Optimal. Leaf size=358 \[ -\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{d}+\frac {2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d}+\frac {a \log (x)}{d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 d}-\frac {b \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\right )}{d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d} \]
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Rubi [A] time = 0.59, antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {36, 29, 31, 1593, 5992, 5912, 6044, 5920, 2402, 2315, 2447} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 d}+\frac {b \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 d}-\frac {b \text {PolyLog}\left (2,1-\frac {2}{c \sqrt {x}+1}\right )}{d}-\frac {b \text {PolyLog}\left (2,-c \sqrt {x}\right )}{d}+\frac {b \text {PolyLog}\left (2,c \sqrt {x}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{d}+\frac {2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d}+\frac {a \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 2315
Rule 2402
Rule 2447
Rule 5912
Rule 5920
Rule 5992
Rule 6044
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x (d+e x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{d x+e x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {e x \left (a+b \tanh ^{-1}(c x)\right )}{d \left (d+e x^2\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 e) \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {a \log (x)}{d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d}-\frac {(2 e) \operatorname {Subst}\left (\int \left (-\frac {a+b \tanh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \tanh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {a \log (x)}{d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d}+\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{d}-\frac {\sqrt {e} \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}+\frac {a \log (x)}{d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d}-2 \frac {(b c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}+\frac {a \log (x)}{d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d}-2 \frac {b \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c \sqrt {x}}\right )}{d}\\ &=\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d}+\frac {a \log (x)}{d}-\frac {b \text {Li}_2\left (1-\frac {2}{1+c \sqrt {x}}\right )}{d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}+\frac {b \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d}-\frac {b \text {Li}_2\left (-c \sqrt {x}\right )}{d}+\frac {b \text {Li}_2\left (c \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 1.15, size = 302, normalized size = 0.84 \[ -\frac {2 a \log (d+e x)-2 a \log (x)+b \text {Li}_2\left (-\frac {\left (d c^2+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2-2 \sqrt {-d} \sqrt {e} c-e}\right )+b \text {Li}_2\left (-\frac {\left (d c^2+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2+2 \sqrt {-d} \sqrt {e} c-e}\right )+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \log \left (\frac {\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{c^2 d-2 c \sqrt {-d} \sqrt {e}-e}+1\right )+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \log \left (\frac {\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{c^2 d+2 c \sqrt {-d} \sqrt {e}-e}+1\right )+2 b \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )-4 b \tanh ^{-1}\left (c \sqrt {x}\right )^2-4 b \tanh ^{-1}\left (c \sqrt {x}\right ) \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{e x^{2} + d x}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 540, normalized size = 1.51 \[ \frac {2 a \ln \left (c \sqrt {x}\right )}{d}-\frac {a \ln \left (c^{2} e x +c^{2} d \right )}{d}+\frac {2 b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )}{d}-\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{d}-\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 d}+\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 d}+\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 d}+\frac {b \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 d}+\frac {b \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 d}+\frac {b \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 d}-\frac {b \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 d}-\frac {b \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 d}-\frac {b \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 d}-\frac {b \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 d}-\frac {b \dilog \left (c \sqrt {x}\right )}{d}-\frac {b \dilog \left (1+c \sqrt {x}\right )}{d}-\frac {b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{d} \]
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 \[ -a {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} + b \int \frac {\log \left (c \sqrt {x} + 1\right )}{2 \, {\left (e x^{\frac {3}{2}} + d \sqrt {x}\right )} \sqrt {x}}\,{d x} - b \int \frac {\log \left (-c \sqrt {x} + 1\right )}{2 \, {\left (e x^{\frac {3}{2}} + d \sqrt {x}\right )} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x\,\left (d+e\,x\right )} \,d x \]
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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