Optimal. Leaf size=139 \[ -\frac {\left (a+b \sqrt {c+d x}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a-b \sqrt {c}}\right )}{\left (a-b \sqrt {c}\right ) (1+p)}-\frac {\left (a+b \sqrt {c+d x}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a+b \sqrt {c}}\right )}{\left (a+b \sqrt {c}\right ) (1+p)} \]
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Rubi [A]
time = 0.10, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {378, 1412, 845,
70} \begin {gather*} -\frac {\left (a+b \sqrt {c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {a+b \sqrt {c+d x}}{a-b \sqrt {c}}\right )}{(p+1) \left (a-b \sqrt {c}\right )}-\frac {\left (a+b \sqrt {c+d x}\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac {a+b \sqrt {c+d x}}{a+b \sqrt {c}}\right )}{(p+1) \left (a+b \sqrt {c}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 378
Rule 845
Rule 1412
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {c+d x}\right )^p}{x} \, dx &=\text {Subst}\left (\int \frac {\left (a+b \sqrt {x}\right )^p}{-c+x} \, dx,x,c+d x\right )\\ &=2 \text {Subst}\left (\int \frac {x (a+b x)^p}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {(a+b x)^p}{2 \left (\sqrt {c}-x\right )}+\frac {(a+b x)^p}{2 \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right )\\ &=-\text {Subst}\left (\int \frac {(a+b x)^p}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )+\text {Subst}\left (\int \frac {(a+b x)^p}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\left (a+b \sqrt {c+d x}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a-b \sqrt {c}}\right )}{\left (a-b \sqrt {c}\right ) (1+p)}-\frac {\left (a+b \sqrt {c+d x}\right )^{1+p} \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a+b \sqrt {c}}\right )}{\left (a+b \sqrt {c}\right ) (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 136, normalized size = 0.98 \begin {gather*} -\frac {\left (a+b \sqrt {c+d x}\right )^{1+p} \left (\left (a+b \sqrt {c}\right ) \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a-b \sqrt {c}}\right )+\left (a-b \sqrt {c}\right ) \, _2F_1\left (1,1+p;2+p;\frac {a+b \sqrt {c+d x}}{a+b \sqrt {c}}\right )\right )}{\left (a-b \sqrt {c}\right ) \left (a+b \sqrt {c}\right ) (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sqrt {d x +c}\right )^{p}}{x}\, dx\]
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 \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 (a + b \sqrt {c + d x}\right )^{p}}{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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\sqrt {c+d\,x}\right )}^p}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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