Optimal. Leaf size=61 \[ \frac {2 x \sqrt {d x} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac {3}{2}+p;\frac {5}{2}+p;-\frac {c x}{b}\right )}{3+2 p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {688, 68, 66}
\begin {gather*} \frac {2 x \sqrt {d x} \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac {3}{2};p+\frac {5}{2};-\frac {c x}{b}\right )}{2 p+3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 66
Rule 68
Rule 688
Rubi steps
\begin {align*} \int \sqrt {d x} \left (b x+c x^2\right )^p \, dx &=\left (x^{-\frac {1}{2}-p} \sqrt {d x} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac {1}{2}+p} (b+c x)^p \, dx\\ &=\left (x^{-\frac {1}{2}-p} \sqrt {d x} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{\frac {1}{2}+p} \left (1+\frac {c x}{b}\right )^p \, dx\\ &=\frac {2 x \sqrt {d x} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac {3}{2}+p;\frac {5}{2}+p;-\frac {c x}{b}\right )}{3+2 p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 58, normalized size = 0.95 \begin {gather*} \frac {x \sqrt {d x} (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \, _2F_1\left (-p,\frac {3}{2}+p;\frac {5}{2}+p;-\frac {c x}{b}\right )}{\frac {3}{2}+p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \sqrt {d x}\, \left (c \,x^{2}+b x \right )^{p}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d x} \left (x \left (b + c x\right )\right )^{p}\, dx \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 {\left (c\,x^2+b\,x\right )}^p\,\sqrt {d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________