Optimal. Leaf size=506 \[ -\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{d^3}+\frac {a e^2 \log (x)}{d^3}-\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{d^3}+\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d^3}+\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 d^3}-\frac {b e^2 \text {PolyLog}\left (2,-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {PolyLog}\left (2,c \sqrt {x}\right )}{d^3} \]
[Out]
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Rubi [A]
time = 0.64, antiderivative size = 506, normalized size of antiderivative = 1.00, number
of steps used = 24, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules
used = {46, 1607, 6129, 6037, 331, 212, 6139, 6031, 6191, 6057, 2449, 2352, 2497}
\begin {gather*} \frac {2 e^2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^3}-\frac {e^2 \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^3}-\frac {e^2 \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^3}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {a e^2 \log (x)}{d^3}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^3}{2 d \sqrt {x}}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {b e^2 \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\right )}{d^3}+\frac {b e^2 \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^3}+\frac {b e^2 \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^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}+\frac {b c e}{d^2 \sqrt {x}}-\frac {b c}{6 d x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 212
Rule 331
Rule 1607
Rule 2352
Rule 2449
Rule 2497
Rule 6031
Rule 6037
Rule 6057
Rule 6129
Rule 6139
Rule 6191
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^3 (d+e x)} \, dx &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{d x^5+e x^7} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^5 \left (d+e x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 e) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (d+e x^2\right )} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )}{2 d}-\frac {(2 e) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )}{2 d}-\frac {(b c e) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (2 e^2\right ) \text {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 )}{d^2}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {\left (b c^5\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 d}-\frac {\left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )}{d^3}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {a e^2 \log (x)}{d^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}-\frac {\left (2 e^3\right ) \text {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^3}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {a e^2 \log (x)}{d^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}+\frac {e^{5/2} \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{d^3}-\frac {e^{5/2} \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{d^3}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d^3}-\frac {e^2 \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^3}-\frac {e^2 \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^3}+\frac {a e^2 \log (x)}{d^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}-2 \frac {\left (b c e^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d^3}+\frac {\left (b c e^2\right ) \text {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^3}+\frac {\left (b c e^2\right ) \text {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^3}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d^3}-\frac {e^2 \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^3}-\frac {e^2 \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^3}+\frac {a e^2 \log (x)}{d^3}+\frac {b e^2 \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^3}+\frac {b e^2 \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^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}-2 \frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c \sqrt {x}}\right )}{d^3}\\ &=-\frac {b c}{6 d x^{3/2}}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c e}{d^2 \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}+\frac {2 e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{d^3}-\frac {e^2 \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^3}-\frac {e^2 \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^3}+\frac {a e^2 \log (x)}{d^3}-\frac {b e^2 \text {Li}_2\left (1-\frac {2}{1+c \sqrt {x}}\right )}{d^3}+\frac {b e^2 \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^3}+\frac {b e^2 \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^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 1.83, size = 394, normalized size = 0.78 \begin {gather*} -\frac {3 a d^2-6 a d e x-6 a e^2 x^2 \log (x)+6 a e^2 x^2 \log (d+e x)+b \left (c d \sqrt {x} \left (d+3 c^2 d x-6 e x\right )-3 \tanh ^{-1}\left (c \sqrt {x}\right ) \left (d \left (-1+c^2 x\right ) \left (d+c^2 d x-2 e x\right )+2 e^2 x^2 \tanh ^{-1}\left (c \sqrt {x}\right )+4 e^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+6 e^2 x^2 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+3 e^2 x^2 \left (2 \tanh ^{-1}\left (c \sqrt {x}\right ) \left (-\tanh ^{-1}\left (c \sqrt {x}\right )+\log \left (1+\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}\right )+\log \left (1+\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}\right )\right )+\text {PolyLog}\left (2,-\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}\right )+\text {PolyLog}\left (2,-\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}\right )\right )\right )}{6 d^3 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.65, size = 808, normalized size = 1.60
method | result | size |
derivativedivides | \(2 c^{4} \left (-\frac {b}{12 d \,c^{3} x^{\frac {3}{2}}}-\frac {a}{4 d \,c^{4} x^{2}}+\frac {b e}{2 c^{3} d^{2} \sqrt {x}}-\frac {b \ln \left (1+c \sqrt {x}\right ) e}{4 c^{2} d^{2}}+\frac {b \ln \left (c \sqrt {x}-1\right ) e}{4 c^{2} d^{2}}-\frac {a \,e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{4} d^{3}}+\frac {a \,e^{2} \ln \left (c \sqrt {x}\right )}{c^{4} d^{3}}+\frac {a e}{2 c^{4} d^{2} x}-\frac {b \,e^{2} \dilog \left (1+c \sqrt {x}\right )}{2 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (c \sqrt {x}\right )}{2 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{4 c^{4} d^{3}}+\frac {b \arctanh \left (c \sqrt {x}\right ) e}{2 c^{4} d^{2} x}-\frac {b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{4} d^{3}}+\frac {b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c \sqrt {x}\right )}{c^{4} d^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{4 d \,c^{4} x^{2}}+\frac {b \ln \left (1+c \sqrt {x}\right )}{8 d}-\frac {b \ln \left (c \sqrt {x}-1\right )}{8 d}-\frac {b}{4 d c \sqrt {x}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{4} d^{3}}+\frac {b \,e^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}\right )\) | \(808\) |
default | \(2 c^{4} \left (-\frac {b}{12 d \,c^{3} x^{\frac {3}{2}}}-\frac {a}{4 d \,c^{4} x^{2}}+\frac {b e}{2 c^{3} d^{2} \sqrt {x}}-\frac {b \ln \left (1+c \sqrt {x}\right ) e}{4 c^{2} d^{2}}+\frac {b \ln \left (c \sqrt {x}-1\right ) e}{4 c^{2} d^{2}}-\frac {a \,e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{4} d^{3}}+\frac {a \,e^{2} \ln \left (c \sqrt {x}\right )}{c^{4} d^{3}}+\frac {a e}{2 c^{4} d^{2} x}-\frac {b \,e^{2} \dilog \left (1+c \sqrt {x}\right )}{2 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (c \sqrt {x}\right )}{2 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{4 c^{4} d^{3}}+\frac {b \arctanh \left (c \sqrt {x}\right ) e}{2 c^{4} d^{2} x}-\frac {b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{2 c^{4} d^{3}}+\frac {b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c \sqrt {x}\right )}{c^{4} d^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{4 d \,c^{4} x^{2}}+\frac {b \ln \left (1+c \sqrt {x}\right )}{8 d}-\frac {b \ln \left (c \sqrt {x}-1\right )}{8 d}-\frac {b}{4 d c \sqrt {x}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{4} d^{3}}+\frac {b \,e^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}+\frac {b \,e^{2} \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 )}{4 c^{4} d^{3}}\right )\) | \(808\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^3\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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