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

Optimal. Leaf size=328 \[ \frac{c x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{c x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{b \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3} \]

[Out]

(c*(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)])/(a^2*(b - Sqrt[b^2 - 4*a*c
])*(1 + (e*x^2)/d)^q) + (c*(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)])/(a
^2*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - ((d + e*x^2)^q*Hypergeometric2F1
[-3/2, -q, -1/2, -((e*x^2)/d)])/(3*a*x^3*(1 + (e*x^2)/d)^q) + (b*(d + e*x^2)^q*H
ypergeometric2F1[-1/2, -q, 1/2, -((e*x^2)/d)])/(a^2*x*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 1.31291, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{c x \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{c x \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{a^2 \left (\sqrt{b^2-4 a c}+b\right )}+\frac{b \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{1}{2},-q;\frac{1}{2};-\frac{e x^2}{d}\right )}{a^2 x}-\frac{\left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (-\frac{3}{2},-q;-\frac{1}{2};-\frac{e x^2}{d}\right )}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*(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)])/(a^2*(b - Sqrt[b^2 - 4*a*c
])*(1 + (e*x^2)/d)^q) + (c*(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)])/(a
^2*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - ((d + e*x^2)^q*Hypergeometric2F1
[-3/2, -q, -1/2, -((e*x^2)/d)])/(3*a*x^3*(1 + (e*x^2)/d)^q) + (b*(d + e*x^2)^q*H
ypergeometric2F1[-1/2, -q, 1/2, -((e*x^2)/d)])/(a^2*x*(1 + (e*x^2)/d)^q)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.295477, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^q}{x^4 \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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^4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q}}{c x^{8} + b x^{6} + a x^{4}}, x\right ) \]

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^4),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)^q/(c*x^8 + b*x^6 + a*x^4), 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**4/(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. \[ \int \frac{{\left (e x^{2} + d\right )}^{q}}{{\left (c x^{4} + b x^{2} + a\right )} x^{4}}\,{d x} \]

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^4),x, algorithm="giac")

[Out]

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