3.10.42 \(\int \frac {x^{-1+m} (2 a m+b (2 m-n) x^n)}{2 (a+b x^n)^{3/2}} \, dx\) [942]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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, 460} \begin {gather*} \frac {x^m}{\sqrt {a+b x^n}} \end {gather*}

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 460

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*}

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Mathematica [A]
time = 0.17, size = 15, normalized size = 1.00 \begin {gather*} \frac {x^m}{\sqrt {a+b x^n}} \end {gather*}

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[a + b*x^n]

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+m} \left (2 a m +b \left (2 m -n \right ) x^{n}\right )}{2 \left (a +b \,x^{n}\right )^{\frac {3}{2}}}\, dx\]

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 = 0.35, size = 13, normalized size = 0.87 \begin {gather*} \frac {x^{m}}{\sqrt {b x^{n} + a}} \end {gather*}

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 = 0.33, size = 16, normalized size = 1.07 \begin {gather*} \frac {x x^{m - 1}}{\sqrt {b x^{n} + a}} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 65.65, size = 100, normalized size = 6.67 \begin {gather*} \frac {m x^{m} \Gamma \left (\frac {m}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} \\ \frac {m}{n} + 1 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 1\right )} + \frac {b x^{m} x^{n} \left (2 m - n\right ) \Gamma \left (\frac {m}{n} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} + 1 \\ \frac {m}{n} + 2 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} n \Gamma \left (\frac {m}{n} + 2\right )} \end {gather*}

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]

m*x**m*gamma(m/n)*hyper((3/2, m/n), (m/n + 1,), b*x**n*exp_polar(I*pi)/a)/(sqrt(a)*n*gamma(m/n + 1)) + b*x**m*
x**n*(2*m - n)*gamma(m/n + 1)*hyper((3/2, m/n + 1), (m/n + 2,), b*x**n*exp_polar(I*pi)/a)/(2*a**(3/2)*n*gamma(
m/n + 2))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 3.68, size = 13, normalized size = 0.87 \begin {gather*} \frac {x^m}{\sqrt {a+b\,x^n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^m/(a + b*x^n)^(1/2)

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