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

Optimal. Leaf size=190 \[ -\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

[Out]

(-2*c*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c
]), -((e*x^2)/d)])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - (2*
c*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]),
-((e*x^2)/d)])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 0.715909, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b - Sqrt[b^2 - 4*a*c
]), -((e*x^2)/d)])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - (2*
c*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]),
-((e*x^2)/d)])/((b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q)

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Rubi in Sympy [A]  time = 61.7594, size = 163, normalized size = 0.86 \[ - \frac{2 c x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} - \frac{2 c x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*appellf1(1/2, 1, -q, 3/2, -2*c*x**2/
(b + sqrt(-4*a*c + b**2)), -e*x**2/d)/(-4*a*c + b**2 + b*sqrt(-4*a*c + b**2)) -
2*c*x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*appellf1(1/2, 1, -q, 3/2, -2*c*x**2/(
b - sqrt(-4*a*c + b**2)), -e*x**2/d)/(-4*a*c + b**2 - b*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.0482326, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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