Optimal. Leaf size=43 \[ -\frac {2 \left (a+b \sqrt {x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {\sqrt {x} b}{a}+1\right )}{a (p+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 65} \[ -\frac {2 \left (a+b \sqrt {x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {\sqrt {x} b}{a}+1\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^p}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \left (a+b \sqrt {x}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b \sqrt {x}}{a}\right )}{a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \[ -\frac {2 \left (a+b \sqrt {x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {\sqrt {x} b}{a}+1\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \sqrt {x} + a\right )}^{p}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \sqrt {x} + a\right )}^{p}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sqrt {x}+a \right )^{p}}{x}\, dx \]
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 \[ \int \frac {{\left (b \sqrt {x} + a\right )}^{p}}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sqrt {x}\right )}^p}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.04, size = 41, normalized size = 0.95 \[ - \frac {2 b^{p} x^{\frac {p}{2}} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b \sqrt {x}}} \right )}}{\Gamma \left (1 - p\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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