3.2699 \(\int (-\frac{b n x^{-1+m+n}}{2 (a+b x^n)^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x^n}}) \, dx\)

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 [C]  time = 0.0724515, antiderivative size = 126, normalized size of antiderivative = 8.4, number of steps used = 5, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {365, 364} \[ \frac{x^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}}-\frac{b n x^{m+n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right )}{2 a (m+n) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (-\frac{b n x^{-1+m+n}}{2 \left (a+b x^n\right )^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x^n}}\right ) \, dx &=m \int \frac{x^{-1+m}}{\sqrt{a+b x^n}} \, dx-\frac{1}{2} (b n) \int \frac{x^{-1+m+n}}{\left (a+b x^n\right )^{3/2}} \, dx\\ &=\frac{\left (m \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{-1+m}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}-\frac{\left (b n \sqrt{1+\frac{b x^n}{a}}\right ) \int \frac{x^{-1+m+n}}{\left (1+\frac{b x^n}{a}\right )^{3/2}} \, dx}{2 a \sqrt{a+b x^n}}\\ &=\frac{x^m \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}}-\frac{b n x^{m+n} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};2+\frac{m}{n};-\frac{b x^n}{a}\right )}{2 a (m+n) \sqrt{a+b x^n}}\\ \end{align*}

Mathematica [C]  time = 0.0580877, 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[-(b*n*x^(-1 + m + n))/(2*(a + b*x^n)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x^n],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., size = 0, normalized size = 0. \begin{align*} \int -{\frac{bn{x}^{-1+m+n}}{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20569, 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*b*n*x^(-1+m+n)/(a+b*x^n)^(3/2)+m*x^(-1+m)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

x^m/sqrt(b*x^n + a)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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