3.2686 \(\int (-\frac {b n x^{-1+m+n}}{2 (a+b x^n)^{3/2}}+\frac {m x^{-1+m}}{\sqrt {a+b x^n}}) \, dx\)

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

[Out]

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

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Rubi [C]  time = 0.08, antiderivative size = 126, normalized size of antiderivative = 8.40, 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 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])

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])

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*}

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Mathematica [C]  time = 0.23, size = 111, normalized size = 7.40 \[ \frac {x^m \sqrt {\frac {b x^n}{a}+1} \left (b (2 m-n) x^n \, _2F_1\left (\frac {3}{2},\frac {m+n}{n};\frac {m}{n}+2;-\frac {b x^n}{a}\right )+2 a (m+n) \, _2F_1\left (\frac {3}{2},\frac {m}{n};\frac {m+n}{n};-\frac {b x^n}{a}\right )\right )}{2 a (m+n) \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[-1/2*(b*n*x^(-1 + m + n))/(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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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

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)

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int -\frac {b n \,x^{m +n -1}}{2 \left (b \,x^{n}+a \right )^{\frac {3}{2}}}+\frac {m \,x^{m -1}}{\sqrt {b \,x^{n}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.89, size = 13, normalized size = 0.87 \[ \frac {x^{m}}{\sqrt {b x^{n} + a}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {m\,x^{m-1}}{\sqrt {a+b\,x^n}}-\frac {b\,n\,x^{m+n-1}}{2\,{\left (a+b\,x^n\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 64.20, size = 94, normalized size = 6.27 \[ \frac {m x^{m} \Gamma \left (\frac {m}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{n} \\ \frac {m}{n} + 1 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt {a} n \Gamma \left (\frac {m}{n} + 1\right )} - \frac {b x^{m} x^{n} \Gamma \left (\frac {m}{n} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {m}{n} + 1 \\ \frac {m}{n} + 2 \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {m}{n} + 2\right )} \]

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]

m*x**m*gamma(m/n)*hyper((1/2, 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*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)*gamma(m/n + 2))

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