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

Optimal. Leaf size=47 \[ -\frac{c^3 (c x)^{m-3} \, _2F_1\left (2,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{b^2 (3-m)} \]

[Out]

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

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Rubi [A]  time = 0.05965, antiderivative size = 47, 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 \[ -\frac{c^3 (c x)^{m-3} \, _2F_1\left (2,\frac{m-3}{2};\frac{m-1}{2};-\frac{c x^2}{b}\right )}{b^2 (3-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.53209, size = 37, normalized size = 0.79 \[ - \frac{c^{3} \left (c x\right )^{m - 3}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} - \frac{3}{2} \\ \frac{m}{2} - \frac{1}{2} \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{b^{2} \left (- m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.131121, size = 109, normalized size = 2.32 \[ \frac{(c x)^m \left (\frac{2 c^2 x^4 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}+\frac{c^2 x^4 \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{c x^2}{b}\right )}{m+1}+b \left (\frac{b}{m-3}-\frac{2 c x^2}{m-1}\right )\right )}{b^4 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{m}}{ \left ( c{x}^{4}+b{x}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/(c*x^4+b*x^2)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*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. \[ \int \frac{\left (c x\right )^{m}}{x^{4} \left (b + c x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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