3.401 \(\int \frac{\sqrt{c x}}{\left (a x+b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{a^{3/2} (1-n) \sqrt{c x}}-\frac{2 \sqrt{c x}}{a (1-n) \sqrt{a x+b x^n}} \]

[Out]

(-2*Sqrt[c*x])/(a*(1 - n)*Sqrt[a*x + b*x^n]) + (2*c*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqr
t[x])/Sqrt[a*x + b*x^n]])/(a^(3/2)*(1 - n)*Sqrt[c*x])

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Rubi [A]  time = 0.235133, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{2 c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{a^{3/2} (1-n) \sqrt{c x}}-\frac{2 \sqrt{c x}}{a (1-n) \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*x]/(a*x + b*x^n)^(3/2),x]

[Out]

(-2*Sqrt[c*x])/(a*(1 - n)*Sqrt[a*x + b*x^n]) + (2*c*Sqrt[x]*ArcTanh[(Sqrt[a]*Sqr
t[x])/Sqrt[a*x + b*x^n]])/(a^(3/2)*(1 - n)*Sqrt[c*x])

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Rubi in Sympy [A]  time = 23.5995, size = 70, normalized size = 0.82 \[ - \frac{2 \sqrt{c x}}{a \left (- n + 1\right ) \sqrt{a x + b x^{n}}} + \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{n}}} \right )}}{a^{\frac{3}{2}} \sqrt{x} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(1/2)/(a*x+b*x**n)**(3/2),x)

[Out]

-2*sqrt(c*x)/(a*(-n + 1)*sqrt(a*x + b*x**n)) + 2*sqrt(c*x)*atanh(sqrt(a)*sqrt(x)
/sqrt(a*x + b*x**n))/(a**(3/2)*sqrt(x)*(-n + 1))

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Mathematica [A]  time = 0.20224, size = 104, normalized size = 1.22 \[ \frac{2 \sqrt{c x} \left (\sqrt{a} \sqrt{x}-\sqrt{b} x^{n/2} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )\right )}{a^{3/2} (n-1) \sqrt{x} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*x]/(a*x + b*x^n)^(3/2),x]

[Out]

(2*Sqrt[c*x]*(Sqrt[a]*Sqrt[x] - Sqrt[b]*x^(n/2)*Sqrt[1 + (a*x^(1 - n))/b]*ArcSin
h[(Sqrt[a]*x^(1/2 - n/2))/Sqrt[b]]))/(a^(3/2)*(-1 + n)*Sqrt[x]*Sqrt[a*x + b*x^n]
)

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt{cx} \left ( ax+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(1/2)/(a*x+b*x^n)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (a x + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2), 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(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\left (a x + b x^{n}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(1/2)/(a*x+b*x**n)**(3/2),x)

[Out]

Integral(sqrt(c*x)/(a*x + b*x**n)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (a x + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2), x)