Optimal. Leaf size=78 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]
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Rubi [A] time = 0.197142, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{1}{3} (2 m+5);-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{b (c x)^{3/2}}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[a + b*(c*x)^(3/2)],x]
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Rubi in Sympy [A] time = 13.8224, size = 80, normalized size = 1.03 \[ \frac{\left (c x\right )^{- m} \left (c x\right )^{m + 1} \left (d x\right )^{m} \sqrt{a + b \left (c x\right )^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2 m}{3} + \frac{2}{3} \\ \frac{2 m}{3} + \frac{5}{3} \end{matrix}\middle |{- \frac{b \left (c x\right )^{\frac{3}{2}}}{a}} \right )}}{c \sqrt{1 + \frac{b \left (c x\right )^{\frac{3}{2}}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(a+b*(c*x)**(3/2))**(1/2),x)
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Mathematica [A] time = 0.0873848, size = 79, normalized size = 1.01 \[ \frac{x (d x)^m \sqrt{a+b (c x)^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{2 (m+1)}{3};\frac{2 (m+1)}{3}+1;-\frac{b (c x)^{3/2}}{a}\right )}{(m+1) \sqrt{\frac{a+b (c x)^{3/2}}{a}}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*Sqrt[a + b*(c*x)^(3/2)],x]
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Maple [F] time = 0.067, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b \left ( cx \right ) ^{{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(a+b*(c*x)^(3/2))^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^(3/2)*b + a)*(d*x)^m,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(sqrt((c*x)^(3/2)*b + a)*(d*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d x\right )^{m} \sqrt{a + b \left (c x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**m*(a+b*(c*x)**(3/2))**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (c x\right )^{\frac{3}{2}} b + a} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((c*x)^(3/2)*b + a)*(d*x)^m,x, algorithm="giac")
[Out]