3.366 \(\int \frac{\sqrt{a x^3+b x^n}}{(c x)^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{2 \sqrt{a} \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt{x}}-\frac{2 \sqrt{a x^3+b x^n}}{c (3-n) (c x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.258849, antiderivative size = 91, 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 \[ \frac{2 \sqrt{a} \sqrt{c x} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt{x}}-\frac{2 \sqrt{a x^3+b x^n}}{c (3-n) (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^3 + b*x^n]/(c*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.1645, size = 76, normalized size = 0.84 \[ \frac{2 \sqrt{a} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{a x^{3} + b x^{n}}} \right )}}{c^{3} \sqrt{x} \left (- n + 3\right )} - \frac{2 \sqrt{a x^{3} + b x^{n}}}{c \left (c x\right )^{\frac{3}{2}} \left (- n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**3+b*x**n)**(1/2)/(c*x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.278467, size = 103, normalized size = 1.13 \[ \frac{2 x \left (-\sqrt{a} \sqrt{b} x^{\frac{n+3}{2}} \sqrt{\frac{a x^{3-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{3}{2}-\frac{n}{2}}}{\sqrt{b}}\right )+a x^3+b x^n\right )}{(n-3) (c x)^{5/2} \sqrt{a x^3+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^3 + b*x^n]/(c*x)^(5/2),x]

[Out]

(2*x*(a*x^3 + b*x^n - Sqrt[a]*Sqrt[b]*x^((3 + n)/2)*Sqrt[1 + (a*x^(3 - n))/b]*Ar
cSinh[(Sqrt[a]*x^(3/2 - n/2))/Sqrt[b]]))/((-3 + n)*(c*x)^(5/2)*Sqrt[a*x^3 + b*x^
n])

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{1\sqrt{a{x}^{3}+b{x}^{n}} \left ( cx \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^3+b*x^n)^(1/2)/(c*x)^(5/2),x)

[Out]

int((a*x^3+b*x^n)^(1/2)/(c*x)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{3} + b x^{n}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^3 + b*x^n)/(c*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^3 + b*x^n)/(c*x)^(5/2), 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^3 + b*x^n)/(c*x)^(5/2),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((a*x**3+b*x**n)**(1/2)/(c*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{3} + b x^{n}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^3 + b*x^n)/(c*x)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*x^3 + b*x^n)/(c*x)^(5/2), x)