Optimal. Leaf size=43 \[ 2 \sqrt {x} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {b x}{a}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, 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*} 2 \sqrt {x} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {b x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 66
Rule 68
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{\sqrt {x}} \, dx &=\left ((a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int \frac {\left (1+\frac {b x}{a}\right )^n}{\sqrt {x}} \, dx\\ &=2 \sqrt {x} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {b x}{a}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 43, normalized size = 1.00 \begin {gather*} 2 \sqrt {x} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};-\frac {b x}{a}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n}}{\sqrt {x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [F]
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]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.36, size = 26, normalized size = 0.60 \begin {gather*} 2 a^{n} \sqrt {x} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________