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

Optimal. Leaf size=87 \[ \frac{2 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c (1-n) \sqrt{c x}}-\frac{2 \sqrt{a x+b x^n}}{c (1-n) \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.234596, antiderivative size = 87, 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 \[ \frac{2 \sqrt{a} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{c (1-n) \sqrt{c x}}-\frac{2 \sqrt{a x+b x^n}}{c (1-n) \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.6288, size = 73, normalized size = 0.84 \[ \frac{2 \sqrt{a} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{n}}} \right )}}{c^{2} \sqrt{x} \left (- n + 1\right )} - \frac{2 \sqrt{a x + b x^{n}}}{c \sqrt{c x} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.250894, size = 100, normalized size = 1.15 \[ \frac{x \left (-2 \sqrt{a} \sqrt{b} x^{\frac{n+1}{2}} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+2 a x+2 b x^n\right )}{(n-1) (c x)^{3/2} \sqrt{a x+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1\sqrt{ax+b{x}^{n}} \left ( cx \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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