3.223 \(\int \frac {1}{x^4 (a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=220 \[ \frac {1616615 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{23/2}}+\frac {1616615 b}{65536 a^{11} x}-\frac {1616615}{196608 a^{10} x^3}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9} \]

[Out]

-1616615/196608/a^10/x^3+1616615/65536*b/a^11/x+1/18/a/x^3/(b*x^2+a)^9+7/96/a^2/x^3/(b*x^2+a)^8+19/192/a^3/x^3
/(b*x^2+a)^7+323/2304/a^4/x^3/(b*x^2+a)^6+323/1536/a^5/x^3/(b*x^2+a)^5+4199/12288/a^6/x^3/(b*x^2+a)^4+46189/73
728/a^7/x^3/(b*x^2+a)^3+46189/32768/a^8/x^3/(b*x^2+a)^2+323323/65536/a^9/x^3/(b*x^2+a)+1616615/65536*b^(3/2)*a
rctan(x*b^(1/2)/a^(1/2))/a^(23/2)

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Rubi [A]  time = 0.14, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac {1616615 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{23/2}}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {1616615 b}{65536 a^{11} x}-\frac {1616615}{196608 a^{10} x^3}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^10),x]

[Out]

-1616615/(196608*a^10*x^3) + (1616615*b)/(65536*a^11*x) + 1/(18*a*x^3*(a + b*x^2)^9) + 7/(96*a^2*x^3*(a + b*x^
2)^8) + 19/(192*a^3*x^3*(a + b*x^2)^7) + 323/(2304*a^4*x^3*(a + b*x^2)^6) + 323/(1536*a^5*x^3*(a + b*x^2)^5) +
4199/(12288*a^6*x^3*(a + b*x^2)^4) + 46189/(73728*a^7*x^3*(a + b*x^2)^3) + 46189/(32768*a^8*x^3*(a + b*x^2)^2
) + 323323/(65536*a^9*x^3*(a + b*x^2)) + (1616615*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(23/2))

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^4 \left (a+b x^2\right )^{10}} \, dx &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7 \int \frac {1}{x^4 \left (a+b x^2\right )^9} \, dx}{6 a}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {133 \int \frac {1}{x^4 \left (a+b x^2\right )^8} \, dx}{96 a^2}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323 \int \frac {1}{x^4 \left (a+b x^2\right )^7} \, dx}{192 a^3}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {1615 \int \frac {1}{x^4 \left (a+b x^2\right )^6} \, dx}{768 a^4}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199 \int \frac {1}{x^4 \left (a+b x^2\right )^5} \, dx}{1536 a^5}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189 \int \frac {1}{x^4 \left (a+b x^2\right )^4} \, dx}{12288 a^6}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189 \int \frac {1}{x^4 \left (a+b x^2\right )^3} \, dx}{8192 a^7}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {323323 \int \frac {1}{x^4 \left (a+b x^2\right )^2} \, dx}{32768 a^8}\\ &=\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac {1616615 \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{65536 a^9}\\ &=-\frac {1616615}{196608 a^{10} x^3}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}-\frac {(1616615 b) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{65536 a^{10}}\\ &=-\frac {1616615}{196608 a^{10} x^3}+\frac {1616615 b}{65536 a^{11} x}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac {\left (1616615 b^2\right ) \int \frac {1}{a+b x^2} \, dx}{65536 a^{11}}\\ &=-\frac {1616615}{196608 a^{10} x^3}+\frac {1616615 b}{65536 a^{11} x}+\frac {1}{18 a x^3 \left (a+b x^2\right )^9}+\frac {7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac {19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac {323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac {323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac {4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac {46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac {46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac {323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac {1616615 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{23/2}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 157, normalized size = 0.71 \[ \frac {\frac {\sqrt {a} \left (-196608 a^{10}+4128768 a^9 b x^2+63897057 a^8 b^2 x^4+318434718 a^7 b^3 x^6+850547502 a^6 b^4 x^8+1404993798 a^5 b^5 x^{10}[phone] a^4 b^6 x^{12}[phone] a^3 b^7 x^{14}[phone] a^2 b^8 x^{16}[phone] a b^9 x^{18}[phone] b^{10} x^{20}\right )}{x^3 \left (a+b x^2\right )^9}[phone] b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 a^{23/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^10),x]

