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

Optimal. Leaf size=98 \[ \frac{x^{m-3} (b B (3-m)-A c (5-m)) \, _2F_1\left (1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{2 b^2 c (3-m)}-\frac{x^{m-3} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

[Out]

-((b*B - A*c)*x^(-3 + m))/(2*b*c*(b + c*x^2)) + ((b*B*(3 - m) - A*c*(5 - m))*x^(
-3 + m)*Hypergeometric2F1[1, (-3 + m)/2, (-1 + m)/2, -((c*x^2)/b)])/(2*b^2*c*(3
- m))

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Rubi [A]  time = 0.165398, antiderivative size = 92, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m-3} \left (\frac{b B}{c}-\frac{A (5-m)}{3-m}\right ) \, _2F_1\left (1,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{2 b^2}-\frac{x^{m-3} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*B - A*c)*x^(-3 + m))/(2*b*c*(b + c*x^2)) + (((b*B)/c - (A*(5 - m))/(3 - m))
*x^(-3 + m)*Hypergeometric2F1[1, (-3 + m)/2, (-1 + m)/2, -((c*x^2)/b)])/(2*b^2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.299578, size = 131, normalized size = 1.34 \[ \frac{x^{m-3} \left (\frac{c x^4 (2 A c-b B) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}+\frac{c x^4 (A c-b B) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}+b \left (\frac{A b}{m-3}-\frac{2 A c x^2}{m-1}+\frac{b B x^2}{m-1}\right )\right )}{b^4} \]

Antiderivative was successfully verified.

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

[Out]

(x^(-3 + m)*(b*((A*b)/(-3 + m) + (b*B*x^2)/(-1 + m) - (2*A*c*x^2)/(-1 + m)) + (c
*(-(b*B) + 2*A*c)*x^4*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((c*x^2)/b)])/
(1 + m) + (c*(-(b*B) + A*c)*x^4*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((c*
x^2)/b)])/(1 + m)))/b^4

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( B{x}^{2}+A \right ) }{ \left ( c{x}^{4}+b{x}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{m}}{{\left (c x^{4} + b x^{2}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} x^{m}}{c^{2} x^{8} + 2 \, b c x^{6} + b^{2} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*x^m/(c^2*x^8 + 2*b*c*x^6 + b^2*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m} \left (A + B x^{2}\right )}{x^{4} \left (b + c x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**m*(A + B*x**2)/(x**4*(b + c*x**2)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{m}}{{\left (c x^{4} + b x^{2}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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