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

Optimal. Leaf size=194 \[ \frac{(f x)^{m+1} \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(f x)^{m+1} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (\sqrt{b^2-4 a c}+b\right )} \]

[Out]

((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f*x)^(1 + m)*Hypergeometric2F1[1, (1 + m
)/2, (3 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*f*
(1 + m)) + ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f*x)^(1 + m)*Hypergeometric2F
1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 -
4*a*c])*f*(1 + m))

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Rubi [A]  time = 0.627567, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(f x)^{m+1} \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(f x)^{m+1} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (\sqrt{b^2-4 a c}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

((e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f*x)^(1 + m)*Hypergeometric2F1[1, (1 + m
)/2, (3 + m)/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*f*
(1 + m)) + ((e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*(f*x)^(1 + m)*Hypergeometric2F
1[1, (1 + m)/2, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 -
4*a*c])*f*(1 + m))

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Rubi in Sympy [A]  time = 48.5086, size = 185, normalized size = 0.95 \[ \frac{\left (f x\right )^{m + 1} \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{f \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} - \frac{\left (f x\right )^{m + 1} \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{f \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.47532, size = 316, normalized size = 1.63 \[ \frac{d (f x)^m \text{RootSum}\left [\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right ]}{2 m}+\frac{e (f x)^m \text{RootSum}\left [\text{$\#$1}^4 c+\text{$\#$1}^2 b+a\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3 c+\text{$\#$1} b}\&\right ]}{2 m (m+1) (m+2)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(d*(f*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) + (e*(f*x)^m*RootSum[a
+ b*#1^2 + c*#1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*Hypergeometri
c2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hypergeometric2F
1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-
m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1)^m)/(b*#1 +
2*c*#1^3) & ])/(2*m*(1 + m)*(2 + m))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) }{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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