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

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)*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))

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Rubi [A]  time = 0.298859, 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, Rules used = {1285, 364} \[ \frac{(f x)^{m+1} \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(f x)^{m+1} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1) \left (\sqrt{b^2-4 a c}+b\right )} \]

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)*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))

Rule 1285

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt
[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d
- b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4*a*c,
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{(f x)^m \left (d+e x^2\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f x)^m}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx+\frac{1}{2} \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{(f x)^m}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx\\ &=\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{\left (b-\sqrt{b^2-4 a c}\right ) f (1+m)}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) (f x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) f (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.268756, size = 156, normalized size = 0.8 \[ \frac{x (f x)^m \left (\left (d \sqrt{b^2-4 a c}-2 a e+b d\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+\left (d \sqrt{b^2-4 a c}+2 a e-b d\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )\right )}{2 a (m+1) \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) }{c{x}^{4}+b{x}^{2}+a}}\, dx \end{align*}

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. \begin{align*} \int \frac{{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{c x^{4} + b x^{2} + a}, x\right ) \end{align*}

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, 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. \begin{align*} \int \frac{\left (f x\right )^{m} \left (d + e x^{2}\right )}{a + b x^{2} + c x^{4}}\, dx \end{align*}

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

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, algorithm="giac")

[Out]

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