3.77 \(\int \frac{x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=95 \[ -\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{7/2}}+\frac{x (9 b B-5 A c)}{8 c^3 \left (b+c x^2\right )}-\frac{b x (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac{B x}{c^3} \]

[Out]

(B*x)/c^3 - (b*(b*B - A*c)*x)/(4*c^3*(b + c*x^2)^2) + ((9*b*B - 5*A*c)*x)/(8*c^3
*(b + c*x^2)) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*Sqrt[b]*c^(7/2)
)

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Rubi [A]  time = 0.225515, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{7/2}}+\frac{x (9 b B-5 A c)}{8 c^3 \left (b+c x^2\right )}-\frac{b x (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac{B x}{c^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(B*x)/c^3 - (b*(b*B - A*c)*x)/(4*c^3*(b + c*x^2)^2) + ((9*b*B - 5*A*c)*x)/(8*c^3
*(b + c*x^2)) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*Sqrt[b]*c^(7/2)
)

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Rubi in Sympy [A]  time = 41.5798, size = 88, normalized size = 0.93 \[ \frac{B x}{c^{3}} + \frac{b x \left (A c - B b\right )}{4 c^{3} \left (b + c x^{2}\right )^{2}} - \frac{x \left (5 A c - 9 B b\right )}{8 c^{3} \left (b + c x^{2}\right )} + \frac{3 \left (A c - 5 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 \sqrt{b} c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

B*x/c**3 + b*x*(A*c - B*b)/(4*c**3*(b + c*x**2)**2) - x*(5*A*c - 9*B*b)/(8*c**3*
(b + c*x**2)) + 3*(A*c - 5*B*b)*atan(sqrt(c)*x/sqrt(b))/(8*sqrt(b)*c**(7/2))

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Mathematica [A]  time = 0.132337, size = 92, normalized size = 0.97 \[ \frac{x \left (b \left (25 B c x^2-3 A c\right )+c^2 x^2 \left (8 B x^2-5 A\right )+15 b^2 B\right )}{8 c^3 \left (b+c x^2\right )^2}-\frac{3 (5 b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 \sqrt{b} c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(x*(15*b^2*B + c^2*x^2*(-5*A + 8*B*x^2) + b*(-3*A*c + 25*B*c*x^2)))/(8*c^3*(b +
c*x^2)^2) - (3*(5*b*B - A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*Sqrt[b]*c^(7/2))

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Maple [A]  time = 0.015, size = 122, normalized size = 1.3 \[{\frac{Bx}{{c}^{3}}}-{\frac{5\,A{x}^{3}}{8\,c \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{9\,B{x}^{3}b}{8\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{3\,Abx}{8\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{7\,xB{b}^{2}}{8\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{3\,A}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{15\,Bb}{8\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

B*x/c^3-5/8/c/(c*x^2+b)^2*A*x^3+9/8/c^2/(c*x^2+b)^2*B*x^3*b-3/8/c^2/(c*x^2+b)^2*
A*x*b+7/8/c^3/(c*x^2+b)^2*x*B*b^2+3/8/c^2/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*A-
15/8/c^3/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*B*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^10/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219783, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, b c x +{\left (c x^{2} - b\right )} \sqrt{-b c}}{c x^{2} + b}\right ) - 2 \,{\left (8 \, B c^{2} x^{5} + 5 \,{\left (5 \, B b c - A c^{2}\right )} x^{3} + 3 \,{\left (5 \, B b^{2} - A b c\right )} x\right )} \sqrt{-b c}}{16 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \sqrt{-b c}}, -\frac{3 \,{\left ({\left (5 \, B b c^{2} - A c^{3}\right )} x^{4} + 5 \, B b^{3} - A b^{2} c + 2 \,{\left (5 \, B b^{2} c - A b c^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) -{\left (8 \, B c^{2} x^{5} + 5 \,{\left (5 \, B b c - A c^{2}\right )} x^{3} + 3 \,{\left (5 \, B b^{2} - A b c\right )} x\right )} \sqrt{b c}}{8 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(3*((5*B*b*c^2 - A*c^3)*x^4 + 5*B*b^3 - A*b^2*c + 2*(5*B*b^2*c - A*b*c^2)
*x^2)*log((2*b*c*x + (c*x^2 - b)*sqrt(-b*c))/(c*x^2 + b)) - 2*(8*B*c^2*x^5 + 5*(
5*B*b*c - A*c^2)*x^3 + 3*(5*B*b^2 - A*b*c)*x)*sqrt(-b*c))/((c^5*x^4 + 2*b*c^4*x^
2 + b^2*c^3)*sqrt(-b*c)), -1/8*(3*((5*B*b*c^2 - A*c^3)*x^4 + 5*B*b^3 - A*b^2*c +
2*(5*B*b^2*c - A*b*c^2)*x^2)*arctan(sqrt(b*c)*x/b) - (8*B*c^2*x^5 + 5*(5*B*b*c
- A*c^2)*x^3 + 3*(5*B*b^2 - A*b*c)*x)*sqrt(b*c))/((c^5*x^4 + 2*b*c^4*x^2 + b^2*c
^3)*sqrt(b*c))]

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Sympy [A]  time = 2.30462, size = 194, normalized size = 2.04 \[ \frac{B x}{c^{3}} + \frac{3 \sqrt{- \frac{1}{b c^{7}}} \left (- A c + 5 B b\right ) \log{\left (- \frac{3 b c^{3} \sqrt{- \frac{1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} - \frac{3 \sqrt{- \frac{1}{b c^{7}}} \left (- A c + 5 B b\right ) \log{\left (\frac{3 b c^{3} \sqrt{- \frac{1}{b c^{7}}} \left (- A c + 5 B b\right )}{- 3 A c + 15 B b} + x \right )}}{16} + \frac{x^{3} \left (- 5 A c^{2} + 9 B b c\right ) + x \left (- 3 A b c + 7 B b^{2}\right )}{8 b^{2} c^{3} + 16 b c^{4} x^{2} + 8 c^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

B*x/c**3 + 3*sqrt(-1/(b*c**7))*(-A*c + 5*B*b)*log(-3*b*c**3*sqrt(-1/(b*c**7))*(-
A*c + 5*B*b)/(-3*A*c + 15*B*b) + x)/16 - 3*sqrt(-1/(b*c**7))*(-A*c + 5*B*b)*log(
3*b*c**3*sqrt(-1/(b*c**7))*(-A*c + 5*B*b)/(-3*A*c + 15*B*b) + x)/16 + (x**3*(-5*
A*c**2 + 9*B*b*c) + x*(-3*A*b*c + 7*B*b**2))/(8*b**2*c**3 + 16*b*c**4*x**2 + 8*c
**5*x**4)

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GIAC/XCAS [A]  time = 0.21574, size = 108, normalized size = 1.14 \[ \frac{B x}{c^{3}} - \frac{3 \,{\left (5 \, B b - A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} c^{3}} + \frac{9 \, B b c x^{3} - 5 \, A c^{2} x^{3} + 7 \, B b^{2} x - 3 \, A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

B*x/c^3 - 3/8*(5*B*b - A*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^3) + 1/8*(9*B*b*c
*x^3 - 5*A*c^2*x^3 + 7*B*b^2*x - 3*A*b*c*x)/((c*x^2 + b)^2*c^3)