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

Optimal. Leaf size=210 \[ -\frac{2 c x^3 \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3 \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

[Out]

(-2*c*x^3*(d + e*x^n)^q*AppellF1[3/n, 1, -q, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2
- 4*a*c]), -((e*x^n)/d)])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^n)/d
)^q) - (2*c*x^3*(d + e*x^n)^q*AppellF1[3/n, 1, -q, (3 + n)/n, (-2*c*x^n)/(b + Sq
rt[b^2 - 4*a*c]), -((e*x^n)/d)])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 + (e*
x^n)/d)^q)

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Rubi [A]  time = 1.15095, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{2 c x^3 \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3 \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{3}{n};1,-q;\frac{n+3}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*x^3*(d + e*x^n)^q*AppellF1[3/n, 1, -q, (3 + n)/n, (-2*c*x^n)/(b - Sqrt[b^2
- 4*a*c]), -((e*x^n)/d)])/(3*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(1 + (e*x^n)/d
)^q) - (2*c*x^3*(d + e*x^n)^q*AppellF1[3/n, 1, -q, (3 + n)/n, (-2*c*x^n)/(b + Sq
rt[b^2 - 4*a*c]), -((e*x^n)/d)])/(3*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*(1 + (e*
x^n)/d)^q)

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Rubi in Sympy [A]  time = 91.3635, size = 173, normalized size = 0.82 \[ - \frac{2 c x^{3} \left (1 + \frac{e x^{n}}{d}\right )^{- q} \left (d + e x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{3}{n},1,- q,\frac{n + 3}{n},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{n}}{d} \right )}}{3 \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} - \frac{2 c x^{3} \left (1 + \frac{e x^{n}}{d}\right )^{- q} \left (d + e x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{3}{n},1,- q,\frac{n + 3}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{n}}{d} \right )}}{3 \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.150858, size = 0, normalized size = 0. \[ \int \frac{x^2 \left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( d+e{x}^{n} \right ) ^{q}}{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x^n + d)^q*x^2/(c*x^(2*n) + b*x^n + 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**2*(d+e*x**n)**q/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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