Optimal. Leaf size=60 \[ \frac{c (b c-a d) \log \left (c+d x^n\right )}{d^3 n}-\frac{x^n (b c-a d)}{d^2 n}+\frac{b x^{2 n}}{2 d n} \]
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Rubi [A] time = 0.153049, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{c (b c-a d) \log \left (c+d x^n\right )}{d^3 n}-\frac{x^n (b c-a d)}{d^2 n}+\frac{b x^{2 n}}{2 d n} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + 2*n)*(a + b*x^n))/(c + d*x^n),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b \int ^{x^{n}} x\, dx}{d n} - \frac{c \left (a d - b c\right ) \log{\left (c + d x^{n} \right )}}{d^{3} n} + \frac{\left (a d - b c\right ) \int ^{x^{n}} \frac{1}{d^{2}}\, dx}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)*(a+b*x**n)/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.0664064, size = 50, normalized size = 0.83 \[ \frac{d x^n \left (2 a d-2 b c+b d x^n\right )+2 c (b c-a d) \log \left (c+d x^n\right )}{2 d^3 n} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + 2*n)*(a + b*x^n))/(c + d*x^n),x]
[Out]
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Maple [A] time = 0.03, size = 87, normalized size = 1.5 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}a}{dn}}-{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}c}{{d}^{2}n}}+{\frac{b \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,dn}}-{\frac{c\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) a}{{d}^{2}n}}+{\frac{{c}^{2}\ln \left ( c+d{{\rm e}^{n\ln \left ( x \right ) }} \right ) b}{{d}^{3}n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)*(a+b*x^n)/(c+d*x^n),x)
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Maxima [A] time = 1.39146, size = 112, normalized size = 1.87 \[ a{\left (\frac{x^{n}}{d n} - \frac{c \log \left (\frac{d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac{1}{2} \, b{\left (\frac{2 \, c^{2} \log \left (\frac{d x^{n} + c}{d}\right )}{d^{3} n} + \frac{d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(2*n - 1)/(d*x^n + c),x, algorithm="maxima")
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Fricas [A] time = 0.233288, size = 76, normalized size = 1.27 \[ \frac{b d^{2} x^{2 \, n} - 2 \,{\left (b c d - a d^{2}\right )} x^{n} + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d x^{n} + c\right )}{2 \, d^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(2*n - 1)/(d*x^n + c),x, algorithm="fricas")
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Sympy [A] time = 53.3224, size = 105, normalized size = 1.75 \[ \begin{cases} \frac{\left (a + b\right ) \log{\left (x \right )}}{c} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\frac{a x^{2 n}}{2 n} + \frac{b x^{3 n}}{3 n}}{c} & \text{for}\: d = 0 \\\frac{\left (a + b\right ) \log{\left (x \right )}}{c + d} & \text{for}\: n = 0 \\- \frac{a c \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{2} n} + \frac{a x^{n}}{d n} + \frac{b c^{2} \log{\left (\frac{c}{d} + x^{n} \right )}}{d^{3} n} - \frac{b c x^{n}}{d^{2} n} + \frac{b x^{2 n}}{2 d n} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+2*n)*(a+b*x**n)/(c+d*x**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )} x^{2 \, n - 1}}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)*x^(2*n - 1)/(d*x^n + c),x, algorithm="giac")
[Out]