\(\int \frac {x^m}{(a+b x)^{3/2}} \, dx\) [711]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m,\frac {1}{2},1+\frac {b x}{a}\right )}{b \sqrt {a+b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {69, 67} \[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m,\frac {1}{2},\frac {b x}{a}+1\right )}{b \sqrt {a+b x}} \]

[In]

Int[x^m/(a + b*x)^(3/2),x]

[Out]

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

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 69

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-b)*(c/d))^IntPart[m]*((b*x)^FracPart[m]/(
(-d)*(x/c))^FracPart[m]), Int[((-d)*(x/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=-\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-m,\frac {1}{2},1+\frac {b x}{a}\right )}{b \sqrt {a+b x}} \]

[In]

Integrate[x^m/(a + b*x)^(3/2),x]

[Out]

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

Maple [F]

\[\int \frac {x^{m}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*x + a)*x^m/(b^2*x^2 + 2*a*b*x + a^2), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=\frac {x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {3}{2}} \Gamma \left (m + 2\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/(b*x + a)^(3/2), x)

Giac [F]

\[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=\int { \frac {x^{m}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x^m/(b*x + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{(a+b x)^{3/2}} \, dx=\int \frac {x^m}{{\left (a+b\,x\right )}^{3/2}} \,d x \]

[In]

int(x^m/(a + b*x)^(3/2),x)

[Out]

int(x^m/(a + b*x)^(3/2), x)