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

Optimal. Leaf size=146 \[ \frac{(b c-a d) (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^n}}{\sqrt{b} \sqrt{c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n}-\frac{(a d+3 b c) \sqrt{a+b x^n} \sqrt{c+d x^n}}{4 b d^2 n}+\frac{\left (a+b x^n\right )^{3/2} \sqrt{c+d x^n}}{2 b d n} \]

[Out]

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

[Out]

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

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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)

[Out]

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

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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]

[Out]

Sqrt[a + b*x^n]*Sqrt[c + d*x^n]*(-(3*b*c - a*d)/(4*b*d^2*n) + x^n/(2*d*n)) + ((b
*c - a*d)*(3*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]])/(8*b^(3/2)*d^(5/2)*n)

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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)

[Out]

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")

[Out]

Exception raised: ValueError

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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")

[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 - 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)*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)))/(sqrt(b
*d)*b*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 - a^2*d^2)*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)))/(s
qrt(-b*d)*b*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+2*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. \[ \int \frac{\sqrt{b x^{n} + a} x^{2 \, n - 1}}{\sqrt{d x^{n} + c}}\,{d x} \]

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]

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