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

Optimal. Leaf size=205 \[ \frac {715 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac {715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac {715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {x}{288 a b \left (a+b x^2\right )^8}-\frac {x}{18 b \left (a+b x^2\right )^9} \]

[Out]

-1/18*x/b/(b*x^2+a)^9+1/288*x/a/b/(b*x^2+a)^8+5/1344*x/a^2/b/(b*x^2+a)^7+65/16128*x/a^3/b/(b*x^2+a)^6+143/3225
6*x/a^4/b/(b*x^2+a)^5+143/28672*x/a^5/b/(b*x^2+a)^4+143/24576*x/a^6/b/(b*x^2+a)^3+715/98304*x/a^7/b/(b*x^2+a)^
2+715/65536*x/a^8/b/(b*x^2+a)+715/65536*arctan(x*b^(1/2)/a^(1/2))/a^(17/2)/b^(3/2)

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Rubi [A]  time = 0.11, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 199, 205} \[ \frac {715 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac {715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac {715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {x}{288 a b \left (a+b x^2\right )^8}-\frac {x}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2)^10,x]

[Out]

-x/(18*b*(a + b*x^2)^9) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x^2)^7) + (65*x)/(16128*a^3*b*(
a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6
*b*(a + b*x^2)^3) + (715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*ArcTan[(Sqr
t[b]*x)/Sqrt[a]])/(65536*a^(17/2)*b^(3/2))

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]

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^2}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {\int \frac {1}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 \int \frac {1}{\left (a+b x^2\right )^8} \, dx}{96 a b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 \int \frac {1}{\left (a+b x^2\right )^7} \, dx}{1344 a^2 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {715 \int \frac {1}{\left (a+b x^2\right )^6} \, dx}{16128 a^3 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^5} \, dx}{3584 a^4 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 \int \frac {1}{\left (a+b x^2\right )^4} \, dx}{4096 a^5 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {715 \int \frac {1}{\left (a+b x^2\right )^3} \, dx}{24576 a^6 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac {715 \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{32768 a^7 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac {715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac {715 \int \frac {1}{a+b x^2} \, dx}{65536 a^8 b}\\ &=-\frac {x}{18 b \left (a+b x^2\right )^9}+\frac {x}{288 a b \left (a+b x^2\right )^8}+\frac {5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac {65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac {143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac {143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac {143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac {715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac {715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac {715 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 a^{17/2} b^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 138, normalized size = 0.67 \[ \frac {\frac {\sqrt {a} \sqrt {b} x \left (-45045 a^8+985866 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 a^{17/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 + 985866*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 4349826*a^5*b^3*x^6 + 4685824*a^4*b
^4*x^8 + 3317886*a^3*b^5*x^10 + 1495494*a^2*b^6*x^12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45
045*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(17/2)*b^(3/2))

