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

Optimal. Leaf size=209 \[ -\frac {230945 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{21/2}}-\frac {230945}{65536 a^{10} x}+\frac {230945}{196608 a^9 x \left (a+b x^2\right )}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {1}{18 a x \left (a+b x^2\right )^9} \]

[Out]

-230945/65536/a^10/x+1/18/a/x/(b*x^2+a)^9+19/288/a^2/x/(b*x^2+a)^8+323/4032/a^3/x/(b*x^2+a)^7+1615/16128/a^4/x
/(b*x^2+a)^6+4199/32256/a^5/x/(b*x^2+a)^5+46189/258048/a^6/x/(b*x^2+a)^4+46189/172032/a^7/x/(b*x^2+a)^3+46189/
98304/a^8/x/(b*x^2+a)^2+230945/196608/a^9/x/(b*x^2+a)-230945/65536*arctan(x*b^(1/2)/a^(1/2))*b^(1/2)/a^(21/2)

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Rubi [A]  time = 0.13, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac {230945}{196608 a^9 x \left (a+b x^2\right )}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}-\frac {230945 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{21/2}}-\frac {230945}{65536 a^{10} x}+\frac {1}{18 a x \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^2)^10),x]

[Out]

-230945/(65536*a^10*x) + 1/(18*a*x*(a + b*x^2)^9) + 19/(288*a^2*x*(a + b*x^2)^8) + 323/(4032*a^3*x*(a + b*x^2)
^7) + 1615/(16128*a^4*x*(a + b*x^2)^6) + 4199/(32256*a^5*x*(a + b*x^2)^5) + 46189/(258048*a^6*x*(a + b*x^2)^4)
+ 46189/(172032*a^7*x*(a + b*x^2)^3) + 46189/(98304*a^8*x*(a + b*x^2)^2) + 230945/(196608*a^9*x*(a + b*x^2))
- (230945*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(21/2))

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]

Rule 290

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

Rule 325

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

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^{10}} \, dx &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19 \int \frac {1}{x^2 \left (a+b x^2\right )^9} \, dx}{18 a}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323 \int \frac {1}{x^2 \left (a+b x^2\right )^8} \, dx}{288 a^2}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615 \int \frac {1}{x^2 \left (a+b x^2\right )^7} \, dx}{1344 a^3}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {20995 \int \frac {1}{x^2 \left (a+b x^2\right )^6} \, dx}{16128 a^4}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189 \int \frac {1}{x^2 \left (a+b x^2\right )^5} \, dx}{32256 a^5}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189 \int \frac {1}{x^2 \left (a+b x^2\right )^4} \, dx}{28672 a^6}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189 \int \frac {1}{x^2 \left (a+b x^2\right )^3} \, dx}{24576 a^7}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {230945 \int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx}{98304 a^8}\\ &=\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {230945}{196608 a^9 x \left (a+b x^2\right )}+\frac {230945 \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{65536 a^9}\\ &=-\frac {230945}{65536 a^{10} x}+\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {230945}{196608 a^9 x \left (a+b x^2\right )}-\frac {(230945 b) \int \frac {1}{a+b x^2} \, dx}{65536 a^{10}}\\ &=-\frac {230945}{65536 a^{10} x}+\frac {1}{18 a x \left (a+b x^2\right )^9}+\frac {19}{288 a^2 x \left (a+b x^2\right )^8}+\frac {323}{4032 a^3 x \left (a+b x^2\right )^7}+\frac {1615}{16128 a^4 x \left (a+b x^2\right )^6}+\frac {4199}{32256 a^5 x \left (a+b x^2\right )^5}+\frac {46189}{258048 a^6 x \left (a+b x^2\right )^4}+\frac {46189}{172032 a^7 x \left (a+b x^2\right )^3}+\frac {46189}{98304 a^8 x \left (a+b x^2\right )^2}+\frac {230945}{196608 a^9 x \left (a+b x^2\right )}-\frac {230945 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{21/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 147, normalized size = 0.70 \[ \frac {-\frac {\sqrt {a} \left (4128768 a^9+63897057 a^8 b x^2+318434718 a^7 b^2 x^4+850547502 a^6 b^3 x^6+1404993798 a^5 b^4 x^8+1513521152 a^4 b^5 x^{10}[phone] a^3 b^6 x^{12}[phone] a^2 b^7 x^{14}[phone] a b^8 x^{16}[phone] b^9 x^{18}\right )}{x \left (a+b x^2\right )^9}-14549535 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 a^{21/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^2)^10),x]

