3.709 \(\int x^m \sqrt {a+b x} \, dx\)

Optimal. Leaf size=48 \[ \frac {2 x^m (a+b x)^{3/2} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};\frac {b x}{a}+1\right )}{3 b} \]

[Out]

2/3*x^m*(b*x+a)^(3/2)*hypergeom([3/2, -m],[5/2],1+b*x/a)/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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {67, 65} \[ \frac {2 x^m (a+b x)^{3/2} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};\frac {b x}{a}+1\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sqrt[a + b*x],x]

[Out]

(2*x^m*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, -m, 5/2, 1 + (b*x)/a])/(3*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 x^m \sqrt {a+b x} \, dx &=\left (x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \left (-\frac {b x}{a}\right )^m \sqrt {a+b x} \, dx\\ &=\frac {2 x^m \left (-\frac {b x}{a}\right )^{-m} (a+b x)^{3/2} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};1+\frac {b x}{a}\right )}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 1.00 \[ \frac {2 x^m (a+b x)^{3/2} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {3}{2},-m;\frac {5}{2};\frac {b x}{a}+1\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sqrt[a + b*x],x]

[Out]

(2*x^m*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, -m, 5/2, 1 + (b*x)/a])/(3*b*(-((b*x)/a))^m)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x + a} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x + a)*x^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x + a)*x^m, x)

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

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 \[ \int \sqrt {b x + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x + a)*x^m, x)

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

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 = 1.68, size = 37, normalized size = 0.77 \[ \frac {\sqrt {a} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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