Optimal. Leaf size=122 \[ -\frac{2 a^{3/2} c^4 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}}+\frac{2 a c^2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{n+3}+\frac{2 (c x)^{9/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \]
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Rubi [A] time = 0.42938, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 a^{3/2} c^4 \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a}}{x^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}\right )}{(n+3) \sqrt{c x}}+\frac{2 a c^2 (c x)^{3/2} \sqrt{\frac{a}{x^3}+b x^n}}{n+3}+\frac{2 (c x)^{9/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]
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Rubi in Sympy [A] time = 33.1237, size = 109, normalized size = 0.89 \[ - \frac{2 a^{\frac{3}{2}} c^{3} \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}} \right )}}{\sqrt{x} \left (n + 3\right )} + \frac{2 a c^{2} \left (c x\right )^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}} + b x^{n}}}{n + 3} + \frac{2 \left (c x\right )^{\frac{9}{2}} \left (\frac{a}{x^{3}} + b x^{n}\right )^{\frac{3}{2}}}{3 c \left (n + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(7/2)*(a/x**3+b*x**n)**(3/2),x)
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Mathematica [A] time = 0.432136, size = 0, normalized size = 0. \[ \int (c x)^{7/2} \left (\frac{a}{x^3}+b x^n\right )^{3/2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]
[Out]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{7}{2}}} \left ({\frac{a}{{x}^{3}}}+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),x, algorithm="maxima")
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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((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),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((c*x)**(7/2)*(a/x**3+b*x**n)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2),x, algorithm="giac")
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