3.789 \(\int \frac{(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=44 \[ \frac{(d x)^{m+1} \, _2F_1\left (4,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^4 d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0519988, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(d x)^{m+1} \, _2F_1\left (4,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^4 d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.6726, size = 34, normalized size = 0.77 \[ \frac{\left (d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 4, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a^{4} d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*hyper((4, m/2 + 1/2), (m/2 + 3/2,), -b*x**2/a)/(a**4*d*(m + 1))

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Mathematica [A]  time = 0.0365097, size = 42, normalized size = 0.95 \[ \frac{x (d x)^m \, _2F_1\left (4,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{a^4 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m/(b^4*x^8 + 4*a*b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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