2431 \int \frac{x^6}{\left (a+b x^2\right )^4} \, dx}{4096 b^6}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}+\frac{12155 \int \frac{x^4}{\left (a+b x^2\right )^3} \, dx}{24576 b^7}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}+\frac{12155 \int \frac{x^2}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}+\frac{12155 \int \frac{1}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac{x^{17}}{18 b \left (a+b x^2\right )^9}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}+\frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 \sqrt{a} b^{19/2}}\\ \end{align*}

Mathematica [A]  time = 0.073698, size = 134, normalized size = 0.68 \[ \frac{\frac{765765 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b} x \left (44765658 a^2 b^6 x^{12}[phone] a^3 b^5 x^{10}[phone] a^4 b^4 x^8+56404062 a^5 b^3 x^6+25423398 a^6 b^2 x^4+6636630 a^7 b x^2+765765 a^8+16759722 a b^7 x^{14}+3363003 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}}{4128768 b^{19/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^18/(a + b*x^2)^10,x]

[Out]

(-((Sqrt[b]*x*(765765*a^8 + 6636630*a^7*b*x^2 + 25423398*a^6*b^2*x^4 + 56404062*a^5*b^3*x^6 + 79659008*a^4*b^4
*x^8 + 73947042*a^3*b^5*x^10 + 44765658*a^2*b^6*x^12 + 16759722*a*b^7*x^14 + 3363003*b^8*x^16))/(a + b*x^2)^9)
+ (765765*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a])/(4128768*b^(19/2))

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Maple [A]  time = 0.014, size = 124, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{12155\,{a}^{8}x}{65536\,{b}^{9}}}-{\frac{158015\,{a}^{7}{x}^{3}}{98304\,{b}^{8}}}-{\frac{201773\,{a}^{6}{x}^{5}}{32768\,{b}^{7}}}-{\frac{3133559\,{a}^{5}{x}^{7}}{229376\,{b}^{6}}}-{\frac{2431\,{a}^{4}{x}^{9}}{126\,{b}^{5}}}-{\frac{4108169\,{a}^{3}{x}^{11}}{229376\,{b}^{4}}}-{\frac{355283\,{a}^{2}{x}^{13}}{32768\,{b}^{3}}}-{\frac{399041\,a{x}^{15}}{98304\,{b}^{2}}}-{\frac{53381\,{x}^{17}}{65536\,b}} \right ) }+{\frac{12155}{65536\,{b}^{9}}\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^18/(b*x^2+a)^10,x)

[Out]

(-12155/65536*a^8/b^9*x-158015/98304*a^7/b^8*x^3-201773/32768*a^6/b^7*x^5-3133559/229376*a^5/b^6*x^7-2431/126*
a^4/b^5*x^9-4108169/229376*a^3/b^4*x^11-355283/32768*a^2/b^3*x^13-399041/98304/b^2*a*x^15-53381/65536/b*x^17)/
(b*x^2+a)^9+12155/65536/b^9/(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^18/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53849, size = 1615, normalized size = 8.2 \begin{align*} \left [-\frac{6726006 \, a b^{9} x^{17} + 33519444 \, a^{2} b^{8} x^{15} + 89531316 \, a^{3} b^{7} x^{13} + 147894084 \, a^{4} b^{6} x^{11} + 159318016 \, a^{5} b^{5} x^{9} + 112808124 \, a^{6} b^{4} x^{7} + 50846796 \, a^{7} b^{3} x^{5} + 13273260 \, a^{8} b^{2} x^{3} + 1531530 \, a^{9} b x + 765765 \,{\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 b^{19} x^{18} + 9 \, a^{2} b^{18} x^{16} + 36 \, a^{3} b^{17} x^{14} + 84 \, a^{4} b^{16} x^{12} + 126 \, a^{5} b^{15} x^{10} + 126 \, a^{6} b^{14} x^{8} + 84 \, a^{7} b^{13} x^{6} + 36 \, a^{8} b^{12} x^{4} + 9 \, a^{9} b^{11} x^{2} + a^{10} b^{10}\right )}}, -\frac{3363003 \, a b^{9} x^{17} + 16759722 \, a^{2} b^{8} x^{15} + 44765658 \, a^{3} b^{7} x^{13} + 73947042 \, a^{4} b^{6} x^{11} + 79659008 \, a^{5} b^{5} x^{9} + 56404062 \, a^{6} b^{4} x^{7} + 25423398 \, a^{7} b^{3} x^{5} + 6636630 \, a^{8} b^{2} x^{3} + 765765 \, a^{9} b x - 765765 \,{\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 b^{19} x^{18} + 9 \, a^{2} b^{18} x^{16} + 36 \, a^{3} b^{17} x^{14} + 84 \, a^{4} b^{16} x^{12} + 126 \, a^{5} b^{15} x^{10} + 126 \, a^{6} b^{14} x^{8} + 84 \, a^{7} b^{13} x^{6} + 36 \, a^{8} b^{12} x^{4} + 9 \, a^{9} b^{11} x^{2} + a^{10} b^{10}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^18/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[-1/8257536*(6726006*a*b^9*x^17 + 33519444*a^2*b^8*x^15 + 89531316*a^3*b^7*x^13 + 147894084*a^4*b^6*x^11 + 159
318016*a^5*b^5*x^9 + 112808124*a^6*b^4*x^7 + 50846796*a^7*b^3*x^5 + 13273260*a^8*b^2*x^3 + 1531530*a^9*b*x + 7
65765*(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*
b^19*x^18 + 9*a^2*b^18*x^16 + 36*a^3*b^17*x^14 + 84*a^4*b^16*x^12 + 126*a^5*b^15*x^10 + 126*a^6*b^14*x^8 + 84*
a^7*b^13*x^6 + 36*a^8*b^12*x^4 + 9*a^9*b^11*x^2 + a^10*b^10), -1/4128768*(3363003*a*b^9*x^17 + 16759722*a^2*b^
8*x^15 + 44765658*a^3*b^7*x^13 + 73947042*a^4*b^6*x^11 + 79659008*a^5*b^5*x^9 + 56404062*a^6*b^4*x^7 + 2542339
8*a^7*b^3*x^5 + 6636630*a^8*b^2*x^3 + 765765*a^9*b*x - 765765*(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*b^19*x^18 + 9*a^2*b^18*x^16 + 36*a^3*b^17*x^14 + 84*a^4*b^16*x^12 + 126*a^5*b^1
5*x^10 + 126*a^6*b^14*x^8 + 84*a^7*b^13*x^6 + 36*a^8*b^12*x^4 + 9*a^9*b^11*x^2 + a^10*b^10)]