[Out]

(-((Sqrt[a]*(4128768*a^9 + 63897057*a^8*b*x^2 + 318434718*a^7*b^2*x^4 + 850547502*a^6*b^3*x^6 + 1404993798*a^5
*b^4*x^8 + 1513521152*a^4*b^5*x^10 + 1071677178*a^3*b^6*x^12 + 483044562*a^2*b^7*x^14 + 126095970*a*b^8*x^16 +
14549535*b^9*x^18))/(x*(a + b*x^2)^9)) - 14549535*Sqrt[b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(21/2))

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fricas [A]  time = 1.10, size = 664, normalized size = 3.18 \[ \left [-\frac {29099070 \, b^{9} x^{18} + 252191940 \, a b^{8} x^{16} + 966089124 \, a^{2} b^{7} x^{14} + 2143354356 \, a^{3} b^{6} x^{12} + 3027042304 \, a^{4} b^{5} x^{10} + 2809987596 \, a^{5} b^{4} x^{8} + 1701095004 \, a^{6} b^{3} x^{6} + 636869436 \, a^{7} b^{2} x^{4} + 127794114 \, a^{8} b x^{2} + 8257536 \, a^{9} - 14549535 \, {\left (b^{9} x^{19} + 9 \, a b^{8} x^{17} + 36 \, a^{2} b^{7} x^{15} + 84 \, a^{3} b^{6} x^{13} + 126 \, a^{4} b^{5} x^{11} + 126 \, a^{5} b^{4} x^{9} + 84 \, a^{6} b^{3} x^{7} + 36 \, a^{7} b^{2} x^{5} + 9 \, a^{8} b x^{3} + a^{9} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{10} b^{9} x^{19} + 9 \, a^{11} b^{8} x^{17} + 36 \, a^{12} b^{7} x^{15} + 84 \, a^{13} b^{6} x^{13} + 126 \, a^{14} b^{5} x^{11} + 126 \, a^{15} b^{4} x^{9} + 84 \, a^{16} b^{3} x^{7} + 36 \, a^{17} b^{2} x^{5} + 9 \, a^{18} b x^{3} + a^{19} x\right )}}, -\frac {14549535 \, b^{9} x^{18} + 126095970 \, a b^{8} x^{16} + 483044562 \, a^{2} b^{7} x^{14} + 1071677178 \, a^{3} b^{6} x^{12} + 1513521152 \, a^{4} b^{5} x^{10} + 1404993798 \, a^{5} b^{4} x^{8} + 850547502 \, a^{6} b^{3} x^{6} + 318434718 \, a^{7} b^{2} x^{4} + 63897057 \, a^{8} b x^{2} + 4128768 \, a^{9} + 14549535 \, {\left (b^{9} x^{19} + 9 \, a b^{8} x^{17} + 36 \, a^{2} b^{7} x^{15} + 84 \, a^{3} b^{6} x^{13} + 126 \, a^{4} b^{5} x^{11} + 126 \, a^{5} b^{4} x^{9} + 84 \, a^{6} b^{3} x^{7} + 36 \, a^{7} b^{2} x^{5} + 9 \, a^{8} b x^{3} + a^{9} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{4128768 \, {\left (a^{10} b^{9} x^{19} + 9 \, a^{11} b^{8} x^{17} + 36 \, a^{12} b^{7} x^{15} + 84 \, a^{13} b^{6} x^{13} + 126 \, a^{14} b^{5} x^{11} + 126 \, a^{15} b^{4} x^{9} + 84 \, a^{16} b^{3} x^{7} + 36 \, a^{17} b^{2} x^{5} + 9 \, a^{18} b x^{3} + a^{19} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8257536*(29099070*b^9*x^18 + 252191940*a*b^8*x^16 + 966089124*a^2*b^7*x^14 + 2143354356*a^3*b^6*x^12 + 302
7042304*a^4*b^5*x^10 + 2809987596*a^5*b^4*x^8 + 1701095004*a^6*b^3*x^6 + 636869436*a^7*b^2*x^4 + 127794114*a^8
*b*x^2 + 8257536*a^9 - 14549535*(b^9*x^19 + 9*a*b^8*x^17 + 36*a^2*b^7*x^15 + 84*a^3*b^6*x^13 + 126*a^4*b^5*x^1
1 + 126*a^5*b^4*x^9 + 84*a^6*b^3*x^7 + 36*a^7*b^2*x^5 + 9*a^8*b*x^3 + a^9*x)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqr
t(-b/a) - a)/(b*x^2 + a)))/(a^10*b^9*x^19 + 9*a^11*b^8*x^17 + 36*a^12*b^7*x^15 + 84*a^13*b^6*x^13 + 126*a^14*b
^5*x^11 + 126*a^15*b^4*x^9 + 84*a^16*b^3*x^7 + 36*a^17*b^2*x^5 + 9*a^18*b*x^3 + a^19*x), -1/4128768*(14549535*
b^9*x^18 + 126095970*a*b^8*x^16 + 483044562*a^2*b^7*x^14 + 1071677178*a^3*b^6*x^12 + 1513521152*a^4*b^5*x^10 +
1404993798*a^5*b^4*x^8 + 850547502*a^6*b^3*x^6 + 318434718*a^7*b^2*x^4 + 63897057*a^8*b*x^2 + 4128768*a^9 + 1
4549535*(b^9*x^19 + 9*a*b^8*x^17 + 36*a^2*b^7*x^15 + 84*a^3*b^6*x^13 + 126*a^4*b^5*x^11 + 126*a^5*b^4*x^9 + 84
*a^6*b^3*x^7 + 36*a^7*b^2*x^5 + 9*a^8*b*x^3 + a^9*x)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^10*b^9*x^19 + 9*a^11*b^
8*x^17 + 36*a^12*b^7*x^15 + 84*a^13*b^6*x^13 + 126*a^14*b^5*x^11 + 126*a^15*b^4*x^9 + 84*a^16*b^3*x^7 + 36*a^1
7*b^2*x^5 + 9*a^18*b*x^3 + a^19*x)]

