Optimal. Leaf size=316 \[ -\frac {2 \sqrt {b} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{\sqrt {d+e x} (b d-a e)}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}} \]
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Rubi [A] time = 0.94, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319} \[ -\frac {2 \sqrt {b} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{\sqrt {d+e x} (b d-a e)}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {2 \sqrt {b} \log (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \log \left (\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2347
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) (d+e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\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 d-a e}-\frac {e \operatorname {Subst}\left (\int \frac {\log (x)}{\left (\frac {b d-a e}{b}+\frac {e x}{b}\right )^{3/2}} \, dx,x,a+b x\right )}{b (b d-a e)}\\ &=\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\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 d-a e}-\frac {2 \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 d-a e}\\ &=\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}+\frac {\left (2 \sqrt {b}\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 d-a e)^{3/2}}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {b d-a e}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}+\frac {\left (4 b^{3/2}\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 )}{(b d-a e)^{3/2}}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}+\frac {\left (4 b^{3/2}\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 )}{(b d-a e)^{3/2}}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {(4 b) \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)^2}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \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 d-a e)^{3/2}}+\frac {(4 b) \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)^2}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \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 d-a e)^{3/2}}-\frac {\left (4 \sqrt {b}\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 d-a e)^{3/2}}\\ &=\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}+\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )^2}{(b d-a e)^{3/2}}+\frac {2 \log (a+b x)}{(b d-a e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right ) \log (a+b x)}{(b d-a e)^{3/2}}-\frac {4 \sqrt {b} \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 d-a e)^{3/2}}-\frac {2 \sqrt {b} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}}\right )}{(b d-a e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 550, normalized size = 1.74 \[ \frac {-2 \sqrt {b} \sqrt {d+e x} \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+2 \sqrt {b} \sqrt {d+e x} \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}+1\right )\right )-\sqrt {b} \sqrt {d+e x} \log ^2\left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+\sqrt {b} \sqrt {d+e x} \log ^2\left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \sqrt {b} \sqrt {d+e x} \log (a+b x) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )-2 \sqrt {b} \sqrt {d+e x} \log \left (\frac {1}{2} \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}+1\right )\right ) \log \left (\sqrt {b d-a e}-\sqrt {b} \sqrt {d+e x}\right )+4 \sqrt {b d-a e} \log (a+b x)-2 \sqrt {b} \sqrt {d+e x} \log (a+b x) \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right )+2 \sqrt {b} \sqrt {d+e x} \log \left (\sqrt {b d-a e}+\sqrt {b} \sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {b} \sqrt {d+e x}}{2 \sqrt {b d-a e}}\right )+8 \sqrt {b} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{2 \sqrt {d+e x} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} \log \left (b x + a\right )}{b e^{2} x^{3} + a d^{2} + {\left (2 \, b d e + a e^{2}\right )} x^{2} + {\left (b d^{2} + 2 \, a d e\right )} x}, x\right ) \]
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 \[ \int \frac {\log \left (b x + a\right )}{{\left (b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (b x +a \right )}{\left (b x +a \right ) \left (e x +d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\ln \left (a+b\,x\right )}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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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