3.900 \(\int \frac{x^{-1+2 n}}{\sqrt{a+b x^n} \sqrt{c+d x^n}} \, dx\)

Optimal. Leaf size=89 \[ \frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b d n}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{3/2} d^{3/2} n} \]

[Out]

(Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b*d*n) - ((b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(3/2)*d^(3/2)*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.241543, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b d n}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{b^{3/2} d^{3/2} n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]

[Out]

(Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b*d*n) - ((b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(3/2)*d^(3/2)*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.8427, size = 75, normalized size = 0.84 \[ \frac{\sqrt{a + b x^{n}} \sqrt{c + d x^{n}}}{b d n} - \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{n}}}{\sqrt{b} \sqrt{c + d x^{n}}} \right )}}{b^{\frac{3}{2}} d^{\frac{3}{2}} n} \]

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)

[Out]

sqrt(a + b*x**n)*sqrt(c + d*x**n)/(b*d*n) - (a*d + b*c)*atanh(sqrt(d)*sqrt(a + b
*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(b**(3/2)*d**(3/2)*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.159944, size = 106, normalized size = 1.19 \[ \frac{\sqrt{a+b x^n} \sqrt{c+d x^n}}{b d n}-\frac{(a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^n} \sqrt{c+d x^n}+a d+b c+2 b d x^n\right )}{2 b^{3/2} d^{3/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]

[Out]

(Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b*d*n) - ((b*c + a*d)*Log[b*c + a*d + 2*b*d*x
^n + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c + d*x^n]])/(2*b^(3/2)*d^(3/2)*n)

_______________________________________________________________________________________

Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{{x}^{-1+2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}{\frac{1}{\sqrt{c+d{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x)

[Out]

int(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x)

_______________________________________________________________________________________

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^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.313606, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} \log \left (8 \, \sqrt{b d} b^{2} d^{2} x^{2 \, n} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} \sqrt{b d} x^{n} - 4 \,{\left (2 \, b^{2} d^{2} x^{n} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b d}\right ) + 4 \, \sqrt{b d} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{4 \, \sqrt{b d} b d n}, -\frac{{\left (b c + a d\right )} \arctan \left (\frac{2 \, \sqrt{-b d} b d x^{n} +{\left (b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{n} + a} \sqrt{d x^{n} + c} b d}\right ) - 2 \, \sqrt{-b d} \sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}{2 \, \sqrt{-b d} b d n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="fricas")

[Out]

[1/4*((b*c + a*d)*log(8*sqrt(b*d)*b^2*d^2*x^(2*n) + 8*(b^2*c*d + a*b*d^2)*sqrt(b
*d)*x^n - 4*(2*b^2*d^2*x^n + b^2*c*d + a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c)
+ (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)) + 4*sqrt(b*d)*sqrt(b*x^n + a)*sqrt(
d*x^n + c))/(sqrt(b*d)*b*d*n), -1/2*((b*c + a*d)*arctan(1/2*(2*sqrt(-b*d)*b*d*x^
n + (b*c + a*d)*sqrt(-b*d))/(sqrt(b*x^n + a)*sqrt(d*x^n + c)*b*d)) - 2*sqrt(-b*d
)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(sqrt(-b*d)*b*d*n)]

_______________________________________________________________________________________

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]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{\sqrt{b x^{n} + a} \sqrt{d x^{n} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)), x)