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