3.906 \(\int \frac{x^{-1+3 n}}{\sqrt{a+b x^n} \sqrt{c+d x^n}} \, dx\)

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]

(-3*(b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(4*b^2*d^2*n) + (x^n*Sqrt[a + b
*x^n]*Sqrt[c + d*x^n])/(2*b*d*n) - ((4*a*b*c*d - 3*(b*c + a*d)^2)*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(4*b^(5/2)*d^(5/2)*n)

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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]

(-3*(b*c + a*d)*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(4*b^2*d^2*n) + (x^n*Sqrt[a + b
*x^n]*Sqrt[c + d*x^n])/(2*b*d*n) - ((4*a*b*c*d - 3*(b*c + a*d)^2)*ArcTanh[(Sqrt[
d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(4*b^(5/2)*d^(5/2)*n)

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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]

x**n*sqrt(a + b*x**n)*sqrt(c + d*x**n)/(2*b*d*n) - 3*sqrt(a + b*x**n)*sqrt(c + d
*x**n)*(a*d + b*c)/(4*b**2*d**2*n) - (a*b*c*d - 3*(a*d + b*c)**2/4)*atanh(sqrt(d
)*sqrt(a + b*x**n)/(sqrt(b)*sqrt(c + d*x**n)))/(b**(5/2)*d**(5/2)*n)

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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]

(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^n]*Sqrt[c + d*x^n]*(-3*b*c - 3*a*d + 2*b*d*x^n)
+ (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*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]])/(8*b^(5/2)*d^(5/2)*n)

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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]

int(x^(-1+3*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(x^(3*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

[1/16*(4*(2*sqrt(b*d)*b*d*x^n - 3*(b*c + a*d)*sqrt(b*d))*sqrt(b*x^n + a)*sqrt(d*
x^n + c) + (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*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)*sqr
t(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)))/(sqrt
(b*d)*b^2*d^2*n), 1/8*(2*(2*sqrt(-b*d)*b*d*x^n - 3*(b*c + a*d)*sqrt(-b*d))*sqrt(
b*x^n + a)*sqrt(d*x^n + c) + (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*arctan(1/2*(2*s
qrt(-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
)))/(sqrt(-b*d)*b^2*d^2*n)]

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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+3*n)/(a+b*x**n)**(1/2)/(c+d*x**n)**(1/2),x)

[Out]

Timed out

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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]

integrate(x^(3*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)), x)