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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]