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

Optimal. Leaf size=220 \[ \frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \left (\frac{2 c f-b g}{\sqrt{b^2-4 a c}}+g\right ) F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b-\sqrt{b^2-4 a c}}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \left (g-\frac{2 c f-b g}{\sqrt{b^2-4 a c}}\right ) F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{\sqrt{b^2-4 a c}+b} \]

[Out]

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

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Rubi [A]  time = 1.00577, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \left (\frac{2 c f-b g}{\sqrt{b^2-4 a c}}+g\right ) F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{b-\sqrt{b^2-4 a c}}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \left (g-\frac{2 c f-b g}{\sqrt{b^2-4 a c}}\right ) F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{\sqrt{b^2-4 a c}+b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.3052, size = 199, normalized size = 0.9 \[ \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b g - 2 c f + g \sqrt{- 4 a c + b^{2}}\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} + \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b g - 2 c f - g \sqrt{- 4 a c + b^{2}}\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*(b*g - 2*c*f + g*sqrt(-4*a*c + b**2))*app
ellf1(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)) + x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*(b*g - 2
*c*f - g*sqrt(-4*a*c + b**2))*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 = 0.157136, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q \left (f+g x^2\right )}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q} \left ( g{x}^{2}+f \right ) }{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((g*x^2 + f)*(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. \[{\rm integral}\left (\frac{{\left (g x^{2} + f\right )}{\left (e x^{2} + d\right )}^{q}}{c x^{4} + b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((g*x^2 + f)*(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*(g*x**2+f)/(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. \[ \int \frac{{\left (g x^{2} + f\right )}{\left (e x^{2} + d\right )}^{q}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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