3.393 \(\int \frac{1}{\sqrt{c x} \sqrt{a x+b x^n}} \, dx\)

Optimal. Leaf size=51 \[ \frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{\sqrt{a} (1-n) \sqrt{c x}} \]

[Out]

(2*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b*x^n]])/(Sqrt[a]*(1 - n)*Sqrt[c
*x])

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Rubi [A]  time = 0.148136, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{\sqrt{a} (1-n) \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[a*x + b*x^n]])/(Sqrt[a]*(1 - n)*Sqrt[c
*x])

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Rubi in Sympy [A]  time = 15.6798, size = 46, normalized size = 0.9 \[ \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{n}}} \right )}}{\sqrt{a} c \sqrt{x} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(1/2)/(a*x+b*x**n)**(1/2),x)

[Out]

2*sqrt(c*x)*atanh(sqrt(a)*sqrt(x)/sqrt(a*x + b*x**n))/(sqrt(a)*c*sqrt(x)*(-n + 1
))

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Mathematica [A]  time = 0.139286, size = 87, normalized size = 1.71 \[ -\frac{2 \sqrt{b} x^{\frac{n+1}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )}{\sqrt{a} (n-1) \sqrt{c x} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b]*x^((1 + n)/2)*Sqrt[1 + (a*x^(1 - n))/b]*ArcSinh[(Sqrt[a]*x^(1/2 - n/
2))/Sqrt[b]])/(Sqrt[a]*(-1 + n)*Sqrt[c*x]*Sqrt[a*x + b*x^n])

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{ax+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(1/2)/(a*x+b*x**n)**(1/2),x)

[Out]

Integral(1/(sqrt(c*x)*sqrt(a*x + b*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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