[Out]

((Sqrt[a]*(-196608*a^10 + 4128768*a^9*b*x^2 + 63897057*a^8*b^2*x^4 + 318434718*a^7*b^3*x^6 + 850547502*a^6*b^4
*x^8 + 1404993798*a^5*b^5*x^10 + 1513521152*a^4*b^6*x^12 + 1071677178*a^3*b^7*x^14 + 483044562*a^2*b^8*x^16 +
126095970*a*b^9*x^18 + 14549535*b^10*x^20))/(x^3*(a + b*x^2)^9) + 14549535*b^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]
)/(589824*a^(23/2))

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fricas [A]  time = 0.97, size = 700, normalized size = 3.18 \[ \left [\frac {29099070 \, b^{10} x^{20} + 252191940 \, a b^{9} x^{18} + 966089124 \, a^{2} b^{8} x^{16} + 2143354356 \, a^{3} b^{7} x^{14} + 3027042304 \, a^{4} b^{6} x^{12} + 2809987596 \, a^{5} b^{5} x^{10} + 1701095004 \, a^{6} b^{4} x^{8} + 636869436 \, a^{7} b^{3} x^{6} + 127794114 \, a^{8} b^{2} x^{4} + 8257536 \, a^{9} b x^{2} - 393216 \, a^{10} + 14549535 \, {\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{1179648 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}, \frac {14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10} + 14549535 \, {\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{589824 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/1179648*(29099070*b^10*x^20 + 252191940*a*b^9*x^18 + 966089124*a^2*b^8*x^16 + 2143354356*a^3*b^7*x^14 + 302
7042304*a^4*b^6*x^12 + 2809987596*a^5*b^5*x^10 + 1701095004*a^6*b^4*x^8 + 636869436*a^7*b^3*x^6 + 127794114*a^
8*b^2*x^4 + 8257536*a^9*b*x^2 - 393216*a^10 + 14549535*(b^10*x^21 + 9*a*b^9*x^19 + 36*a^2*b^8*x^17 + 84*a^3*b^
7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b^5*x^11 + 84*a^6*b^4*x^9 + 36*a^7*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*b*x^3)*sq
rt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17
+ 84*a^14*b^6*x^15 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5
+ a^20*x^3), 1/589824*(14549535*b^10*x^20 + 126095970*a*b^9*x^18 + 483044562*a^2*b^8*x^16 + 1071677178*a^3*b^7
*x^14 + 1513521152*a^4*b^6*x^12 + 1404993798*a^5*b^5*x^10 + 850547502*a^6*b^4*x^8 + 318434718*a^7*b^3*x^6 + 63
897057*a^8*b^2*x^4 + 4128768*a^9*b*x^2 - 196608*a^10 + 14549535*(b^10*x^21 + 9*a*b^9*x^19 + 36*a^2*b^8*x^17 +
84*a^3*b^7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b^5*x^11 + 84*a^6*b^4*x^9 + 36*a^7*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*
b*x^3)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84*a^14*b^6*x^15 +
126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5 + a^20*x^3)]

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giac [A]  time = 0.63, size = 148, normalized size = 0.67 \[ \frac {1616615 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{11}} + \frac {30 \, b x^{2} - a}{3 \, a^{11} x^{3}} + \frac {8651295 \, b^{10} x^{17} + 73208418 \, a b^{9} x^{15} + 272477394 \, a^{2} b^{8} x^{13} + 583302906 \, a^{3} b^{7} x^{11} + 786857984 \, a^{4} b^{6} x^{9} + 686588166 \, a^{5} b^{5} x^{7} + 379867950 \, a^{6} b^{4} x^{5} + 122613150 \, a^{7} b^{3} x^{3} + 17890785 \, a^{8} b^{2} x}{589824 \, {\left (b x^{2} + a\right )}^{9} a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^10,x, algorithm="giac")

[Out]