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giac [A]  time = 0.63, size = 134, normalized size = 0.64 \[ -\frac {230945 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{10}} - \frac {1}{a^{10} x} - \frac {10420767 \, b^{9} x^{17} + 88937058 \, a b^{8} x^{15} + 334408914 \, a^{2} b^{7} x^{13} + 724860666 \, a^{3} b^{6} x^{11} + 993296384 \, a^{4} b^{5} x^{9} + 884769030 \, a^{5} b^{4} x^{7} + 503730990 \, a^{6} b^{3} x^{5} + 169799070 \, a^{7} b^{2} x^{3} + 26738145 \, a^{8} b x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-230945/65536*b*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^10) - 1/(a^10*x) - 1/4128768*(10420767*b^9*x^17 + 88937058*
a*b^8*x^15 + 334408914*a^2*b^7*x^13 + 724860666*a^3*b^6*x^11 + 993296384*a^4*b^5*x^9 + 884769030*a^5*b^4*x^7 +
503730990*a^6*b^3*x^5 + 169799070*a^7*b^2*x^3 + 26738145*a^8*b*x)/((b*x^2 + a)^9*a^10)

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maple [A]  time = 0.02, size = 206, normalized size = 0.99 \[ -\frac {165409 b^{9} x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} a^{10}}-\frac {2117549 b^{8} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} a^{9}}-\frac {2654039 b^{7} x^{13}}{32768 \left (b \,x^{2}+a \right )^{9} a^{8}}-\frac {40270037 b^{6} x^{11}}{229376 \left (b \,x^{2}+a \right )^{9} a^{7}}-\frac {30313 b^{5} x^{9}}{126 \left (b \,x^{2}+a \right )^{9} a^{6}}-\frac {49153835 b^{4} x^{7}}{229376 \left (b \,x^{2}+a \right )^{9} a^{5}}-\frac {3997865 b^{3} x^{5}}{32768 \left (b \,x^{2}+a \right )^{9} a^{4}}-\frac {4042835 b^{2} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} a^{3}}-\frac {424415 b x}{65536 \left (b \,x^{2}+a \right )^{9} a^{2}}-\frac {230945 b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{10}}-\frac {1}{a^{10} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^10/x-424415/65536/a^2*b/(b*x^2+a)^9*x-4042835/98304/a^3*b^2/(b*x^2+a)^9*x^3-3997865/32768/a^4*b^3/(b*x^2+
a)^9*x^5-49153835/229376/a^5*b^4/(b*x^2+a)^9*x^7-30313/126/a^6*b^5/(b*x^2+a)^9*x^9-40270037/229376/a^7*b^6/(b*
x^2+a)^9*x^11-2654039/32768/a^8*b^7/(b*x^2+a)^9*x^13-2117549/98304/a^9*b^8/(b*x^2+a)^9*x^15-165409/65536/a^10*
