3.1108 \(\int \frac{(d x)^m}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.514942, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.4358, size = 148, normalized size = 0.86 \[ - \frac{2 c \left (d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{d \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{2 c \left (d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{d \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0883319, size = 82, normalized size = 0.47 \[ \frac{(d x)^m \text{RootSum}\left [\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right ]}{2 m} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^4+b*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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