Optimal. Leaf size=242 \[ -\frac {2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e}} \]
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Rubi [A] time = 0.65, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2411, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315} \[ -\frac {2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} \sqrt {b d-a e}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2348
Rule 2402
Rule 2411
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \frac {\log (a+b x)}{(a+b x) \sqrt {d+e x}} \, dx &=\frac {\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}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\frac {\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}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}+\frac {2 \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 )}{\sqrt {b} \sqrt {b d-a e}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}+\frac {\left (4 \sqrt {b}\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 d-a e}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}+\frac {\left (4 \sqrt {b}\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 d-a e}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\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 d-a e}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \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 )}{\sqrt {b} \sqrt {b d-a e}}+\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 d-a e}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \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 )}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \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 )}{\sqrt {b} \sqrt {b d-a e}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{\sqrt {b} \sqrt {b d-a e}}-\frac {4 \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 )}{\sqrt {b} \sqrt {b d-a e}}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{\sqrt {b} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 2.89, size = 239, normalized size = 0.99 \[ \frac {\frac {\sqrt {\frac {b (d+e x)}{b d-a e}} \left (-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {\frac {b (d+e x)}{b d-a e}}\right )+\log ^2\left (\frac {e (a+b x)}{a e-b d}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {\frac {b (d+e x)}{b d-a e}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {\frac {b (d+e x)}{b d-a e}}+1\right )\right ) \log \left (\frac {e (a+b x)}{a e-b d}\right )\right )}{2 \sqrt {d+e x}}-\frac {2 \left (\log (a+b x)-\log \left (\frac {e (a+b x)}{a e-b d}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-\frac {a e}{b}}}\right )}{\sqrt {d-\frac {a e}{b}}}}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} \log \left (b x + a\right )}{b e x^{2} + a d + {\left (b d + a e\right )} x}, x\right ) \]
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 \[ \int \frac {\log \left (b x + a\right )}{{\left (b x + a\right )} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (b x +a \right )}{\left (b x +a \right ) \sqrt {e x +d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (a+b\,x\right )}{\left (a+b\,x\right )\,\sqrt {d+e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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