3.2263 \(\int (a+b \sqrt{x})^p x^m \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 x^{m+1} \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (1,2 m+p+3;p+2;\frac{a+b \sqrt{x}}{a}\right )}{a (p+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0264015, antiderivative size = 63, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {341, 66, 64} \[ \frac{x^{m+1} \left (a+b \sqrt{x}\right )^p \left (\frac{b \sqrt{x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 m+3;-\frac{b \sqrt{x}}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^p*x^m,x]

[Out]

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

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 66

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])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

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

Mathematica [A]  time = 0.0191747, size = 65, normalized size = 1.25 \[ \frac{x^{m+1} \left (a+b \sqrt{x}\right )^p \left (\frac{b \sqrt{x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 (m+1)+1;-\frac{b \sqrt{x}}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^p*x^m,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sqrt(x) + a)^p*x^m, x)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sqrt(x) + a)^p*x^m, x)