Optimal. Leaf size=146 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}-\frac{(a d+3 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b d^2 n}+\frac{\left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{2 b d n} \]
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Rubi [A] time = 0.335273, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}-\frac{(a d+3 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b d^2 n}+\frac{\left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{2 b d n} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + 2*n)*Sqrt[a + b*x^n])/Sqrt[c + d*x^n],x]
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Rubi in Sympy [A] time = 28.4802, size = 124, normalized size = 0.85 \[ \frac{\left (a + b x^{n}\right )^{\frac{3}{2}} \sqrt{c + d x^{n}}}{2 b d n} - \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d + 3 b c\right )}{4 b d^{2} n} - \frac{\left (a d - b c\right ) \left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x^{n}}}{\sqrt{d} \sqrt{a + b x^{n}}} \right )}}{4 b^{\frac{3}{2}} d^{\frac{5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)*(a+b*x**n)**(1/2)/(c+d*x**n)**(1/2),x)
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Mathematica [A] time = 0.301757, size = 142, normalized size = 0.97 \[ \frac{(b c-a d) (a d+3 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{8 b^{3/2} d^{5/2} n}+\sqrt{a+b x^n} \sqrt{c+d x^n} \left (\frac{x^n}{2 d n}-\frac{3 b c-a d}{4 b d^2 n}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + 2*n)*Sqrt[a + b*x^n])/Sqrt[c + d*x^n],x]
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Maple [F] time = 0.085, size = 0, normalized size = 0. \[ \int{{x}^{-1+2\,n}\sqrt{a+b{x}^{n}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)*(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*n - 1)/sqrt(d*x^n + c),x, algorithm="maxima")
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Fricas [A] time = 0.311709, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, \sqrt{b d} b d x^{n} -{\left (3 \, b c - a d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} -{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} - 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b d^{2} n}, \frac{2 \,{\left (2 \, \sqrt{-b d} b d x^{n} -{\left (3 \, b c - a d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{8 \, \sqrt{-b d} b d^{2} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*n - 1)/sqrt(d*x^n + c),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+2*n)*(a+b*x**n)**(1/2)/(c+d*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*n - 1)/sqrt(d*x^n + c),x, algorithm="giac")
[Out]