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

Optimal. Leaf size=71 \[ \frac{x^{m-1} (b B-A c) \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b c (1-m)}-\frac{B x^{m-1}}{c (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.119387, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m-1} (b B-A c) \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b c (1-m)}-\frac{B x^{m-1}}{c (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 92.699, size = 51, normalized size = 0.72 \[ - \frac{A x^{m - 1}}{b \left (- m + 1\right )} - \frac{x^{m + 1} \left (A c - B b\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{b^{2} \left (m + 1\right )} \]

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),x)

[Out]

-A*x**(m - 1)/(b*(-m + 1)) - x**(m + 1)*(A*c - B*b)*hyper((1, m/2 + 1/2), (m/2 +
3/2,), -c*x**2/b)/(b**2*(m + 1))

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Mathematica [A]  time = 0.0953437, size = 60, normalized size = 0.85 \[ \frac{x^{m-1} \left (\frac{x^2 (b B-A c) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}+\frac{A b}{m-1}\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( B{x}^{2}+A \right ) }{c{x}^{4}+b{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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