3.723 \(\int \frac {(-x)^m}{\sqrt {a+b x}} \, dx\)

Optimal. Leaf size=48 \[ \frac {2 (-x)^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b x}{a}+1\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {67, 65} \[ \frac {2 (-x)^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b x}{a}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[(-x)^m/Sqrt[a + b*x],x]

[Out]

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

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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 48, normalized size = 1.00 \[ \frac {2 (-x)^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b x}{a}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(-x)^m/Sqrt[a + b*x],x]

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (-x\right )^{m}}{\sqrt {b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-x)^m/sqrt(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x)^m/sqrt(b*x + a), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x \right )^{m}}{\sqrt {b x +a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x)^m/sqrt(b*x + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (-x\right )}^m}{\sqrt {a+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x)^m/(a + b*x)^(1/2),x)

[Out]

int((-x)^m/(a + b*x)^(1/2), x)

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sympy [C]  time = 1.54, size = 42, normalized size = 0.88 \[ \frac {x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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