3.183 \(\int \frac {x^8}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=85 \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {35 a x}{8 b^4}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {35 x^3}{24 b^3} \]

[Out]

-35/8*a*x/b^4+35/24*x^3/b^3-1/4*x^7/b/(b*x^2+a)^2-7/8*x^5/b^2/(b*x^2+a)+35/8*a^(3/2)*arctan(x*b^(1/2)/a^(1/2))
/b^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac {35 a x}{8 b^4}-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {35 x^3}{24 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^2)^3,x]

[Out]

(-35*a*x)/(8*b^4) + (35*x^3)/(24*b^3) - x^7/(4*b*(a + b*x^2)^2) - (7*x^5)/(8*b^2*(a + b*x^2)) + (35*a^(3/2)*Ar
cTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(9/2))

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]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin {align*} \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {7 \int \frac {x^6}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 \int \frac {x^4}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=-\frac {35 a x}{8 b^4}+\frac {35 x^3}{24 b^3}-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {\left (35 a^2\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^4}\\ &=-\frac {35 a x}{8 b^4}+\frac {35 x^3}{24 b^3}-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 77, normalized size = 0.91 \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {105 a^3 x+175 a^2 b x^3+56 a b^2 x^5-8 b^3 x^7}{24 b^4 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^2)^3,x]

[Out]

-1/24*(105*a^3*x + 175*a^2*b*x^3 + 56*a*b^2*x^5 - 8*b^3*x^7)/(b^4*(a + b*x^2)^2) + (35*a^(3/2)*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(8*b^(9/2))

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 230, normalized size = 2.71 \[ \left [\frac {16 \, b^{3} x^{7} - 112 \, a b^{2} x^{5} - 350 \, a^{2} b x^{3} - 210 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{48 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {8 \, b^{3} x^{7} - 56 \, a b^{2} x^{5} - 175 \, a^{2} b x^{3} - 105 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{24 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/48*(16*b^3*x^7 - 112*a*b^2*x^5 - 350*a^2*b*x^3 - 210*a^3*x + 105*(a*b^2*x^4 + 2*a^2*b*x^2 + a^3)*sqrt(-a/b)
*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4), 1/24*(8*b^3*x^7 - 56*a*b^
2*x^5 - 175*a^2*b*x^3 - 105*a^3*x + 105*(a*b^2*x^4 + 2*a^2*b*x^2 + a^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^
6*x^4 + 2*a*b^5*x^2 + a^2*b^4)]

________________________________________________________________________________________

giac [A]  time = 0.63, size = 73, normalized size = 0.86 \[ \frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{4}} + \frac {b^{6} x^{3} - 9 \, a b^{5} x}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^3,x, algorithm="giac")

[Out]

35/8*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) - 1/8*(13*a^2*b*x^3 + 11*a^3*x)/((b*x^2 + a)^2*b^4) + 1/3*(b^6*
x^3 - 9*a*b^5*x)/b^9

________________________________________________________________________________________

maple [A]  time = 0.01, size = 77, normalized size = 0.91 \[ -\frac {13 a^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {11 a^{3} x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {x^{3}}{3 b^{3}}+\frac {35 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {3 a x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^2+a)^3,x)

[Out]

1/3*x^3/b^3-3*a*x/b^4-13/8/b^3*a^2/(b*x^2+a)^2*x^3-11/8/b^4*a^3/(b*x^2+a)^2*x+35/8/b^4*a^2/(a*b)^(1/2)*arctan(
1/(a*b)^(1/2)*b*x)

________________________________________________________________________________________

maxima [A]  time = 2.97, size = 82, normalized size = 0.96 \[ -\frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} + \frac {b x^{3} - 9 \, a x}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*(13*a^2*b*x^3 + 11*a^3*x)/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4) + 35/8*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b
^4) + 1/3*(b*x^3 - 9*a*x)/b^4

________________________________________________________________________________________

mupad [B]  time = 4.72, size = 77, normalized size = 0.91 \[ \frac {x^3}{3\,b^3}-\frac {\frac {11\,a^3\,x}{8}+\frac {13\,b\,a^2\,x^3}{8}}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{9/2}}-\frac {3\,a\,x}{b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x^2)^3,x)

[Out]

x^3/(3*b^3) - ((11*a^3*x)/8 + (13*a^2*b*x^3)/8)/(a^2*b^4 + b^6*x^4 + 2*a*b^5*x^2) + (35*a^(3/2)*atan((b^(1/2)*
x)/a^(1/2)))/(8*b^(9/2)) - (3*a*x)/b^4

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 133, normalized size = 1.56 \[ - \frac {3 a x}{b^{4}} - \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {- 11 a^{3} x - 13 a^{2} b x^{3}}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} + \frac {x^{3}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**2+a)**3,x)

[Out]

-3*a*x/b**4 - 35*sqrt(-a**3/b**9)*log(x - b**4*sqrt(-a**3/b**9)/a)/16 + 35*sqrt(-a**3/b**9)*log(x + b**4*sqrt(
-a**3/b**9)/a)/16 + (-11*a**3*x - 13*a**2*b*x**3)/(8*a**2*b**4 + 16*a*b**5*x**2 + 8*b**6*x**4) + x**3/(3*b**3)

________________________________________________________________________________________