b^9/(b*x^2+a)^9*x^17-230945/65536/a^10*b/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.13, size = 225, normalized size = 1.08 \[ -\frac {14549535 \, b^{9} x^{18} + 126095970 \, a b^{8} x^{16} + 483044562 \, a^{2} b^{7} x^{14} + 1071677178 \, a^{3} b^{6} x^{12} + 1513521152 \, a^{4} b^{5} x^{10} + 1404993798 \, a^{5} b^{4} x^{8} + 850547502 \, a^{6} b^{3} x^{6} + 318434718 \, a^{7} b^{2} x^{4} + 63897057 \, a^{8} b x^{2} + 4128768 \, a^{9}}{4128768 \, {\left (a^{10} b^{9} x^{19} + 9 \, a^{11} b^{8} x^{17} + 36 \, a^{12} b^{7} x^{15} + 84 \, a^{13} b^{6} x^{13} + 126 \, a^{14} b^{5} x^{11} + 126 \, a^{15} b^{4} x^{9} + 84 \, a^{16} b^{3} x^{7} + 36 \, a^{17} b^{2} x^{5} + 9 \, a^{18} b x^{3} + a^{19} x\right )}} - \frac {230945 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4128768*(14549535*b^9*x^18 + 126095970*a*b^8*x^16 + 483044562*a^2*b^7*x^14 + 1071677178*a^3*b^6*x^12 + 1513
521152*a^4*b^5*x^10 + 1404993798*a^5*b^4*x^8 + 850547502*a^6*b^3*x^6 + 318434718*a^7*b^2*x^4 + 63897057*a^8*b*
x^2 + 4128768*a^9)/(a^10*b^9*x^19 + 9*a^11*b^8*x^17 + 36*a^12*b^7*x^15 + 84*a^13*b^6*x^13 + 126*a^14*b^5*x^11
+ 126*a^15*b^4*x^9 + 84*a^16*b^3*x^7 + 36*a^17*b^2*x^5 + 9*a^18*b*x^3 + a^19*x) - 230945/65536*b*arctan(b*x/sq
rt(a*b))/(sqrt(a*b)*a^10)

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mupad [B]  time = 5.09, size = 220, normalized size = 1.05 \[ -\frac {\frac {1}{a}+\frac {1014239\,b\,x^2}{65536\,a^2}+\frac {7581779\,b^2\,x^4}{98304\,a^3}+\frac {6750377\,b^3\,x^6}{32768\,a^4}+\frac {78055211\,b^4\,x^8}{229376\,a^5}+\frac {46189\,b^5\,x^{10}}{126\,a^6}+\frac {59537621\,b^6\,x^{12}}{229376\,a^7}+\frac {3833687\,b^7\,x^{14}}{32768\,a^8}+\frac {3002285\,b^8\,x^{16}}{98304\,a^9}+\frac {230945\,b^9\,x^{18}}{65536\,a^{10}}}{a^9\,x+9\,a^8\,b\,x^3+36\,a^7\,b^2\,x^5+84\,a^6\,b^3\,x^7+126\,a^5\,b^4\,x^9+126\,a^4\,b^5\,x^{11}+84\,a^3\,b^6\,x^{13}+36\,a^2\,b^7\,x^{15}+9\,a\,b^8\,x^{17}+b^9\,x^{19}}-\frac {230945\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{21/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^2)^10),x)

