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

Optimal. Leaf size=79 \[ -\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}+\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 x^5}{10 b^2} \]

[Out]

7/2*a^2*x/b^4-7/6*a*x^3/b^3+7/10*x^5/b^2-1/2*x^7/b/(b*x^2+a)-7/2*a^(5/2)*arctan(x*b^(1/2)/a^(1/2))/b^(9/2)

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Rubi [A]  time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {7 a^2 x}{2 b^4}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}-\frac {7 a x^3}{6 b^3}-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 x^5}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^2)^2,x]

[Out]

(7*a^2*x)/(2*b^4) - (7*a*x^3)/(6*b^3) + (7*x^5)/(10*b^2) - x^7/(2*b*(a + b*x^2)) - (7*a^(5/2)*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/(2*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 )^2} \, dx &=-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 \int \frac {x^6}{a+b x^2} \, dx}{2 b}\\ &=-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 \int \left (\frac {a^2}{b^3}-\frac {a x^2}{b^2}+\frac {x^4}{b}-\frac {a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {\left (7 a^3\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^4}\\ &=\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 71, normalized size = 0.90 \[ \frac {x \left (\frac {15 a^3}{a+b x^2}+90 a^2-20 a b x^2+6 b^2 x^4\right )}{30 b^4}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^2)^2,x]

[Out]

(x*(90*a^2 - 20*a*b*x^2 + 6*b^2*x^4 + (15*a^3)/(a + b*x^2)))/(30*b^4) - (7*a^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]
)/(2*b^(9/2))

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fricas [A]  time = 0.95, size = 190, normalized size = 2.41 \[ \left [\frac {12 \, b^{3} x^{7} - 28 \, a b^{2} x^{5} + 140 \, a^{2} b x^{3} + 210 \, a^{3} x + 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} x - 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(12*b^3*x^7 - 28*a*b^2*x^5 + 140*a^2*b*x^3 + 210*a^3*x + 105*(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^5*x^2 + a*b^4), 1/30*(6*b^3*x^7 - 14*a*b^2*x^5 + 70*a^2*b*x^3 + 105*a^3*
x - 105*(a^2*b*x^2 + a^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^5*x^2 + a*b^4)]

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giac [A]  time = 0.63, size = 73, normalized size = 0.92 \[ -\frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {a^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} x^{5} - 10 \, a b^{7} x^{3} + 45 \, a^{2} b^{6} x}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/2*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/2*a^3*x/((b*x^2 + a)*b^4) + 1/15*(3*b^8*x^5 - 10*a*b^7*x^3
+ 45*a^2*b^6*x)/b^10

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maple [A]  time = 0.01, size = 68, normalized size = 0.86 \[ \frac {x^{5}}{5 b^{2}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {a^{3} x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}+\frac {3 a^{2} x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 71, normalized size = 0.90 \[ \frac {a^{3} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} x^{5} - 10 \, a b x^{3} + 45 \, a^{2} x}{15 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*x/(b^5*x^2 + a*b^4) - 7/2*a^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/15*(3*b^2*x^5 - 10*a*b*x^3 + 4
5*a^2*x)/b^4

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mupad [B]  time = 4.59, size = 66, normalized size = 0.84 \[ \frac {x^5}{5\,b^2}-\frac {2\,a\,x^3}{3\,b^3}+\frac {3\,a^2\,x}{b^4}-\frac {7\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{9/2}}+\frac {a^3\,x}{2\,\left (b^5\,x^2+a\,b^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x^2)^2,x)

[Out]

x^5/(5*b^2) - (2*a*x^3)/(3*b^3) + (3*a^2*x)/b^4 - (7*a^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(2*b^(9/2)) + (a^3*x)/
(2*(a*b^4 + b^5*x^2))

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sympy [A]  time = 0.31, size = 124, normalized size = 1.57 \[ \frac {a^{3} x}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {3 a^{2} x}{b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} + \frac {x^{5}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x/(2*a*b**4 + 2*b**5*x**2) + 3*a**2*x/b**4 - 2*a*x**3/(3*b**3) + 7*sqrt(-a**5/b**9)*log(x - b**4*sqrt(-a*
*5/b**9)/a**2)/4 - 7*sqrt(-a**5/b**9)*log(x + b**4*sqrt(-a**5/b**9)/a**2)/4 + x**5/(5*b**2)

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