3.218 \(\int \frac{x^6}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=203 \[ \frac{55 x}{65536 a^6 b^3 \left (a+b x^2\right )}+\frac{55 x}{98304 a^5 b^3 \left (a+b x^2\right )^2}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{13/2} b^{7/2}}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}-\frac{x^5}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^5/(18*b*(a + b*x^2)^9) - (5*x^3)/(288*b^2*(a + b*x^2)^8) - (5*x)/(1344*b^3*(a + b*x^2)^7) + (5*x)/(16128*a*
b^3*(a + b*x^2)^6) + (11*x)/(32256*a^2*b^3*(a + b*x^2)^5) + (11*x)/(28672*a^3*b^3*(a + b*x^2)^4) + (11*x)/(245
76*a^4*b^3*(a + b*x^2)^3) + (55*x)/(98304*a^5*b^3*(a + b*x^2)^2) + (55*x)/(65536*a^6*b^3*(a + b*x^2)) + (55*Ar
cTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(13/2)*b^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.111235, antiderivative size = 203, normalized size of antiderivative = 1., 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} \[ \frac{55 x}{65536 a^6 b^3 \left (a+b x^2\right )}+\frac{55 x}{98304 a^5 b^3 \left (a+b x^2\right )^2}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{13/2} b^{7/2}}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}-\frac{x^5}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^2)^10,x]

[Out]

-x^5/(18*b*(a + b*x^2)^9) - (5*x^3)/(288*b^2*(a + b*x^2)^8) - (5*x)/(1344*b^3*(a + b*x^2)^7) + (5*x)/(16128*a*
b^3*(a + b*x^2)^6) + (11*x)/(32256*a^2*b^3*(a + b*x^2)^5) + (11*x)/(28672*a^3*b^3*(a + b*x^2)^4) + (11*x)/(245
76*a^4*b^3*(a + b*x^2)^3) + (55*x)/(98304*a^5*b^3*(a + b*x^2)^2) + (55*x)/(65536*a^6*b^3*(a + b*x^2)) + (55*Ar
cTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(13/2)*b^(7/2))

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]

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{x^6}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}+\frac{5 \int \frac{x^4}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}+\frac{5 \int \frac{x^2}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 \int \frac{1}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{55 \int \frac{1}{\left (a+b x^2\right )^6} \, dx}{16128 a b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 \int \frac{1}{\left (a+b x^2\right )^5} \, dx}{3584 a^2 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 \int \frac{1}{\left (a+b x^2\right )^4} \, dx}{4096 a^3 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{55 \int \frac{1}{\left (a+b x^2\right )^3} \, dx}{24576 a^4 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^5 b^3 \left (a+b x^2\right )^2}+\frac{55 \int \frac{1}{\left (a+b x^2\right )^2} \, dx}{32768 a^5 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^5 b^3 \left (a+b x^2\right )^2}+\frac{55 x}{65536 a^6 b^3 \left (a+b x^2\right )}+\frac{55 \int \frac{1}{a+b x^2} \, dx}{65536 a^6 b^3}\\ &=-\frac{x^5}{18 b \left (a+b x^2\right )^9}-\frac{5 x^3}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x}{1344 b^3 \left (a+b x^2\right )^7}+\frac{5 x}{16128 a b^3 \left (a+b x^2\right )^6}+\frac{11 x}{32256 a^2 b^3 \left (a+b x^2\right )^5}+\frac{11 x}{28672 a^3 b^3 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a^4 b^3 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^5 b^3 \left (a+b x^2\right )^2}+\frac{55 x}{65536 a^6 b^3 \left (a+b x^2\right )}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{13/2} b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0614697, size = 138, normalized size = 0.68 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (115038 a^2 b^6 x^{12}+255222 a^3 b^5 x^{10}+360448 a^4 b^4 x^8+334602 a^5 b^3 x^6-115038 a^6 b^2 x^4-30030 a^7 b x^2-3465 a^8+30030 a b^7 x^{14}+3465 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+3465 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{13/2} b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-3465*a^8 - 30030*a^7*b*x^2 - 115038*a^6*b^2*x^4 + 334602*a^5*b^3*x^6 + 360448*a^4*b^4*x^
8 + 255222*a^3*b^5*x^10 + 115038*a^2*b^6*x^12 + 30030*a*b^7*x^14 + 3465*b^8*x^16))/(a + b*x^2)^9 + 3465*ArcTan
[(Sqrt[b]*x)/Sqrt[a]])/(4128768*a^(13/2)*b^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 122, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{55\,{a}^{2}x}{65536\,{b}^{3}}}-{\frac{715\,a{x}^{3}}{98304\,{b}^{2}}}-{\frac{913\,{x}^{5}}{32768\,b}}+{\frac{18589\,{x}^{7}}{229376\,a}}+{\frac{11\,b{x}^{9}}{126\,{a}^{2}}}+{\frac{14179\,{b}^{2}{x}^{11}}{229376\,{a}^{3}}}+{\frac{913\,{b}^{3}{x}^{13}}{32768\,{a}^{4}}}+{\frac{715\,{b}^{4}{x}^{15}}{98304\,{a}^{5}}}+{\frac{55\,{b}^{5}{x}^{17}}{65536\,{a}^{6}}} \right ) }+{\frac{55}{65536\,{a}^{6}{b}^{3}}\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(x^6/(b*x^2+a)^10,x)

