Optimal. Leaf size=61 \[ \frac {2 x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac {1}{2}+p;\frac {3}{2}+p;-\frac {c x}{b}\right )}{(1+2 p) \sqrt {d x}} \]
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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 \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+\frac {1}{2};p+\frac {3}{2};-\frac {c x}{b}\right )}{(2 p+1) \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rule 688
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^p}{\sqrt {d x}} \, dx &=\frac {\left (x^{\frac {1}{2}-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-\frac {1}{2}+p} (b+c x)^p \, dx}{\sqrt {d x}}\\ &=\frac {\left (x^{\frac {1}{2}-p} \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}{\sqrt {d x}}\\ &=\frac {2 x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,\frac {1}{2}+p;\frac {3}{2}+p;-\frac {c x}{b}\right )}{(1+2 p) \sqrt {d x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 58, normalized size = 0.95 \begin {gather*} \frac {x (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \, _2F_1\left (-p,\frac {1}{2}+p;\frac {3}{2}+p;-\frac {c x}{b}\right )}{\left (\frac {1}{2}+p\right ) \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+b x \right )^{p}}{\sqrt {d x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{p}}{\sqrt {d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\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.
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