Optimal. Leaf size=323 \[ -\frac{2 \sqrt{b d-a e} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}-\frac{2 \sqrt{b d-a e} \log (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}-\frac{4 \sqrt{b d-a e} \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{4 \sqrt{d+e x}}{b} \]
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Rubi [A] time = 0.913033, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {2411, 2346, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 50} \[ -\frac{2 \sqrt{b d-a e} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}-\frac{2 \sqrt{b d-a e} \log (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}-\frac{4 \sqrt{b d-a e} \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{4 \sqrt{d+e x}}{b} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 2346
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rule 50
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x} \log (a+b x)}{a+b x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{\frac{b d-a e}{b}+\frac{e x}{b}} \log (x)}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{\log (x)}{\sqrt{\frac{b d-a e}{b}+\frac{e x}{b}}} \, dx,x,a+b x\right )}{b^2}+\frac{(b d-a e) \operatorname{Subst}\left (\int \frac{\log (x)}{x \sqrt{\frac{b d-a e}{b}+\frac{e x}{b}}} \, dx,x,a+b x\right )}{b^2}\\ &=\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{\frac{b d-a e}{b}+\frac{e x}{b}}}{x} \, dx,x,a+b x\right )}{b}-\frac{(b d-a e) \operatorname{Subst}\left (\int -\frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d-\frac{a e}{b}+\frac{e x}{b}}}{\sqrt{b d-a e}}\right )}{\sqrt{b d-a e} x} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac{\left (2 \sqrt{b d-a e}\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d-\frac{a e}{b}+\frac{e x}{b}}}{\sqrt{b d-a e}}\right )}{x} \, dx,x,a+b x\right )}{b^{3/2}}-\frac{(2 (b d-a e)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{b d-a e}{b}+\frac{e x}{b}}} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac{\left (4 \sqrt{b d-a e}\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b d-a e}}\right )}{a e+b \left (-d+x^2\right )} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{b}}-\frac{(4 (b d-a e)) \operatorname{Subst}\left (\int \frac{1}{-\frac{b d-a e}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b e}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}+\frac{\left (4 \sqrt{b d-a e}\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b d-a e}}\right )}{-b d+a e+b x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{b}}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac{4 \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b d-a e}}\right )}{1-\frac{\sqrt{b} x}{\sqrt{b d-a e}}} \, dx,x,\sqrt{d+e x}\right )}{b}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}+\frac{4 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{b} x}{\sqrt{b d-a e}}}\right )}{1-\frac{b x^2}{b d-a e}} \, dx,x,\sqrt{d+e x}\right )}{b}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}-\frac{\left (4 \sqrt{b d-a e}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}\\ &=-\frac{4 \sqrt{d+e x}}{b}+\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}+\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )^2}{b^{3/2}}+\frac{2 \sqrt{d+e x} \log (a+b x)}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log (a+b x)}{b^{3/2}}-\frac{4 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}-\frac{2 \sqrt{b d-a e} \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.356524, size = 534, normalized size = 1.65 \[ \frac{-2 \sqrt{b d-a e} \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{b} \sqrt{d+e x}}{2 \sqrt{b d-a e}}\right )+2 \sqrt{b d-a e} \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}+1\right )\right )-\sqrt{b d-a e} \log ^2\left (\sqrt{b d-a e}-\sqrt{b} \sqrt{d+e x}\right )+\sqrt{b d-a e} \log ^2\left (\sqrt{b d-a e}+\sqrt{b} \sqrt{d+e x}\right )+2 \sqrt{b d-a e} \log (a+b x) \log \left (\sqrt{b d-a e}-\sqrt{b} \sqrt{d+e x}\right )-2 \sqrt{b d-a e} \log \left (\frac{1}{2} \left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}+1\right )\right ) \log \left (\sqrt{b d-a e}-\sqrt{b} \sqrt{d+e x}\right )+4 \sqrt{b} \sqrt{d+e x} \log (a+b x)-2 \sqrt{b d-a e} \log (a+b x) \log \left (\sqrt{b d-a e}+\sqrt{b} \sqrt{d+e x}\right )+2 \sqrt{b d-a e} \log \left (\sqrt{b d-a e}+\sqrt{b} \sqrt{d+e x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{b} \sqrt{d+e x}}{2 \sqrt{b d-a e}}\right )+8 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )-8 \sqrt{b} \sqrt{d+e x}}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.835, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( bx+a \right ) }{bx+a}\sqrt{ex+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d} \log \left (b x + a\right )}{b x + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d} \log \left (b x + a\right )}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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