Optimal. Leaf size=121 \[ \frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};-\frac {1}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {145, 144, 143}
\begin {gather*} \frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};-\frac {1}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 143
Rule 144
Rule 145
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x} (e+f x)^n}{\sqrt {a+b x}} \, dx &=\frac {\sqrt {c+d x} \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} (e+f x)^n}{\sqrt {a+b x}} \, dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {\left (\sqrt {c+d x} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n}\right ) \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^n}{\sqrt {a+b x}} \, dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\\ &=\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};-\frac {1}{2},-n;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.75, size = 119, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^n \left (\frac {b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac {1}{2};-\frac {1}{2},-n;\frac {3}{2};\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{n} \sqrt {d x +c}}{\sqrt {b x +a}}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^n\,\sqrt {c+d\,x}}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________