1616615/65536*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^11) + 1/3*(30*b*x^2 - a)/(a^11*x^3) + 1/589824*(8651295*b
^10*x^17 + 73208418*a*b^9*x^15 + 272477394*a^2*b^8*x^13 + 583302906*a^3*b^7*x^11 + 786857984*a^4*b^6*x^9 + 686
588166*a^5*b^5*x^7 + 379867950*a^6*b^4*x^5 + 122613150*a^7*b^3*x^3 + 17890785*a^8*b^2*x)/((b*x^2 + a)^9*a^11)

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maple [A]  time = 0.02, size = 219, normalized size = 1.00 \[ \frac {961255 b^{10} x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} a^{11}}+\frac {12201403 b^{9} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} a^{10}}+\frac {15137633 b^{8} x^{13}}{32768 \left (b \,x^{2}+a \right )^{9} a^{9}}+\frac {32405717 b^{7} x^{11}}{32768 \left (b \,x^{2}+a \right )^{9} a^{8}}+\frac {24013 b^{6} x^{9}}{18 \left (b \,x^{2}+a \right )^{9} a^{7}}+\frac {38143787 b^{5} x^{7}}{32768 \left (b \,x^{2}+a \right )^{9} a^{6}}+\frac {21103775 b^{4} x^{5}}{32768 \left (b \,x^{2}+a \right )^{9} a^{5}}+\frac {20435525 b^{3} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} a^{4}}+\frac {1987865 b^{2} x}{65536 \left (b \,x^{2}+a \right )^{9} a^{3}}+\frac {1616615 b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{11}}+\frac {10 b}{a^{11} x}-\frac {1}{3 a^{10} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^10,x)

[Out]

-1/3/a^10/x^3+10*b/a^11/x+1987865/65536/a^3*b^2/(b*x^2+a)^9*x+20435525/98304/a^4*b^3/(b*x^2+a)^9*x^3+21103775/
32768/a^5*b^4/(b*x^2+a)^9*x^5+38143787/32768/a^6*b^5/(b*x^2+a)^9*x^7+24013/18/a^7*b^6/(b*x^2+a)^9*x^9+32405717
/32768/a^8*b^7/(b*x^2+a)^9*x^11+15137633/32768/a^9*b^8/(b*x^2+a)^9*x^13+12201403/98304/a^10*b^9/(b*x^2+a)^9*x^
15+961255/65536/a^11*b^10/(b*x^2+a)^9*x^17+1616615/65536/a^11*b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.30, size = 240, normalized size = 1.09 \[ \frac {14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10}}{589824 \, {\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}} + \frac {1616615 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/589824*(14549535*b^10*x^20 + 126095970*a*b^9*x^18 + 483044562*a^2*b^8*x^16 + 1071677178*a^3*b^7*x^14 + 15135
21152*a^4*b^6*x^12 + 1404993798*a^5*b^5*x^10 + 850547502*a^6*b^4*x^8 + 318434718*a^7*b^3*x^6 + 63897057*a^8*b^
2*x^4 + 4128768*a^9*b*x^2 - 196608*a^10)/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84*a^14*b^6*x^1
5 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^18*b^2*x^7 + 9*a^19*b*x^5 + a^20*x^3) + 161
6615/65536*b^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^11)

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mupad [B]  time = 5.11, size = 234, normalized size = 1.06 \[ \frac {\frac {7\,b\,x^2}{a^2}-\frac {1}{3\,a}+\frac {7099673\,b^2\,x^4}{65536\,a^3}+\frac {53072453\,b^3\,x^6}{98304\,a^4}+\frac {47252639\,b^4\,x^8}{32768\,a^5}+\frac {78055211\,b^5\,x^{10}}{32768\,a^6}+\frac {46189\,b^6\,x^{12}}{18\,a^7}+\frac {59537621\,b^7\,x^{14}}{32768\,a^8}+\frac {26835809\,b^8\,x^{16}}{32768\,a^9}+\frac {21015995\,b^9\,x^{18}}{98304\,a^{10}}+\frac {1616615\,b^{10}\,x^{20}}{65536\,a^{11}}}{a^9\,x^3+9\,a^8\,b\,x^5+36\,a^7\,b^2\,x^7+84\,a^6\,b^3\,x^9+126\,a^5\,b^4\,x^{11}+126\,a^4\,b^5\,x^{13}+84\,a^3\,b^6\,x^{15}+36\,a^2\,b^7\,x^{17}+9\,a\,b^8\,x^{19}+b^9\,x^{21}}+\frac {1616615\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{23/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^10),x)

