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

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

[Out]

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

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Rubi [A]  time = 0.084008, 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 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.4334, size = 44, normalized size = 0.75 \[ \frac{x \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{n},1,- p,1 + \frac{1}{n},- \frac{d x^{n}}{c},- \frac{b x^{n}}{a} \right )}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.418044, size = 180, normalized size = 3.05 \[ \frac{a c (n+1) x \left (a+b x^n\right )^p F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{\left (c+d x^n\right ) \left (b c n p x^n F_1\left (1+\frac{1}{n};1-p,1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a d n x^n F_1\left (1+\frac{1}{n};-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a c (n+1) F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{c+d{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p/(c+d*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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