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

Optimal. Leaf size=273 \[ -\frac{x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c} \]

[Out]

-(((b - (b^2 - 2*a*c)/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)])/(c*(b - Sqrt[b^2 - 4*a*c])
*(1 + (e*x^2)/d)^q)) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*Ap
pellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(c*(b
+ Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) + (x*(d + e*x^2)^q*Hypergeometric2F1[1/
2, -q, 3/2, -((e*x^2)/d)])/(c*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 1.19441, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (b-\sqrt{b^2-4 a c}\right )}-\frac{x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

-(((b - (b^2 - 2*a*c)/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)])/(c*(b - Sqrt[b^2 - 4*a*c])
*(1 + (e*x^2)/d)^q)) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*x*(d + e*x^2)^q*Ap
pellF1[1/2, 1, -q, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(c*(b
+ Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) + (x*(d + e*x^2)^q*Hypergeometric2F1[1/
2, -q, 3/2, -((e*x^2)/d)])/(c*(1 + (e*x^2)/d)^q)

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Rubi in Sympy [A]  time = 150.094, size = 240, normalized size = 0.88 \[ \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{e x^{2}}{d}} \right )}}{c} - \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} - \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*hyper((-q, 1/2), (3/2,), -e*x**2/d)/c - x
*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*(-2*a*c + b**2 + b*sqrt(-4*a*c + b**2))*ap
pellf1(1/2, 1, -q, 3/2, -2*c*x**2/(b + sqrt(-4*a*c + b**2)), -e*x**2/d)/(c*(-4*a
*c + b**2 + b*sqrt(-4*a*c + b**2))) - x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*(-2
*a*c + b**2 - b*sqrt(-4*a*c + b**2))*appellf1(1/2, 1, -q, 3/2, -2*c*x**2/(b - sq
rt(-4*a*c + b**2)), -e*x**2/d)/(c*(-4*a*c + b**2 - b*sqrt(-4*a*c + b**2)))

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Mathematica [A]  time = 0.136161, size = 0, normalized size = 0. \[ \int \frac{x^4 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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