3.8.46 \(\int \sqrt {x} (a+b x)^n \, dx\) [746]

Optimal. Leaf size=45 \[ \frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, 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 = {68, 66} \begin {gather*} \frac {2}{3} x^{3/2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x)^n,x]

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x)^n,x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 5.18, size = 26, normalized size = 0.58 \begin {gather*} \frac {2 x^{\frac {3}{2}} a^n \text {hyper}\left [\left \{\frac {3}{2},-n\right \},\left \{\frac {5}{2}\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(1/2)*(a + b*x)^n,x]')

[Out]

2 x ^ (3 / 2) a ^ n hyper[{3 / 2, -n}, {5 / 2}, b x exp_polar[I Pi] / a] / 3

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {x}\, \left (b x +a \right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*sqrt(x), x)

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Fricas [F]
time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n*sqrt(x), x)

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Sympy [C] Result contains complex when optimal does not.
time = 3.77, size = 27, normalized size = 0.60 \begin {gather*} \frac {2 a^{n} x^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - n \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**n*x**(3/2)*hyper((3/2, -n), (5/2,), b*x*exp_polar(I*pi)/a)/3

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {x}\,{\left (a+b\,x\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x)^n,x)

[Out]

int(x^(1/2)*(a + b*x)^n, x)

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