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

Optimal. Leaf size=59 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2} \]

[Out]

(x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 2, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]
)/(c^2*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.0831748, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 2, 1 + n^(-1), -((b*x^n)/a), -((d*x^n)/c)]
)/(c^2*(1 + (b*x^n)/a)^p)

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Rubi in Sympy [A]  time = 19.9456, size = 46, normalized size = 0.78 \[ \frac{x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{n},2,- p,1 + \frac{1}{n},- \frac{d x^{n}}{c},- \frac{b x^{n}}{a} \right )}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*appellf1(1/n, 2, -p, 1 + 1/n, -d*x**n/c,
-b*x**n/a)/c**2

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Mathematica [B]  time = 0.473617, size = 180, normalized size = 3.05 \[ \frac{a c (n+1) x \left (a+b x^n\right )^p F_1\left (\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{\left (c+d x^n\right )^2 \left (b c n p x^n F_1\left (1+\frac{1}{n};1-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-2 a d n x^n F_1\left (1+\frac{1}{n};-p,3;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a c (n+1) F_1\left (\frac{1}{n};-p,2;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*(1 + n)*x*(a + b*x^n)^p*AppellF1[n^(-1), -p, 2, 1 + n^(-1), -((b*x^n)/a), -
((d*x^n)/c)])/((c + d*x^n)^2*(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, 2, 2 + n^(
-1), -((b*x^n)/a), -((d*x^n)/c)] - 2*a*d*n*x^n*AppellF1[1 + n^(-1), -p, 3, 2 + n
^(-1), -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, 2, 1 + n^(
-1), -((b*x^n)/a), -((d*x^n)/c)]))

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p/(c+d*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**p/(c+d*x**n)**2,x)

[Out]

Integral((a + b*x**n)**p/(c + d*x**n)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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