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

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

[Out]

(2*(c*x)^(3/2)*Sqrt[a/x^3 + b*x^n])/(c*(3 + n)) - (2*Sqrt[a]*c*Sqrt[x]*ArcTanh[S
qrt[a]/(x^(3/2)*Sqrt[a/x^3 + b*x^n])])/((3 + n)*Sqrt[c*x])

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Rubi [A]  time = 0.328368, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{c (n+3)}-\frac{2 \sqrt{a} c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*x)^(3/2)*Sqrt[a/x^3 + b*x^n])/(c*(3 + n)) - (2*Sqrt[a]*c*Sqrt[x]*ArcTanh[S
qrt[a]/(x^(3/2)*Sqrt[a/x^3 + b*x^n])])/((3 + n)*Sqrt[c*x])

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Rubi in Sympy [A]  time = 23.6595, size = 73, normalized size = 0.86 \[ - \frac{2 \sqrt{a} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}} \right )}}{\sqrt{x} \left (n + 3\right )} + \frac{2 \left (c x\right )^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}}{c \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.153404, size = 102, normalized size = 1.2 \[ \frac{2 x \sqrt{c x} \sqrt{\frac{a}{x^3}+b x^n} \left (\sqrt{a+b x^{n+3}}-\sqrt{a} \log \left (\sqrt{a} \sqrt{a+b x^{n+3}}+a\right )+\sqrt{a} \log \left (x^{\frac{n+3}{2}}\right )\right )}{(n+3) \sqrt{a+b x^{n+3}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int \sqrt{cx}\sqrt{{\frac{a}{{x}^{3}}}+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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