3.399 \(\int \frac{x^6 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=339 \[ \frac{x \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{b x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c^2}+\frac{x^3 \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{3}{2},-q;\frac{5}{2};-\frac{e x^2}{d}\right )}{3 c} \]

[Out]

((b^2 - a*c - (b*(b^2 - 3*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)])/(c^2*(b - Sqrt[b
^2 - 4*a*c])*(1 + (e*x^2)/d)^q) + ((b^2 - a*c + (b*(b^2 - 3*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)])/(c^2*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - (b*x*(d + e*
x^2)^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(c^2*(1 + (e*x^2)/d)^q) +
(x^3*(d + e*x^2)^q*Hypergeometric2F1[3/2, -q, 5/2, -((e*x^2)/d)])/(3*c*(1 + (e*x
^2)/d)^q)

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Rubi [A]  time = 1.41638, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{x \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{b x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c^2}+\frac{x^3 \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{3}{2},-q;\frac{5}{2};-\frac{e x^2}{d}\right )}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]

[Out]

((b^2 - a*c - (b*(b^2 - 3*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)])/(c^2*(b - Sqrt[b
^2 - 4*a*c])*(1 + (e*x^2)/d)^q) + ((b^2 - a*c + (b*(b^2 - 3*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)])/(c^2*(b + Sqrt[b^2 - 4*a*c])*(1 + (e*x^2)/d)^q) - (b*x*(d + e*
x^2)^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(c^2*(1 + (e*x^2)/d)^q) +
(x^3*(d + e*x^2)^q*Hypergeometric2F1[3/2, -q, 5/2, -((e*x^2)/d)])/(3*c*(1 + (e*x
^2)/d)^q)

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Rubi in Sympy [A]  time = 163.142, size = 308, normalized size = 0.91 \[ - \frac{b x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{e x^{2}}{d}} \right )}}{c^{2}} + \frac{x^{3} \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{e x^{2}}{d}} \right )}}{3 c} + \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b \left (- 3 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c^{2} \left (b + \sqrt{- 4 a c + b^{2}}\right ) \sqrt{- 4 a c + b^{2}}} - \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b \left (- 3 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c^{2} \left (b - \sqrt{- 4 a c + b^{2}}\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(e*x**2+d)**q/(c*x**4+b*x**2+a),x)

[Out]

-b*x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*hyper((-q, 1/2), (3/2,), -e*x**2/d)/c*
*2 + x**3*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*hyper((-q, 3/2), (5/2,), -e*x**2/
d)/(3*c) + x*(1 + e*x**2/d)**(-q)*(d + e*x**2)**q*(b*(-3*a*c + b**2) + sqrt(-4*a
*c + b**2)*(-a*c + b**2))*appellf1(1/2, 1, -q, 3/2, -2*c*x**2/(b + sqrt(-4*a*c +
b**2)), -e*x**2/d)/(c**2*(b + sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2)) - x*(1
+ e*x**2/d)**(-q)*(d + e*x**2)**q*(b*(-3*a*c + b**2) - sqrt(-4*a*c + b**2)*(-a*c
+ b**2))*appellf1(1/2, 1, -q, 3/2, -2*c*x**2/(b - sqrt(-4*a*c + b**2)), -e*x**2
/d)/(c**2*(b - sqrt(-4*a*c + b**2))*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.31375, size = 0, normalized size = 0. \[ \int \frac{x^6 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]

[Out]

Integrate[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x]

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{{x}^{6} \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a), x)