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fricas [A]  time = 0.84, size = 654, normalized size = 3.19 \[ \left [\frac {90090 \, a b^{9} x^{17} + 780780 \, a^{2} b^{8} x^{15} + 2990988 \, a^{3} b^{7} x^{13} + 6635772 \, a^{4} b^{6} x^{11} + 9371648 \, a^{5} b^{5} x^{9} + 8699652 \, a^{6} b^{4} x^{7} + 5266548 \, a^{7} b^{3} x^{5} + 1971732 \, a^{8} b^{2} x^{3} - 90090 \, a^{9} b x - 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{8257536 \, {\left (a^{9} b^{11} x^{18} + 9 \, a^{10} b^{10} x^{16} + 36 \, a^{11} b^{9} x^{14} + 84 \, a^{12} b^{8} x^{12} + 126 \, a^{13} b^{7} x^{10} + 126 \, a^{14} b^{6} x^{8} + 84 \, a^{15} b^{5} x^{6} + 36 \, a^{16} b^{4} x^{4} + 9 \, a^{17} b^{3} x^{2} + a^{18} b^{2}\right )}}, \frac {45045 \, a b^{9} x^{17} + 390390 \, a^{2} b^{8} x^{15} + 1495494 \, a^{3} b^{7} x^{13} + 3317886 \, a^{4} b^{6} x^{11} + 4685824 \, a^{5} b^{5} x^{9} + 4349826 \, a^{6} b^{4} x^{7} + 2633274 \, a^{7} b^{3} x^{5} + 985866 \, a^{8} b^{2} x^{3} - 45045 \, a^{9} b x + 45045 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{4128768 \, {\left (a^{9} b^{11} x^{18} + 9 \, a^{10} b^{10} x^{16} + 36 \, a^{11} b^{9} x^{14} + 84 \, a^{12} b^{8} x^{12} + 126 \, a^{13} b^{7} x^{10} + 126 \, a^{14} b^{6} x^{8} + 84 \, a^{15} b^{5} x^{6} + 36 \, a^{16} b^{4} x^{4} + 9 \, a^{17} b^{3} x^{2} + a^{18} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8257536*(90090*a*b^9*x^17 + 780780*a^2*b^8*x^15 + 2990988*a^3*b^7*x^13 + 6635772*a^4*b^6*x^11 + 9371648*a^5
*b^5*x^9 + 8699652*a^6*b^4*x^7 + 5266548*a^7*b^3*x^5 + 1971732*a^8*b^2*x^3 - 90090*a^9*b*x - 45045*(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^9*b^11*x^18 + 9*a
^10*b^10*x^16 + 36*a^11*b^9*x^14 + 84*a^12*b^8*x^12 + 126*a^13*b^7*x^10 + 126*a^14*b^6*x^8 + 84*a^15*b^5*x^6 +
36*a^16*b^4*x^4 + 9*a^17*b^3*x^2 + a^18*b^2), 1/4128768*(45045*a*b^9*x^17 + 390390*a^2*b^8*x^15 + 1495494*a^3
*b^7*x^13 + 3317886*a^4*b^6*x^11 + 4685824*a^5*b^5*x^9 + 4349826*a^6*b^4*x^7 + 2633274*a^7*b^3*x^5 + 985866*a^
8*b^2*x^3 - 45045*a^9*b*x + 45045*(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^9*b^11*x^18 + 9*a^10*b^10*x^16 + 36*a^11*b^9*x^14 + 84*a^12*b^8*x^12 + 126*a^13*b^7*x^10 + 126*a^14*b^6*x^8
+ 84*a^15*b^5*x^6 + 36*a^16*b^4*x^4 + 9*a^17*b^3*x^2 + a^18*b^2)]

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giac [A]  time = 0.62, size = 128, normalized size = 0.62 \[ \frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{8} b} + \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} a^{8} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

715/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8*b) + 1/4128768*(45045*b^8*x^17 + 390390*a*b^7*x^15 + 1495494*a^
2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 4685824*a^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a
^7*b*x^3 - 45045*a^8*x)/((b*x^2 + a)^9*a^8*b)

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maple [A]  time = 0.02, size = 124, normalized size = 0.60 \[ \frac {715 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{8} b}+\frac {\frac {715 b^{7} x^{17}}{65536 a^{8}}+\frac {9295 b^{6} x^{15}}{98304 a^{7}}+\frac {11869 b^{5} x^{13}}{32768 a^{6}}+\frac {184327 b^{4} x^{11}}{229376 a^{5}}+\frac {143 b^{3} x^{9}}{126 a^{4}}+\frac {241657 b^{2} x^{7}}{229376 a^{3}}+\frac {20899 b \,x^{5}}{32768 a^{2}}+\frac {23473 x^{3}}{98304 a}-\frac {715 x}{65536 b}}{\left (b \,x^{2}+a \right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-715/65536/b*x+23473/98304/a*x^3+20899/32768/a^2*b*x^5+241657/229376*b^2/a^3*x^7+143/126*b^3/a^4*x^9+184327/2
29376*b^4/a^5*x^11+11869/32768/a^6*b^5*x^13+9295/98304/a^7*b^6*x^15+715/65536/a^8*b^7*x^17)/(b*x^2+a)^9+715/65
536/a^8/b/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.09, size = 219, normalized size = 1.07 \[ \frac {45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \, {\left (a^{8} b^{10} x^{18} + 9 \, a^{9} b^{9} x^{16} + 36 \, a^{10} b^{8} x^{14} + 84 \, a^{11} b^{7} x^{12} + 126 \, a^{12} b^{6} x^{10} + 126 \, a^{13} b^{5} x^{8} + 84 \, a^{14} b^{4} x^{6} + 36 \, a^{15} b^{3} x^{4} + 9 \, a^{16} b^{2} x^{2} + a^{17} b\right )}} + \frac {715 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{8} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4128768*(45045*b^8*x^17 + 390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 4685824*a^4*b^4*
x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a^7*b*x^3 - 45045*a^8*x)/(a^8*b^10*x^18 + 9*a^9*b^9*x
^16 + 36*a^10*b^8*x^14 + 84*a^11*b^7*x^12 + 126*a^12*b^6*x^10 + 126*a^13*b^5*x^8 + 84*a^14*b^4*x^6 + 36*a^15*b
^3*x^4 + 9*a^16*b^2*x^2 + a^17*b) + 715/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8*b)

