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

Optimal. Leaf size=197 \[ \frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]

[Out]

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

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Rubi [A]  time = 0.820165, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{c \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(q+1) \sqrt{b^2-4 a c} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 71.8548, size = 175, normalized size = 0.89 \[ \frac{c \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{c \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.485026, size = 261, normalized size = 1.32 \[ \frac{e 2^{-q-1} \left (d+e x^2\right )^q \left (\left (\frac{c \left (d+e x^2\right )}{-\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{-2 c e x^2-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-\left (\frac{c \left (d+e x^2\right )}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x^2}\right )^{-q} \, _2F_1\left (-q,-q;1-q;\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c e x^2+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{q \sqrt{e^2 \left (b^2-4 a c\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{x \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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