Optimal. Leaf size=84 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]
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Rubi [A] time = 0.300232, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[a + b/Sqrt[c*x^3]],x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{c x^{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(a+b/(c*x**3)**(1/2))**(1/2),x)
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Mathematica [A] time = 0.107233, size = 89, normalized size = 1.06 \[ \frac{4 x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},\frac{2 m}{3}+\frac{1}{6};\frac{2 m}{3}+\frac{7}{6};-\frac{a \sqrt{c x^3}}{b}\right )}{(4 m+1) \sqrt{\frac{a \sqrt{c x^3}}{b}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*Sqrt[a + b/Sqrt[c*x^3]],x]
[Out]
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Maple [F] time = 0.067, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{c{x}^{3}}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(a+b/(c*x^3)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{c x^{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m*sqrt(a + b/sqrt(c*x^3)),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((d*x)^m*sqrt(a + b/sqrt(c*x^3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{c x^{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(a+b/(c*x**3)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + \frac{b}{\sqrt{c x^{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m*sqrt(a + b/sqrt(c*x^3)),x, algorithm="giac")
[Out]