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

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

[Out]

-1/18*x^17/b/(b*x^2+a)^9-17/288*x^15/b^2/(b*x^2+a)^8-85/1344*x^13/b^3/(b*x^2+a)^7-1105/16128*x^11/b^4/(b*x^2+a
)^6-2431/32256*x^9/b^5/(b*x^2+a)^5-2431/28672*x^7/b^6/(b*x^2+a)^4-2431/24576*x^5/b^7/(b*x^2+a)^3-12155/98304*x
^3/b^8/(b*x^2+a)^2-12155/65536*x/b^9/(b*x^2+a)+12155/65536*arctan(x*b^(1/2)/a^(1/2))/b^(19/2)/a^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 197, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.08, size = 134, normalized size = 0.68 \[ \frac {\frac {765765 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {b} x \left (765765 a^8+6636630 a^7 b x^2+25423398 a^6 b^2 x^4+56404062 a^5 b^3 x^6+79659008 a^4 b^4 x^8+73947042 a^3 b^5 x^{10}[phone] a^2 b^6 x^{12}[phone] a b^7 x^{14}+3363003 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}}{4128768 b^{19/2}} \]

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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fricas [A]  time = 0.88, size = 650, normalized size = 3.30 \[ \left [-\frac {6726006 \, a b^{9} x^{17} + 33519444 \, a^{2} b^{8} x^{15} + 89531316 \, a^{3} b^{7} x^{13} + 147894084 \, a^{4} b^{6} x^{11} + 159318016 \, a^{5} b^{5} x^{9} + 112808124 \, a^{6} b^{4} x^{7} + 50846796 \, a^{7} b^{3} x^{5} + 13273260 \, a^{8} b^{2} x^{3} + 1531530 \, a^{9} b x + 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a b^{19} x^{18} + 9 \, a^{2} b^{18} x^{16} + 36 \, a^{3} b^{17} x^{14} + 84 \, a^{4} b^{16} x^{12} + 126 \, a^{5} b^{15} x^{10} + 126 \, a^{6} b^{14} x^{8} + 84 \, a^{7} b^{13} x^{6} + 36 \, a^{8} b^{12} x^{4} + 9 \, a^{9} b^{11} x^{2} + a^{10} b^{10}\right )}}, -\frac {3363003 \, a b^{9} x^{17} + 16759722 \, a^{2} b^{8} x^{15} + 44765658 \, a^{3} b^{7} x^{13} + 73947042 \, a^{4} b^{6} x^{11} + 79659008 \, a^{5} b^{5} x^{9} + 56404062 \, a^{6} b^{4} x^{7} + 25423398 \, a^{7} b^{3} x^{5} + 6636630 \, a^{8} b^{2} x^{3} + 765765 \, a^{9} b x - 765765 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a b^{19} x^{18} + 9 \, a^{2} b^{18} x^{16} + 36 \, a^{3} b^{17} x^{14} + 84 \, a^{4} b^{16} x^{12} + 126 \, a^{5} b^{15} x^{10} + 126 \, a^{6} b^{14} x^{8} + 84 \, a^{7} b^{13} x^{6} + 36 \, a^{8} b^{12} x^{4} + 9 \, a^{9} b^{11} x^{2} + a^{10} b^{10}\right )}}\right ] \]

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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giac [A]  time = 0.62, size = 122, normalized size = 0.62 \[ \frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{9}} - \frac {3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} b^{9}} \]

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)

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maple [A]  time = 0.02, size = 124, normalized size = 0.63 \[ \frac {12155 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, b^{9}}+\frac {-\frac {53381 x^{17}}{65536 b}-\frac {399041 a \,x^{15}}{98304 b^{2}}-\frac {355283 a^{2} x^{13}}{32768 b^{3}}-\frac {4108169 a^{3} x^{11}}{229376 b^{4}}-\frac {2431 a^{4} x^{9}}{126 b^{5}}-\frac {3133559 a^{5} x^{7}}{229376 b^{6}}-\frac {201773 a^{6} x^{5}}{32768 b^{7}}-\frac {158015 a^{7} x^{3}}{98304 b^{8}}-\frac {12155 a^{8} x}{65536 b^{9}}}{\left (b \,x^{2}+a \right )^{9}} \]

