3.8.13 \(\int \frac {x^{2+m}}{\sqrt {a+b x}} \, dx\) [713]

Optimal. Leaf size=51 \[ \frac {2 a^2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-2-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 51, 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 = {69, 67} \begin {gather*} \frac {2 a^2 x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m-2;\frac {3}{2};\frac {b x}{a}+1\right )}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(2 + m)/Sqrt[a + b*x],x]

[Out]

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

Rule 67

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

Rule 69

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

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Mathematica [A]
time = 0.07, size = 51, normalized size = 1.00 \begin {gather*} \frac {2 a^2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-2-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(2 + m)/Sqrt[a + b*x],x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 3.25, size = 34, normalized size = 0.67 \begin {gather*} \frac {x^{3+m} \text {hyper}\left [\left \{\frac {1}{2},3+m\right \},\left \{4+m\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{\sqrt {a} \left (3+m\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

x ^ (3 + m) hyper[{1 / 2, 3 + m}, {4 + m}, b x exp_polar[I Pi] / a] / (Sqrt[a] (3 + m))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2+m}}{\sqrt {b x +a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(x^(m + 2)/sqrt(b*x + a), x)

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Sympy [C] Result contains complex when optimal does not.
time = 1.73, size = 37, normalized size = 0.73 \begin {gather*} \frac {x^{3} x^{m} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*x**m*gamma(m + 3)*hyper((1/2, m + 3), (m + 4,), b*x*exp_polar(I*pi)/a)/(sqrt(a)*gamma(m + 4))

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

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{m+2}}{\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m + 2)/(a + b*x)^(1/2),x)

[Out]

int(x^(m + 2)/(a + b*x)^(1/2), x)

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