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

Optimal. Leaf size=46 \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.010546, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {67, 65} \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[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] :> 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]

Rule 65

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

Rubi steps

\begin{align*} \int \frac{x^m}{(a+b x)^{3/2}} \, dx &=\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]  time = 0.0349451, size = 46, normalized size = 1. \[ -\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (-\frac{1}{2},-m;\frac{1}{2};\frac{b x}{a}+1\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[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])

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} x^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [C]  time = 1.95313, size = 36, normalized size = 0.78 \begin{align*} \frac{x x^{m} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/(b*x + a)^(3/2), x)