[Out]

- (1/a + (1014239*b*x^2)/(65536*a^2) + (7581779*b^2*x^4)/(98304*a^3) + (6750377*b^3*x^6)/(32768*a^4) + (780552
11*b^4*x^8)/(229376*a^5) + (46189*b^5*x^10)/(126*a^6) + (59537621*b^6*x^12)/(229376*a^7) + (3833687*b^7*x^14)/
(32768*a^8) + (3002285*b^8*x^16)/(98304*a^9) + (230945*b^9*x^18)/(65536*a^10))/(a^9*x + b^9*x^19 + 9*a^8*b*x^3
+ 9*a*b^8*x^17 + 36*a^7*b^2*x^5 + 84*a^6*b^3*x^7 + 126*a^5*b^4*x^9 + 126*a^4*b^5*x^11 + 84*a^3*b^6*x^13 + 36*
a^2*b^7*x^15) - (230945*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(21/2))

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sympy [A]  time = 1.35, size = 282, normalized size = 1.35 \[ \frac {230945 \sqrt {- \frac {b}{a^{21}}} \log {\left (- \frac {a^{11} \sqrt {- \frac {b}{a^{21}}}}{b} + x \right )}}{131072} - \frac {230945 \sqrt {- \frac {b}{a^{21}}} \log {\left (\frac {a^{11} \sqrt {- \frac {b}{a^{21}}}}{b} + x \right )}}{131072} + \frac {- 4128768 a^{9} - 63897057 a^{8} b x^{2} - 318434718 a^{7} b^{2} x^{4} - 850547502 a^{6} b^{3} x^{6} - 1404993798 a^{5} b^{4} x^{8} - 1513521152 a^{4} b^{5} x^{10} - 1071677178 a^{3} b^{6} x^{12} - 483044562 a^{2} b^{7} x^{14} - 126095970 a b^{8} x^{16} - 14549535 b^{9} x^{18}}{4128768 a^{19} x + 37158912 a^{18} b x^{3} + 148635648 a^{17} b^{2} x^{5} + 346816512 a^{16} b^{3} x^{7} + 520224768 a^{15} b^{4} x^{9} + 520224768 a^{14} b^{5} x^{11} + 346816512 a^{13} b^{6} x^{13} + 148635648 a^{12} b^{7} x^{15} + 37158912 a^{11} b^{8} x^{17} + 4128768 a^{10} b^{9} x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

230945*sqrt(-b/a**21)*log(-a**11*sqrt(-b/a**21)/b + x)/131072 - 230945*sqrt(-b/a**21)*log(a**11*sqrt(-b/a**21)
/b + x)/131072 + (-4128768*a**9 - 63897057*a**8*b*x**2 - 318434718*a**7*b**2*x**4 - 850547502*a**6*b**3*x**6 -
1404993798*a**5*b**4*x**8 - 1513521152*a**4*b**5*x**10 - 1071677178*a**3*b**6*x**12 - 483044562*a**2*b**7*x**
14 - 126095970*a*b**8*x**16 - 14549535*b**9*x**18)/(4128768*a**19*x + 37158912*a**18*b*x**3 + 148635648*a**17*
b**2*x**5 + 346816512*a**16*b**3*x**7 + 520224768*a**15*b**4*x**9 + 520224768*a**14*b**5*x**11 + 346816512*a**
13*b**6*x**13 + 148635648*a**12*b**7*x**15 + 37158912*a**11*b**8*x**17 + 4128768*a**10*b**9*x**19)

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