Optimal. Leaf size=73 \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]
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Rubi [A] time = 0.0910764, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x (b c-a d (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d n}-\frac{x (b c-a d)}{c d n \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)/(c + d*x^n)^2,x]
[Out]
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Rubi in Sympy [A] time = 9.66263, size = 53, normalized size = 0.73 \[ \frac{x \left (a d - b c\right )}{c d n \left (c + d x^{n}\right )} + \frac{x \left (- a d \left (- n + 1\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)/(c+d*x**n)**2,x)
[Out]
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Mathematica [A] time = 0.0790748, size = 68, normalized size = 0.93 \[ \frac{x \left (\left (c+d x^n\right ) (a d (n-1)+b c) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+c (a d-b c)\right )}{c^2 d n \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)/(c + d*x^n)^2,x]
[Out]
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Maple [F] time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)/(c+d*x^n)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (a d{\left (n - 1\right )} + b c\right )} \int \frac{1}{c d^{2} n x^{n} + c^{2} d n}\,{d x} - \frac{{\left (b c - a d\right )} x}{c d^{2} n x^{n} + c^{2} d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{n} + a}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)/(c+d*x**n)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{n} + a}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)/(d*x^n + c)^2,x, algorithm="giac")
[Out]