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

Optimal. Leaf size=243 \[ \frac{2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

[Out]

(2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1[(1 + m)/2, 1, -q, (3 + m)/2, (-2*c*x^2
)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a
*c])*f*(1 + m)*(1 + (e*x^2)/d)^q) - (2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1[(1
+ m)/2, 1, -q, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(S
qrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*f*(1 + m)*(1 + (e*x^2)/d)^q)

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Rubi [A]  time = 1.4892, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{m+1}{2};1,-q;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1[(1 + m)/2, 1, -q, (3 + m)/2, (-2*c*x^2
)/(b - Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(Sqrt[b^2 - 4*a*c]*(b - Sqrt[b^2 - 4*a
*c])*f*(1 + m)*(1 + (e*x^2)/d)^q) - (2*c*(f*x)^(1 + m)*(d + e*x^2)^q*AppellF1[(1
+ m)/2, 1, -q, (3 + m)/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), -((e*x^2)/d)])/(S
qrt[b^2 - 4*a*c]*(b + Sqrt[b^2 - 4*a*c])*f*(1 + m)*(1 + (e*x^2)/d)^q)

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Rubi in Sympy [A]  time = 114.954, size = 202, normalized size = 0.83 \[ - \frac{2 c \left (f x\right )^{m + 1} \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},1,- q,\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{f \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{2 c \left (f x\right )^{m + 1} \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{m}{2} + \frac{1}{2},1,- q,\frac{m}{2} + \frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{f \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*(f*x)**(m + 1)*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*appellf1(m/2 + 1/2, 1,
-q, m/2 + 3/2, -2*c*x**2/(b + sqrt(-4*a*c + b**2)), -e*x**2/d)/(f*(b + sqrt(-4*a
*c + b**2))*(m + 1)*sqrt(-4*a*c + b**2)) + 2*c*(f*x)**(m + 1)*(1 + e*x**2/d)**(-
q)*(d + e*x**2)**q*appellf1(m/2 + 1/2, 1, -q, m/2 + 3/2, -2*c*x**2/(b - sqrt(-4*
a*c + b**2)), -e*x**2/d)/(f*(b - sqrt(-4*a*c + b**2))*(m + 1)*sqrt(-4*a*c + b**2
))

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Mathematica [A]  time = 0.106386, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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