3.221 \(\int \frac{1}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=181 \[ \frac{12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac{12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac{2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac{2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac{2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac{1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac{85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac{17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{19/2} \sqrt{b}}+\frac{x}{18 a \left (a+b x^2\right )^9} \]

[Out]

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

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Rubi [A]  time = 0.0996333, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {199, 205} \[ \frac{12155 x}{65536 a^9 \left (a+b x^2\right )}+\frac{12155 x}{98304 a^8 \left (a+b x^2\right )^2}+\frac{2431 x}{24576 a^7 \left (a+b x^2\right )^3}+\frac{2431 x}{28672 a^6 \left (a+b x^2\right )^4}+\frac{2431 x}{32256 a^5 \left (a+b x^2\right )^5}+\frac{1105 x}{16128 a^4 \left (a+b x^2\right )^6}+\frac{85 x}{1344 a^3 \left (a+b x^2\right )^7}+\frac{17 x}{288 a^2 \left (a+b x^2\right )^8}+\frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{19/2} \sqrt{b}}+\frac{x}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-10),x]

[Out]

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

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

Mathematica [A]  time = 0.0974405, size = 131, normalized size = 0.72 \[ \frac{\frac{25423398 a^2 b^6 x^{13}[phone] a^3 b^5 x^{11}[phone] a^4 b^4 x^9+73947042 a^5 b^3 x^7+44765658 a^6 b^2 x^5+16759722 a^7 b x^3+3363003 a^8 x+6636630 a b^7 x^{15}+765765 b^8 x^{17}}{a^9 \left (a+b x^2\right )^9}+\frac{765765 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{19/2} \sqrt{b}}}{4128768} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-10),x]

[Out]

((3363003*a^8*x + 16759722*a^7*b*x^3 + 44765658*a^6*b^2*x^5 + 73947042*a^5*b^3*x^7 + 79659008*a^4*b^4*x^9 + 56
404062*a^3*b^5*x^11 + 25423398*a^2*b^6*x^13 + 6636630*a*b^7*x^15 + 765765*b^8*x^17)/(a^9*(a + b*x^2)^9) + (765
765*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(19/2)*Sqrt[b]))/4128768

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Maple [A]  time = 0.005, size = 156, normalized size = 0.9 \begin{align*}{\frac{x}{18\,a \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{17\,x}{288\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{85\,x}{1344\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{1105\,x}{16128\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{2431\,x}{32256\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{2431\,x}{28672\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{2431\,x}{24576\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{12155\,x}{98304\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{12155\,x}{65536\,{a}^{9} \left ( b{x}^{2}+a \right ) }}+{\frac{12155}{65536\,{a}^{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(1/(b*x^2+a)^10,x)

[Out]

