3.406 \(\int \frac{(c x)^m}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=45 \[ -\frac{c (c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0633662, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{c (c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/(b*x^2 + c*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.41014, size = 34, normalized size = 0.76 \[ - \frac{c \left (c x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} - \frac{1}{2} \\ \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{b \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0712378, size = 56, normalized size = 1.24 \[ \frac{(c x)^m \left (\frac{b}{m-1}-\frac{c x^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}\right )}{b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m}}{c{x}^{4}+b{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(c*x^4+b*x^2),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m/(c*x^4 + b*x^2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x)^m/(c*x^4 + b*x^2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x)^m/(c*x^4 + b*x^2), x)