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

Optimal. Leaf size=212 \[ \frac{2 c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.07128, antiderivative size = 212, 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 \[ \frac{2 c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{1}{n};1,-q;-\frac{1-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 91.4398, size = 168, normalized size = 0.79 \[ \frac{2 c \left (1 + \frac{e x^{n}}{d}\right )^{- q} \left (d + e x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{1}{n},1,- q,\frac{n - 1}{n},- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{n}}{d} \right )}}{x \left (- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right )} + \frac{2 c \left (1 + \frac{e x^{n}}{d}\right )^{- q} \left (d + e x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{1}{n},1,- q,\frac{n - 1}{n},- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{n}}{d} \right )}}{x \left (- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0921769, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^n\right )^q}{x^2 \left (a+b x^n+c x^{2 n}\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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