1/18*x/a/(b*x^2+a)^9+17/288*x/a^2/(b*x^2+a)^8+85/1344*x/a^3/(b*x^2+a)^7+1105/16128*x/a^4/(b*x^2+a)^6+2431/3225
6*x/a^5/(b*x^2+a)^5+2431/28672*x/a^6/(b*x^2+a)^4+2431/24576*x/a^7/(b*x^2+a)^3+12155/98304*x/a^8/(b*x^2+a)^2+12
155/65536*x/a^9/(b*x^2+a)+12155/65536/a^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(1/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.33232, size = 1612, normalized size = 8.91 \begin{align*} \left [\frac{1531530 \, a b^{9} x^{17} + 13273260 \, a^{2} b^{8} x^{15} + 50846796 \, a^{3} b^{7} x^{13} + 112808124 \, a^{4} b^{6} x^{11} + 159318016 \, a^{5} b^{5} x^{9} + 147894084 \, a^{6} b^{4} x^{7} + 89531316 \, a^{7} b^{3} x^{5} + 33519444 \, a^{8} b^{2} x^{3} + 6726006 \, 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^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}, \frac{765765 \, a b^{9} x^{17} + 6636630 \, a^{2} b^{8} x^{15} + 25423398 \, a^{3} b^{7} x^{13} + 56404062 \, a^{4} b^{6} x^{11} + 79659008 \, a^{5} b^{5} x^{9} + 73947042 \, a^{6} b^{4} x^{7} + 44765658 \, a^{7} b^{3} x^{5} + 16759722 \, a^{8} b^{2} x^{3} + 3363003 \, 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^{10} b^{10} x^{18} + 9 \, a^{11} b^{9} x^{16} + 36 \, a^{12} b^{8} x^{14} + 84 \, a^{13} b^{7} x^{12} + 126 \, a^{14} b^{6} x^{10} + 126 \, a^{15} b^{5} x^{8} + 84 \, a^{16} b^{4} x^{6} + 36 \, a^{17} b^{3} x^{4} + 9 \, a^{18} b^{2} x^{2} + a^{19} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/8257536*(1531530*a*b^9*x^17 + 13273260*a^2*b^8*x^15 + 50846796*a^3*b^7*x^13 + 112808124*a^4*b^6*x^11 + 1593
18016*a^5*b^5*x^9 + 147894084*a^6*b^4*x^7 + 89531316*a^7*b^3*x^5 + 33519444*a^8*b^2*x^3 + 6726006*a^9*b*x - 76
5765*(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^1
0*b^10*x^18 + 9*a^11*b^9*x^16 + 36*a^12*b^8*x^14 + 84*a^13*b^7*x^12 + 126*a^14*b^6*x^10 + 126*a^15*b^5*x^8 + 8
4*a^16*b^4*x^6 + 36*a^17*b^3*x^4 + 9*a^18*b^2*x^2 + a^19*b), 1/4128768*(765765*a*b^9*x^17 + 6636630*a^2*b^8*x^
15 + 25423398*a^3*b^7*x^13 + 56404062*a^4*b^6*x^11 + 79659008*a^5*b^5*x^9 + 73947042*a^6*b^4*x^7 + 44765658*a^
7*b^3*x^5 + 16759722*a^8*b^2*x^3 + 3363003*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^10*b^10*x^18 + 9*a^11*b^9*x^16 + 36*a^12*b^8*x^14 + 84*a^13*b^7*x^12 + 126*a^14*b
^6*x^10 + 126*a^15*b^5*x^8 + 84*a^16*b^4*x^6 + 36*a^17*b^3*x^4 + 9*a^18*b^2*x^2 + a^19*b)]

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Sympy [A]  time = 7.302, size = 272, normalized size = 1.5 \begin{align*} - \frac{12155 \sqrt{- \frac{1}{a^{19} b}} \log{\left (- a^{10} \sqrt{- \frac{1}{a^{19} b}} + x \right )}}{131072} + \frac{12155 \sqrt{- \frac{1}{a^{19} b}} \log{\left (a^{10} \sqrt{- \frac{1}{a^{19} b}} + x \right )}}{131072} + \frac{3363003 a^{8} x + 16759722 a^{7} b x^{3} + 44765658 a^{6} b^{2} x^{5} + 73947042 a^{5} b^{3} x^{7} + 79659008 a^{4} b^{4} x^{9} + 56404062 a^{3} b^{5} x^{11} + 25423398 a^{2} b^{6} x^{13} + 6636630 a b^{7} x^{15} + 765765 b^{8} x^{17}}{4128768 a^{18} + 37158912 a^{17} b x^{2} + 148635648 a^{16} b^{2} x^{4} + 346816512 a^{15} b^{3} x^{6} + 520224768 a^{14} b^{4} x^{8} + 520224768 a^{13} b^{5} x^{10} + 346816512 a^{12} b^{6} x^{12} + 148635648 a^{11} b^{7} x^{14} + 37158912 a^{10} b^{8} x^{16} + 4128768 a^{9} b^{9} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**10,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.79747, size = 165, normalized size = 0.91 \begin{align*} \frac{12155 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{9}} + \frac{765765 \, b^{8} x^{17} + 6636630 \, a b^{7} x^{15} + 25423398 \, a^{2} b^{6} x^{13} + 56404062 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 73947042 \, a^{5} b^{3} x^{7} + 44765658 \, a^{6} b^{2} x^{5} + 16759722 \, a^{7} b x^{3} + 3363003 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^10,x, algorithm="giac")

[Out]

12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^9) + 1/4128768*(765765*b^8*x^17 + 6636630*a*b^7*x^15 + 25423398
*a^2*b^6*x^13 + 56404062*a^3*b^5*x^11 + 79659008*a^4*b^4*x^9 + 73947042*a^5*b^3*x^7 + 44765658*a^6*b^2*x^5 + 1
6759722*a^7*b*x^3 + 3363003*a^8*x)/((b*x^2 + a)^9*a^9)