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 = 0.0062557, 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, Rules used = {12, 74} \[ \frac{x^m}{\sqrt{a+b x}} \]

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int \frac{x^{-1+m} (2 a m+b (-1+2 m) x)}{2 (a+b x)^{3/2}} \, dx &=\frac{1}{2} \int \frac{x^{-1+m} (2 a m+b (-1+2 m) x)}{(a+b x)^{3/2}} \, dx\\ &=\frac{x^m}{\sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0793584, size = 13, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x}} \]

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]

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Maple [A]  time = 0.006, size = 12, normalized size = 0.9 \begin{align*}{{x}^{m}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

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 = 1.25709, size = 15, normalized size = 1.15 \begin{align*} \frac{x^{m}}{\sqrt{b x + a}} \end{align*}

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, algorithm="maxima")

[Out]

x^m/sqrt(b*x + a)

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Fricas [A]  time = 1.96447, size = 36, normalized size = 2.77 \begin{align*} \frac{x x^{m - 1}}{\sqrt{b x + a}} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, algorithm="giac")

[Out]

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