[Out]

((7*b*x^2)/a^2 - 1/(3*a) + (7099673*b^2*x^4)/(65536*a^3) + (53072453*b^3*x^6)/(98304*a^4) + (47252639*b^4*x^8)
/(32768*a^5) + (78055211*b^5*x^10)/(32768*a^6) + (46189*b^6*x^12)/(18*a^7) + (59537621*b^7*x^14)/(32768*a^8) +
(26835809*b^8*x^16)/(32768*a^9) + (21015995*b^9*x^18)/(98304*a^10) + (1616615*b^10*x^20)/(65536*a^11))/(a^9*x
^3 + b^9*x^21 + 9*a^8*b*x^5 + 9*a*b^8*x^19 + 36*a^7*b^2*x^7 + 84*a^6*b^3*x^9 + 126*a^5*b^4*x^11 + 126*a^4*b^5*
x^13 + 84*a^3*b^6*x^15 + 36*a^2*b^7*x^17) + (1616615*b^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(23/2))

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sympy [A]  time = 1.45, size = 304, normalized size = 1.38 \[ - \frac {1616615 \sqrt {- \frac {b^{3}}{a^{23}}} \log {\left (- \frac {a^{12} \sqrt {- \frac {b^{3}}{a^{23}}}}{b^{2}} + x \right )}}{131072} + \frac {1616615 \sqrt {- \frac {b^{3}}{a^{23}}} \log {\left (\frac {a^{12} \sqrt {- \frac {b^{3}}{a^{23}}}}{b^{2}} + x \right )}}{131072} + \frac {- 196608 a^{10} + 4128768 a^{9} b x^{2} + 63897057 a^{8} b^{2} x^{4} + 318434718 a^{7} b^{3} x^{6} + 850547502 a^{6} b^{4} x^{8} + 1404993798 a^{5} b^{5} x^{10} + 1513521152 a^{4} b^{6} x^{12} + 1071677178 a^{3} b^{7} x^{14} + 483044562 a^{2} b^{8} x^{16} + 126095970 a b^{9} x^{18} + 14549535 b^{10} x^{20}}{589824 a^{20} x^{3} + 5308416 a^{19} b x^{5} + 21233664 a^{18} b^{2} x^{7} + 49545216 a^{17} b^{3} x^{9} + 74317824 a^{16} b^{4} x^{11} + 74317824 a^{15} b^{5} x^{13} + 49545216 a^{14} b^{6} x^{15} + 21233664 a^{13} b^{7} x^{17} + 5308416 a^{12} b^{8} x^{19} + 589824 a^{11} b^{9} x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**10,x)

[Out]

-1616615*sqrt(-b**3/a**23)*log(-a**12*sqrt(-b**3/a**23)/b**2 + x)/131072 + 1616615*sqrt(-b**3/a**23)*log(a**12
*sqrt(-b**3/a**23)/b**2 + x)/131072 + (-196608*a**10 + 4128768*a**9*b*x**2 + 63897057*a**8*b**2*x**4 + 3184347
18*a**7*b**3*x**6 + 850547502*a**6*b**4*x**8 + 1404993798*a**5*b**5*x**10 + 1513521152*a**4*b**6*x**12 + 10716
77178*a**3*b**7*x**14 + 483044562*a**2*b**8*x**16 + 126095970*a*b**9*x**18 + 14549535*b**10*x**20)/(589824*a**
20*x**3 + 5308416*a**19*b*x**5 + 21233664*a**18*b**2*x**7 + 49545216*a**17*b**3*x**9 + 74317824*a**16*b**4*x**
11 + 74317824*a**15*b**5*x**13 + 49545216*a**14*b**6*x**15 + 21233664*a**13*b**7*x**17 + 5308416*a**12*b**8*x*
*19 + 589824*a**11*b**9*x**21)

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