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

Optimal. Leaf size=261 \[ \frac{c \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 a (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 a (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{\left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a d (q+1)} \]

[Out]

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

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Rubi [A]  time = 1.21648, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{c \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{2 a (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^2+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 a (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{\left (d+e x^2\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^2}{d}+1\right )}{2 a d (q+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 105.931, size = 243, normalized size = 0.93 \[ \frac{c \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )} + \frac{c \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x^{2}\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{2 a \left (q + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{\left (d + e x^{2}\right )^{q + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, q + 1 \\ q + 2 \end{matrix}\middle |{1 + \frac{e x^{2}}{d}} \right )}}{2 a d \left (q + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.110124, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{x \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{ \left ( e{x}^{2}+d \right ) ^{q}}{x \left ( c{x}^{4}+b{x}^{2}+a \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q}}{c x^{5} + b x^{3} + a x}, x\right ) \]

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),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)^q/(c*x^5 + b*x^3 + a*x), 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/x/(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. \[ \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x}\,{d x} \]

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),x, algorithm="giac")

[Out]

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