Optimal. Leaf size=60 \[ \frac{4 x^{m+1} \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \, _2F_1\left (1,\frac{1}{2} (-4 m-1);\frac{5}{2};\frac{a+b \sqrt{\frac{c}{x}}}{a}\right )}{3 a} \]
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Rubi [A] time = 0.221122, antiderivative size = 80, normalized size of antiderivative = 1.33, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{4 b^2 c (d x)^m \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \, _2F_1\left (\frac{3}{2},2 m+3;\frac{5}{2};\frac{\sqrt{\frac{c}{x}} b}{a}+1\right )}{3 a^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*Sqrt[c/x]]*(d*x)^m,x]
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Rubi in Sympy [A] time = 15.8198, size = 65, normalized size = 1.08 \[ \frac{4 b^{2} c \left (d x\right )^{m} \left (- \frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} 2 m + 3, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b \sqrt{\frac{c}{x}}}{a}} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(a+b*(c/x)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.068796, size = 78, normalized size = 1.3 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{\frac{c}{x}}} \, _2F_1\left (-\frac{1}{2},-2 (m+1);-2 m-1;-\frac{b \sqrt{\frac{c}{x}}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{\frac{c}{x}}}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*Sqrt[c/x]]*(d*x)^m,x]
[Out]
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Maple [F] time = 0.056, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(a+b*(c/x)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \sqrt{\frac{c}{x}} + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(c/x) + a)*(d*x)^m,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(b*sqrt(c/x) + a)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \sqrt{\frac{c}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(a+b*(c/x)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b \sqrt{\frac{c}{x}} + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*sqrt(c/x) + a)*(d*x)^m,x, algorithm="giac")
[Out]