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

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

[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 = 0.166138, 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 \[ \frac{b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b^2 x (b B-A c)}{c^4}+\frac{b x^3 (b B-A c)}{3 c^3}-\frac{x^5 (b B-A c)}{5 c^2}+\frac{B x^7}{7 c} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{7}}{7 c} - \frac{b^{\frac{5}{2}} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{9}{2}}} - \frac{b x^{3} \left (A c - B b\right )}{3 c^{3}} + \frac{x^{5} \left (A c - B b\right )}{5 c^{2}} + \frac{\left (A c - B b\right ) \int b^{2}\, dx}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109151, size = 98, normalized size = 1. \[ \frac{b^{5/2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{9/2}}-\frac{b^2 x (b B-A c)}{c^4}+\frac{b x^3 (b B-A c)}{3 c^3}+\frac{x^5 (A c-b B)}{5 c^2}+\frac{B x^7}{7 c} \]

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 = 0.005, size = 116, normalized size = 1.2 \[{\frac{B{x}^{7}}{7\,c}}+{\frac{A{x}^{5}}{5\,c}}-{\frac{B{x}^{5}b}{5\,{c}^{2}}}-{\frac{Ab{x}^{3}}{3\,{c}^{2}}}+{\frac{B{x}^{3}{b}^{2}}{3\,{c}^{3}}}+{\frac{A{b}^{2}x}{{c}^{3}}}-{\frac{Bx{b}^{3}}{{c}^{4}}}-{\frac{{b}^{3}A}{{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{B{b}^{4}}{{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234015, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, B c^{3} x^{7} - 42 \,{\left (B b c^{2} - A c^{3}\right )} x^{5} + 70 \,{\left (B b^{2} c - A b c^{2}\right )} x^{3} - 105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 210 \,{\left (B b^{3} - A b^{2} c\right )} x}{210 \, c^{4}}, \frac{15 \, B c^{3} x^{7} - 21 \,{\left (B b c^{2} - A c^{3}\right )} x^{5} + 35 \,{\left (B b^{2} c - A b c^{2}\right )} x^{3} + 105 \,{\left (B b^{3} - A b^{2} c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - 105 \,{\left (B b^{3} - A b^{2} c\right )} x}{105 \, c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(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 - 1
05*(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(x/sqrt(b/c)
) - 105*(B*b^3 - A*b^2*c)*x)/c^4]

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Sympy [A]  time = 0.983519, size = 173, normalized size = 1.77 \[ \frac{B x^{7}}{7 c} - \frac{\sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{b^{5}}{c^{9}}} \left (- A c + B b\right )}{- A b^{2} c + B b^{3}} + x \right )}}{2} - \frac{x^{5} \left (- A c + B b\right )}{5 c^{2}} + \frac{x^{3} \left (- A b c + B b^{2}\right )}{3 c^{3}} - \frac{x \left (- A b^{2} c + B b^{3}\right )}{c^{4}} \]

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/XCAS [A]  time = 0.212885, size = 146, normalized size = 1.49 \[ \frac{{\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{4}} + \frac{15 \, B c^{6} x^{7} - 21 \, B b c^{5} x^{5} + 21 \, A c^{6} x^{5} + 35 \, B b^{2} c^{4} x^{3} - 35 \, A b c^{5} x^{3} - 105 \, B b^{3} c^{3} x + 105 \, A b^{2} c^{4} x}{105 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(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