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*a/b^2*x^15-53381/65536/b*x^17)/
(b*x^2+a)^9+12155/65536/b^9/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.17, size = 213, normalized size = 1.08 \[ -\frac {3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x}{4128768 \, {\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )}} + \frac {12155 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4128768*(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)/(b^18*x^18 + 9*a
*b^17*x^16 + 36*a^2*b^16*x^14 + 84*a^3*b^15*x^12 + 126*a^4*b^14*x^10 + 126*a^5*b^13*x^8 + 84*a^6*b^12*x^6 + 36
*a^7*b^11*x^4 + 9*a^8*b^10*x^2 + a^9*b^9) + 12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^9)

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mupad [B]  time = 4.96, size = 210, normalized size = 1.07 \[ \frac {12155\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,\sqrt {a}\,b^{19/2}}-\frac {\frac {53381\,x^{17}}{65536\,b}+\frac {399041\,a\,x^{15}}{98304\,b^2}+\frac {12155\,a^8\,x}{65536\,b^9}+\frac {355283\,a^2\,x^{13}}{32768\,b^3}+\frac {4108169\,a^3\,x^{11}}{229376\,b^4}+\frac {2431\,a^4\,x^9}{126\,b^5}+\frac {3133559\,a^5\,x^7}{229376\,b^6}+\frac {201773\,a^6\,x^5}{32768\,b^7}+\frac {158015\,a^7\,x^3}{98304\,b^8}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^18/(a + b*x^2)^10,x)

[Out]

(12155*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(1/2)*b^(19/2)) - ((53381*x^17)/(65536*b) + (399041*a*x^15)/(98304*
b^2) + (12155*a^8*x)/(65536*b^9) + (355283*a^2*x^13)/(32768*b^3) + (4108169*a^3*x^11)/(229376*b^4) + (2431*a^4
*x^9)/(126*b^5) + (3133559*a^5*x^7)/(229376*b^6) + (201773*a^6*x^5)/(32768*b^7) + (158015*a^7*x^3)/(98304*b^8)
)/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 126*a^5*b^4*x^8 + 126*a^4*b
^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14)

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sympy [A]  time = 1.55, size = 277, normalized size = 1.41 \[ - \frac {12155 \sqrt {- \frac {1}{a b^{19}}} \log {\left (- a b^{9} \sqrt {- \frac {1}{a b^{19}}} + x \right )}}{131072} + \frac {12155 \sqrt {- \frac {1}{a b^{19}}} \log {\left (a b^{9} \sqrt {- \frac {1}{a b^{19}}} + x \right )}}{131072} + \frac {- 765765 a^{8} x - 6636630 a^{7} b x^{3} - 25423398 a^{6} b^{2} x^{5} - 56404062 a^{5} b^{3} x^{7} - 79659008 a^{4} b^{4} x^{9} - 73947042 a^{3} b^{5} x^{11} - 44765658 a^{2} b^{6} x^{13} - 16759722 a b^{7} x^{15} - 3363003 b^{8} x^{17}}{4128768 a^{9} b^{9} + 37158912 a^{8} b^{10} x^{2} + 148635648 a^{7} b^{11} x^{4} + 346816512 a^{6} b^{12} x^{6} + 520224768 a^{5} b^{13} x^{8} + 520224768 a^{4} b^{14} x^{10} + 346816512 a^{3} b^{15} x^{12} + 148635648 a^{2} b^{16} x^{14} + 37158912 a b^{17} x^{16} + 4128768 b^{18} x^{18}} \]

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 + 34681651
2*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 + 1486
35648*a**2*b**16*x**14 + 37158912*a*b**17*x**16 + 4128768*b**18*x**18)

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