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mupad [B]  time = 0.17, size = 206, normalized size = 1.00 \[ \frac {\frac {23473\,x^3}{98304\,a}-\frac {715\,x}{65536\,b}+\frac {20899\,b\,x^5}{32768\,a^2}+\frac {241657\,b^2\,x^7}{229376\,a^3}+\frac {143\,b^3\,x^9}{126\,a^4}+\frac {184327\,b^4\,x^{11}}{229376\,a^5}+\frac {11869\,b^5\,x^{13}}{32768\,a^6}+\frac {9295\,b^6\,x^{15}}{98304\,a^7}+\frac {715\,b^7\,x^{17}}{65536\,a^8}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {715\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{17/2}\,b^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^2)^10,x)

[Out]

((23473*x^3)/(98304*a) - (715*x)/(65536*b) + (20899*b*x^5)/(32768*a^2) + (241657*b^2*x^7)/(229376*a^3) + (143*
b^3*x^9)/(126*a^4) + (184327*b^4*x^11)/(229376*a^5) + (11869*b^5*x^13)/(32768*a^6) + (9295*b^6*x^15)/(98304*a^
7) + (715*b^7*x^17)/(65536*a^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) + (715*atan((b^(1/2)*x)/a^(1/2)))/
(65536*a^(17/2)*b^(3/2))

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sympy [A]  time = 1.03, size = 286, normalized size = 1.40 \[ - \frac {715 \sqrt {- \frac {1}{a^{17} b^{3}}} \log {\left (- a^{9} b \sqrt {- \frac {1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac {715 \sqrt {- \frac {1}{a^{17} b^{3}}} \log {\left (a^{9} b \sqrt {- \frac {1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac {- 45045 a^{8} x + 985866 a^{7} b x^{3} + 2633274 a^{6} b^{2} x^{5} + 4349826 a^{5} b^{3} x^{7} + 4685824 a^{4} b^{4} x^{9} + 3317886 a^{3} b^{5} x^{11} + 1495494 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{17} b + 37158912 a^{16} b^{2} x^{2} + 148635648 a^{15} b^{3} x^{4} + 346816512 a^{14} b^{4} x^{6} + 520224768 a^{13} b^{5} x^{8} + 520224768 a^{12} b^{6} x^{10} + 346816512 a^{11} b^{7} x^{12} + 148635648 a^{10} b^{8} x^{14} + 37158912 a^{9} b^{9} x^{16} + 4128768 a^{8} b^{10} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-715*sqrt(-1/(a**17*b**3))*log(-a**9*b*sqrt(-1/(a**17*b**3)) + x)/131072 + 715*sqrt(-1/(a**17*b**3))*log(a**9*
b*sqrt(-1/(a**17*b**3)) + x)/131072 + (-45045*a**8*x + 985866*a**7*b*x**3 + 2633274*a**6*b**2*x**5 + 4349826*a
**5*b**3*x**7 + 4685824*a**4*b**4*x**9 + 3317886*a**3*b**5*x**11 + 1495494*a**2*b**6*x**13 + 390390*a*b**7*x**
15 + 45045*b**8*x**17)/(4128768*a**17*b + 37158912*a**16*b**2*x**2 + 148635648*a**15*b**3*x**4 + 346816512*a**
14*b**4*x**6 + 520224768*a**13*b**5*x**8 + 520224768*a**12*b**6*x**10 + 346816512*a**11*b**7*x**12 + 148635648
*a**10*b**8*x**14 + 37158912*a**9*b**9*x**16 + 4128768*a**8*b**10*x**18)

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