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Sympy [A]  time = 7.77787, size = 275, normalized size = 1.4 \begin{align*} - \frac{12155 \sqrt{- \frac{1}{a b^{19}}} \log{\left (- a b^{9} \sqrt{- \frac{1}{a b^{19}}} + x \right )}}{131072} + \frac{12155 \sqrt{- \frac{1}{a b^{19}}} \log{\left (a b^{9} \sqrt{- \frac{1}{a b^{19}}} + x \right )}}{131072} - \frac{765765 a^{8} x + 6636630 a^{7} b x^{3} + 25423398 a^{6} b^{2} x^{5} + 56404062 a^{5} b^{3} x^{7} + 79659008 a^{4} b^{4} x^{9} + 73947042 a^{3} b^{5} x^{11} + 44765658 a^{2} b^{6} x^{13} + 16759722 a b^{7} x^{15} + 3363003 b^{8} x^{17}}{4128768 a^{9} b^{9} + 37158912 a^{8} b^{10} x^{2} + 148635648 a^{7} b^{11} x^{4} + 346816512 a^{6} b^{12} x^{6} + 520224768 a^{5} b^{13} x^{8} + 520224768 a^{4} b^{14} x^{10} + 346816512 a^{3} b^{15} x^{12} + 148635648 a^{2} b^{16} x^{14} + 37158912 a b^{17} x^{16} + 4128768 b^{18} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**18/(b*x**2+a)**10,x)

[Out]

-12155*sqrt(-1/(a*b**19))*log(-a*b**9*sqrt(-1/(a*b**19)) + x)/131072 + 12155*sqrt(-1/(a*b**19))*log(a*b**9*sqr
t(-1/(a*b**19)) + x)/131072 - (765765*a**8*x + 6636630*a**7*b*x**3 + 25423398*a**6*b**2*x**5 + 56404062*a**5*b
**3*x**7 + 79659008*a**4*b**4*x**9 + 73947042*a**3*b**5*x**11 + 44765658*a**2*b**6*x**13 + 16759722*a*b**7*x**
15 + 3363003*b**8*x**17)/(4128768*a**9*b**9 + 37158912*a**8*b**10*x**2 + 148635648*a**7*b**11*x**4 + 346816512
*a**6*b**12*x**6 + 520224768*a**5*b**13*x**8 + 520224768*a**4*b**14*x**10 + 346816512*a**3*b**15*x**12 + 14863
5648*a**2*b**16*x**14 + 37158912*a*b**17*x**16 + 4128768*b**18*x**18)

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Giac [A]  time = 2.5509, size = 165, normalized size = 0.84 \begin{align*} \frac{12155 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{9}} - \frac{3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^18/(b*x^2+a)^10,x, algorithm="giac")

[Out]

12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^9) - 1/4128768*(3363003*b^8*x^17 + 16759722*a*b^7*x^15 + 447656
58*a^2*b^6*x^13 + 73947042*a^3*b^5*x^11 + 79659008*a^4*b^4*x^9 + 56404062*a^5*b^3*x^7 + 25423398*a^6*b^2*x^5 +
6636630*a^7*b*x^3 + 765765*a^8*x)/((b*x^2 + a)^9*b^9)