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 = 0.140995, 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, Rules used = {1375, 364} \[ \frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b-\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 c x^4}{b+\sqrt{b^2-4 a c}}\right )}{(m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

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))

Rule 1375

Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]
}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{x^m}{a+b x^4+c x^8} \, dx &=\frac{c \int \frac{x^m}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{\sqrt{b^2-4 a c}}-\frac{c \int \frac{x^m}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{2 c x^{1+m} \, _2F_1\left (1,\frac{1+m}{4};\frac{5+m}{4};-\frac{2 c x^4}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) (1+m)}-\frac{2 c x^{1+m} \, _2F_1\left (1,\frac{1+m}{4};\frac{5+m}{4};-\frac{2 c x^4}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) (1+m)}\\ \end{align*}

Mathematica [C]  time = 0.0435743, size = 58, normalized size = 0.36 \[ \frac{x^{m+1} \text{RootSum}\left [\text{$\#$1}^4 b+\text{$\#$1}^8 c+a\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^4 b+2 a}\& \right ]}{4 (m+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^m/(a + b*x^4 + c*x^8),x]

[Out]

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

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Maple [F]  time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{c{x}^{8}+b{x}^{4}+a}}\, dx \end{align*}

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. \begin{align*} \int \frac{x^{m}}{c x^{8} + b x^{4} + a}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left (\frac{x^{m}}{c x^{8} + b x^{4} + a}, x\right ) \end{align*}

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. \begin{align*} \text{Timed out} \end{align*}

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

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)