3.309 \(\int \frac{x^m}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=163 \[ \frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b-\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b+\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

[Out]

(2*c*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*c*x^4)/(b - Sqrt[b
^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*(1 + m)) - (2*c*x^(1 +
m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*c*x^4)/(b + Sqrt[b^2 - 4*a*c]
)])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*(1 + m))

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Rubi [A]  time = 0.324964, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b-\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b+\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^4 + c*x^8),x]

[Out]

(2*c*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*c*x^4)/(b - Sqrt[b
^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a*c])*(1 + m)) - (2*c*x^(1 +
m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*c*x^4)/(b + Sqrt[b^2 - 4*a*c]
)])/(Sqrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*(1 + m))

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Rubi in Sympy [A]  time = 32.074, size = 141, normalized size = 0.87 \[ - \frac{2 c x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{2 c x^{4}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{2 c x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{2 c x^{4}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(c*x**8+b*x**4+a),x)

[Out]

-2*c*x**(m + 1)*hyper((1, m/4 + 1/4), (m/4 + 5/4,), -2*c*x**4/(b + sqrt(-4*a*c +
b**2)))/((b + sqrt(-4*a*c + b**2))*(m + 1)*sqrt(-4*a*c + b**2)) + 2*c*x**(m + 1
)*hyper((1, m/4 + 1/4), (m/4 + 5/4,), -2*c*x**4/(b - sqrt(-4*a*c + b**2)))/((b -
sqrt(-4*a*c + b**2))*(m + 1)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0835056, size = 82, normalized size = 0.5 \[ \frac{x^m \text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]}{4 m} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^m/(a + b*x^4 + c*x^8),x]

[Out]

(x^m*RootSum[a + b*#1^4 + c*#1^8 & , Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x -
#1))]/((x/(x - #1))^m*(b*#1^3 + 2*c*#1^7)) & ])/(4*m)

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{c{x}^{8}+b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(c*x^8+b*x^4+a),x)

[Out]

int(x^m/(c*x^8+b*x^4+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{c x^{8} + b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(c*x^8 + b*x^4 + a),x, algorithm="maxima")

[Out]

integrate(x^m/(c*x^8 + b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(c*x^8 + b*x^4 + a),x, algorithm="fricas")

[Out]

integral(x^m/(c*x^8 + b*x^4 + a), 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(x**m/(c*x**8+b*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{c x^{8} + b x^{4} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(c*x^8 + b*x^4 + a),x, algorithm="giac")

[Out]

integrate(x^m/(c*x^8 + b*x^4 + a), x)