Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]
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Rubi [A] time = 0.0199807, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{x^m}{\sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + m)*(2*a*m + b*(-1 + 2*m)*x))/(2*(a + b*x)^(3/2)),x]
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Rubi in Sympy [A] time = 4.31975, size = 10, normalized size = 0.77 \[ \frac{x^{m}}{\sqrt{a + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/2*x**(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)**(3/2),x)
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Mathematica [C] time = 0.151444, size = 100, normalized size = 7.69 \[ \frac{x^m \sqrt{a+b x} \left (2 a (m+1) \, _2F_1\left (-\frac{1}{2},m;m+1;-\frac{b x}{a}\right )-b x \left (2 m \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{b x}{a}\right )+\, _2F_1\left (\frac{3}{2},m+1;m+2;-\frac{b x}{a}\right )\right )\right )}{2 a^2 (m+1) \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + m)*(2*a*m + b*(-1 + 2*m)*x))/(2*(a + b*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.01, size = 12, normalized size = 0.9 \[{{x}^{m}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/2*x^(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.69125, size = 15, normalized size = 1.15 \[ \frac{x^{m}}{\sqrt{b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - 1)*x + 2*a*m)*x^(m - 1)/(b*x + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228145, size = 19, normalized size = 1.46 \[ \frac{x x^{m - 1}}{\sqrt{b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - 1)*x + 2*a*m)*x^(m - 1)/(b*x + a)^(3/2),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(1/2*x**(-1+m)*(2*a*m+b*(-1+2*m)*x)/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b{\left (2 \, m - 1\right )} x + 2 \, a m\right )} x^{m - 1}}{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - 1)*x + 2*a*m)*x^(m - 1)/(b*x + a)^(3/2),x, algorithm="giac")
[Out]