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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]