3.368 \(\int \frac{x^{-1+m} (2 a m+b (-1+2 m) x)}{2 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=13 \[ \frac{x^m}{\sqrt{a+b x}} \]

[Out]

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]

[Out]

x^m/Sqrt[a + b*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)

[Out]

x**m/sqrt(a + b*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]

(x^m*Sqrt[a + b*x]*(2*a*(1 + m)*Hypergeometric2F1[-1/2, m, 1 + m, -((b*x)/a)] -
b*x*(2*m*Hypergeometric2F1[1/2, 1 + m, 2 + m, -((b*x)/a)] + Hypergeometric2F1[3/
2, 1 + m, 2 + m, -((b*x)/a)])))/(2*a^2*(1 + m)*Sqrt[1 + (b*x)/a])

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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]

x^m/(b*x+a)^(1/2)

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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]

x^m/sqrt(b*x + a)

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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]

x*x^(m - 1)/sqrt(b*x + a)

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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]

Timed 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]

integrate(1/2*(b*(2*m - 1)*x + 2*a*m)*x^(m - 1)/(b*x + a)^(3/2), x)