[Out]

(-55/65536*a^2*x/b^3-715/98304*a*x^3/b^2-913/32768*x^5/b+18589/229376/a*x^7+11/126*b/a^2*x^9+14179/229376*b^2/
a^3*x^11+913/32768*b^3/a^4*x^13+715/98304*b^4/a^5*x^15+55/65536/a^6*b^5*x^17)/(b*x^2+a)^9+55/65536/a^6/b^3/(a*
b)^(1/2)*arctan(b*x/(a*b)^(1/2))

________________________________________________________________________________________

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(x^6/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.34863, size = 1551, normalized size = 7.64 \begin{align*} \left [\frac{6930 \, a b^{9} x^{17} + 60060 \, a^{2} b^{8} x^{15} + 230076 \, a^{3} b^{7} x^{13} + 510444 \, a^{4} b^{6} x^{11} + 720896 \, a^{5} b^{5} x^{9} + 669204 \, a^{6} b^{4} x^{7} - 230076 \, a^{7} b^{3} x^{5} - 60060 \, a^{8} b^{2} x^{3} - 6930 \, a^{9} b x - 3465 \,{\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^{7} b^{13} x^{18} + 9 \, a^{8} b^{12} x^{16} + 36 \, a^{9} b^{11} x^{14} + 84 \, a^{10} b^{10} x^{12} + 126 \, a^{11} b^{9} x^{10} + 126 \, a^{12} b^{8} x^{8} + 84 \, a^{13} b^{7} x^{6} + 36 \, a^{14} b^{6} x^{4} + 9 \, a^{15} b^{5} x^{2} + a^{16} b^{4}\right )}}, \frac{3465 \, a b^{9} x^{17} + 30030 \, a^{2} b^{8} x^{15} + 115038 \, a^{3} b^{7} x^{13} + 255222 \, a^{4} b^{6} x^{11} + 360448 \, a^{5} b^{5} x^{9} + 334602 \, a^{6} b^{4} x^{7} - 115038 \, a^{7} b^{3} x^{5} - 30030 \, a^{8} b^{2} x^{3} - 3465 \, a^{9} b x + 3465 \,{\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^{7} b^{13} x^{18} + 9 \, a^{8} b^{12} x^{16} + 36 \, a^{9} b^{11} x^{14} + 84 \, a^{10} b^{10} x^{12} + 126 \, a^{11} b^{9} x^{10} + 126 \, a^{12} b^{8} x^{8} + 84 \, a^{13} b^{7} x^{6} + 36 \, a^{14} b^{6} x^{4} + 9 \, a^{15} b^{5} x^{2} + a^{16} b^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/8257536*(6930*a*b^9*x^17 + 60060*a^2*b^8*x^15 + 230076*a^3*b^7*x^13 + 510444*a^4*b^6*x^11 + 720896*a^5*b^5*
x^9 + 669204*a^6*b^4*x^7 - 230076*a^7*b^3*x^5 - 60060*a^8*b^2*x^3 - 6930*a^9*b*x - 3465*(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^7*b^13*x^18 + 9*a^8*b^12*x^1
6 + 36*a^9*b^11*x^14 + 84*a^10*b^10*x^12 + 126*a^11*b^9*x^10 + 126*a^12*b^8*x^8 + 84*a^13*b^7*x^6 + 36*a^14*b^
6*x^4 + 9*a^15*b^5*x^2 + a^16*b^4), 1/4128768*(3465*a*b^9*x^17 + 30030*a^2*b^8*x^15 + 115038*a^3*b^7*x^13 + 25
5222*a^4*b^6*x^11 + 360448*a^5*b^5*x^9 + 334602*a^6*b^4*x^7 - 115038*a^7*b^3*x^5 - 30030*a^8*b^2*x^3 - 3465*a^
9*b*x + 3465*(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^7*b^13*x^18 + 9*a^
8*b^12*x^16 + 36*a^9*b^11*x^14 + 84*a^10*b^10*x^12 + 126*a^11*b^9*x^10 + 126*a^12*b^8*x^8 + 84*a^13*b^7*x^6 +
36*a^14*b^6*x^4 + 9*a^15*b^5*x^2 + a^16*b^4)]

