Optimal. Leaf size=150 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}-\frac{3 (a d+b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.440354, antiderivative size = 150, 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 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n}-\frac{3 (a d+b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b^2 d^2 n}+\frac{x^n \sqrt{a+b x^n} \sqrt{c+d x^n}}{2 b d n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 3*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 32.2895, size = 131, normalized size = 0.87 \[ \frac{x^{n} \sqrt{a + b x^{n}} \sqrt{c + d x^{n}}}{2 b d n} - \frac{3 \sqrt{a + b x^{n}} \sqrt{c + d x^{n}} \left (a d + b c\right )}{4 b^{2} d^{2} n} - \frac{\left (a b c d - \frac{3 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{b^{\frac{5}{2}} d^{\frac{5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+3*n)/(a+b*x**n)**(1/2)/(c+d*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.325203, size = 141, normalized size = 0.94 \[ \frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )+2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n} \left (-3 a d-3 b c+2 b d x^n\right )}{8 b^{5/2} d^{5/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 3*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.083, size = 0, normalized size = 0. \[ \int{{x}^{-1+3\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x)
[Out]
_______________________________________________________________________________________
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(x^(3*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.362179, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, \sqrt{b d} b d x^{n} - 3 \,{\left (b c + a d\right )} \sqrt{b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} + 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right )}{16 \, \sqrt{b d} b^{2} d^{2} n}, \frac{2 \,{\left (2 \, \sqrt{-b d} b d x^{n} - 3 \,{\left (b c + a d\right )} \sqrt{-b d}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right )}{8 \, \sqrt{-b d} b^{2} d^{2} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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+3*n)/(a+b*x**n)**(1/2)/(c+d*x**n)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3 \, n - 1}}{\sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="giac")
[Out]