3.43 \(\int \frac{x^8 (A+B x^2)}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=98 \[ -\frac{b^2 x (b B-A c)}{c^4}+\frac{b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{x^5 (b B-A c)}{5 c^2}+\frac{b x^3 (b B-A c)}{3 c^3}+\frac{B x^7}{7 c} \]

[Out]

-((b^2*(b*B - A*c)*x)/c^4) + (b*(b*B - A*c)*x^3)/(3*c^3) - ((b*B - A*c)*x^5)/(5*c^2) + (B*x^7)/(7*c) + (b^(5/2
)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/2)

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Rubi [A]  time = 0.0762564, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1584, 459, 302, 205} \[ -\frac{b^2 x (b B-A c)}{c^4}+\frac{b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{x^5 (b B-A c)}{5 c^2}+\frac{b x^3 (b B-A c)}{3 c^3}+\frac{B x^7}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

-((b^2*(b*B - A*c)*x)/c^4) + (b*(b*B - A*c)*x^3)/(3*c^3) - ((b*B - A*c)*x^5)/(5*c^2) + (B*x^7)/(7*c) + (b^(5/2
)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/2)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

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]

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

Mathematica [A]  time = 0.0618947, size = 98, normalized size = 1. \[ -\frac{b^2 x (b B-A c)}{c^4}+\frac{b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}+\frac{x^5 (A c-b B)}{5 c^2}+\frac{b x^3 (b B-A c)}{3 c^3}+\frac{B x^7}{7 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4),x]

[Out]

-((b^2*(b*B - A*c)*x)/c^4) + (b*(b*B - A*c)*x^3)/(3*c^3) + ((-(b*B) + A*c)*x^5)/(5*c^2) + (B*x^7)/(7*c) + (b^(
5/2)*(b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(9/2)

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Maple [A]  time = 0.004, size = 116, normalized size = 1.2 \begin{align*}{\frac{B{x}^{7}}{7\,c}}+{\frac{A{x}^{5}}{5\,c}}-{\frac{B{x}^{5}b}{5\,{c}^{2}}}-{\frac{Ab{x}^{3}}{3\,{c}^{2}}}+{\frac{B{x}^{3}{b}^{2}}{3\,{c}^{3}}}+{\frac{A{b}^{2}x}{{c}^{3}}}-{\frac{B{b}^{3}x}{{c}^{4}}}-{\frac{{b}^{3}A}{{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B{b}^{4}}{{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(B*x^2+A)/(c*x^4+b*x^2),x)

[Out]

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

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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^8*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.556467, size = 487, normalized size = 4.97 \begin{align*} \left [\frac{30 \, B c^{3} x^{7} - 42 \,{\left (B b c^{2} - A c^{3}\right )} x^{5} + 70 \,{\left (B b^{2} c - A b c^{2}\right )} x^{3} - 105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 210 \,{\left (B b^{3} - A b^{2} c\right )} x}{210 \, c^{4}}, \frac{15 \, B c^{3} x^{7} - 21 \,{\left (B b c^{2} - A c^{3}\right )} x^{5} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{3} + 105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{c x \sqrt{\frac{b}{c}}}{b}\right ) - 105 \,{\left (B b^{3} - A b^{2} c\right )} x}{105 \, c^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/210*(30*B*c^3*x^7 - 42*(B*b*c^2 - A*c^3)*x^5 + 70*(B*b^2*c - A*b*c^2)*x^3 - 105*(B*b^3 - A*b^2*c)*sqrt(-b/c
)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) - 210*(B*b^3 - A*b^2*c)*x)/c^4, 1/105*(15*B*c^3*x^7 - 21*(B*
b*c^2 - A*c^3)*x^5 + 35*(B*b^2*c - A*b*c^2)*x^3 + 105*(B*b^3 - A*b^2*c)*sqrt(b/c)*arctan(c*x*sqrt(b/c)/b) - 10
5*(B*b^3 - A*b^2*c)*x)/c^4]

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Sympy [A]  time = 0.50813, size = 173, normalized size = 1.77 \begin{align*} \frac{B x^{7}}{7 c} - \frac{\sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} - \frac{x^{5} \left (- A c + B b\right )}{5 c^{2}} + \frac{x^{3} \left (- A b c + B b^{2}\right )}{3 c^{3}} - \frac{x \left (- A b^{2} c + B b^{3}\right )}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.22651, size = 146, normalized size = 1.49 \begin{align*} \frac{{\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{4}} + \frac{15 \, B c^{6} x^{7} - 21 \, B b c^{5} x^{5} + 21 \, A c^{6} x^{5} + 35 \, B b^{2} c^{4} x^{3} - 35 \, A b c^{5} x^{3} - 105 \, B b^{3} c^{3} x + 105 \, A b^{2} c^{4} x}{105 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

(B*b^4 - A*b^3*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^4) + 1/105*(15*B*c^6*x^7 - 21*B*b*c^5*x^5 + 21*A*c^6*x^5
+ 35*B*b^2*c^4*x^3 - 35*A*b*c^5*x^3 - 105*B*b^3*c^3*x + 105*A*b^2*c^4*x)/c^7