________________________________________________________________________________________

Sympy [A]  time = 6.999, size = 291, normalized size = 1.43 \begin{align*} - \frac{55 \sqrt{- \frac{1}{a^{13} b^{7}}} \log{\left (- a^{7} b^{3} \sqrt{- \frac{1}{a^{13} b^{7}}} + x \right )}}{131072} + \frac{55 \sqrt{- \frac{1}{a^{13} b^{7}}} \log{\left (a^{7} b^{3} \sqrt{- \frac{1}{a^{13} b^{7}}} + x \right )}}{131072} + \frac{- 3465 a^{8} x - 30030 a^{7} b x^{3} - 115038 a^{6} b^{2} x^{5} + 334602 a^{5} b^{3} x^{7} + 360448 a^{4} b^{4} x^{9} + 255222 a^{3} b^{5} x^{11} + 115038 a^{2} b^{6} x^{13} + 30030 a b^{7} x^{15} + 3465 b^{8} x^{17}}{4128768 a^{15} b^{3} + 37158912 a^{14} b^{4} x^{2} + 148635648 a^{13} b^{5} x^{4} + 346816512 a^{12} b^{6} x^{6} + 520224768 a^{11} b^{7} x^{8} + 520224768 a^{10} b^{8} x^{10} + 346816512 a^{9} b^{9} x^{12} + 148635648 a^{8} b^{10} x^{14} + 37158912 a^{7} b^{11} x^{16} + 4128768 a^{6} b^{12} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**2+a)**10,x)

[Out]

-55*sqrt(-1/(a**13*b**7))*log(-a**7*b**3*sqrt(-1/(a**13*b**7)) + x)/131072 + 55*sqrt(-1/(a**13*b**7))*log(a**7
*b**3*sqrt(-1/(a**13*b**7)) + x)/131072 + (-3465*a**8*x - 30030*a**7*b*x**3 - 115038*a**6*b**2*x**5 + 334602*a
**5*b**3*x**7 + 360448*a**4*b**4*x**9 + 255222*a**3*b**5*x**11 + 115038*a**2*b**6*x**13 + 30030*a*b**7*x**15 +
3465*b**8*x**17)/(4128768*a**15*b**3 + 37158912*a**14*b**4*x**2 + 148635648*a**13*b**5*x**4 + 346816512*a**12
*b**6*x**6 + 520224768*a**11*b**7*x**8 + 520224768*a**10*b**8*x**10 + 346816512*a**9*b**9*x**12 + 148635648*a*
*8*b**10*x**14 + 37158912*a**7*b**11*x**16 + 4128768*a**6*b**12*x**18)

________________________________________________________________________________________

Giac [A]  time = 2.23001, size = 173, normalized size = 0.85 \begin{align*} \frac{55 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{6} b^{3}} + \frac{3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} + 255222 \, a^{3} b^{5} x^{11} + 360448 \, a^{4} b^{4} x^{9} + 334602 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{6} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^2+a)^10,x, algorithm="giac")

[Out]

55/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^6*b^3) + 1/4128768*(3465*b^8*x^17 + 30030*a*b^7*x^15 + 115038*a^2*
b^6*x^13 + 255222*a^3*b^5*x^11 + 360448*a^4*b^4*x^9 + 334602*a^5*b^3*x^7 - 115038*a^6*b^2*x^5 - 30030*a^7*b*x^
3 - 3465*a^8*x)/((b*x^2 + a)^9*a^6*b^3)