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

Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]

[Out]

x^m/Sqrt[a + b*x^n]

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Rubi [A]  time = 0.0181481, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {12, 449} \[ \frac{x^m}{\sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^m/Sqrt[a + b*x^n]

Rule 12

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

Rule 449

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

Rubi steps

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

Mathematica [C]  time = 0.177732, size = 111, normalized size = 7.4 \[ \frac{x^m \sqrt{\frac{b x^n}{a}+1} \left (b (2 m-n) x^n \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right )+2 a (m+n) \, _2F_1\left (\frac{3}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )\right )}{2 a (m+n) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^m*Sqrt[1 + (b*x^n)/a]*(2*a*(m + n)*Hypergeometric2F1[3/2, m/n, (m + n)/n, -((b*x^n)/a)] + b*(2*m - n)*x^n*H
ypergeometric2F1[3/2, (m + n)/n, 2 + m/n, -((b*x^n)/a)]))/(2*a*(m + n)*Sqrt[a + b*x^n])

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{-1+m} \left ( 2\,am+b \left ( 2\,m-n \right ){x}^{n} \right ) }{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.24732, size = 18, normalized size = 1.2 \begin{align*} \frac{x^{m}}{\sqrt{b x^{n} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x^(-1+m)*(2*a*m+b*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x, algorithm="maxima")

[Out]

x^m/sqrt(b*x^n + a)

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Fricas [A]  time = 1.53544, size = 39, normalized size = 2.6 \begin{align*} \frac{x x^{m - 1}}{\sqrt{b x^{n} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*x^(-1+m)*(2*a*m+b*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x, algorithm="fricas")

[Out]

x*x^(m - 1)/sqrt(b*x^n + 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*(2*m-n)*x**n)/(a+b*x**n)**(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 - n\right )} x^{n} + 2 \, a m\right )} x^{m - 1}}{2 \,{\left (b x^{n} + 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*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x, algorithm="giac")

[Out]

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