Optimal. Leaf size=50 \[ -\frac {\left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-1+p,-p;p;-\frac {c x}{b}\right )}{(1-p) x} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {688, 68, 66}
\begin {gather*} -\frac {\left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (p-1,-p;p;-\frac {c x}{b}\right )}{(1-p) 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}{x^2} \, dx &=\left (x^{-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-2+p} (b+c x)^p \, dx\\ &=\left (x^{-p} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{-2+p} \left (1+\frac {c x}{b}\right )^p \, dx\\ &=-\frac {\left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-1+p,-p;p;-\frac {c x}{b}\right )}{(1-p) x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.90 \begin {gather*} \frac {(x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \, _2F_1\left (-1+p,-p;p;-\frac {c x}{b}\right )}{(-1+p) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (c \,x^{2}+b x \right )^{p}}{x^{2}}\, dx\]
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}}{x^{2}}\, 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}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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