Optimal. Leaf size=506 \[ \frac {2 e^2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{d^3}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {a e^2 \log (x)}{d^3}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}-\frac {b c^3}{2 d \sqrt {x}}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {b e^2 \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 d^3}+\frac {b e^2 \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (\sqrt {-d} c+\sqrt {e}\right ) \left (\sqrt {x} c+1\right )}\right )}{2 d^3}-\frac {b e^2 \text {Li}_2\left (-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {Li}_2\left (c \sqrt {x}\right )}{d^3}+\frac {b c e}{d^2 \sqrt {x}}-\frac {b c}{6 d x^{3/2}} \]
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Rubi [A] time = 0.87, 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 = {44, 1593, 5982, 5916, 325, 206, 5992, 5912, 6044, 5920, 2402, 2315, 2447} \[ -\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2}{c \sqrt {x}+1}\right )}{d^3}+\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{2 d^3}+\frac {b e^2 \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{2 d^3}-\frac {b e^2 \text {PolyLog}\left (2,-c \sqrt {x}\right )}{d^3}+\frac {b e^2 \text {PolyLog}\left (2,c \sqrt {x}\right )}{d^3}+\frac {2 e^2 \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}-\sqrt {e}\right )}\right )}{d^3}-\frac {e^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {x}+1\right ) \left (c \sqrt {-d}+\sqrt {e}\right )}\right )}{d^3}+\frac {e \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d^2 x}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d x^2}+\frac {a e^2 \log (x)}{d^3}-\frac {b c^2 e \tanh ^{-1}\left (c \sqrt {x}\right )}{d^2}-\frac {b c^3}{2 d \sqrt {x}}+\frac {b c^4 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 d}+\frac {b c e}{d^2 \sqrt {x}}-\frac {b c}{6 d x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 325
Rule 1593
Rule 2315
Rule 2402
Rule 2447
Rule 5912
Rule 5916
Rule 5920
Rule 5982
Rule 5992
Rule 6044
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^3 (d+e x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{d x^5+e x^7} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {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 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^5} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 e) \operatorname {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) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )}{2 d}-\frac {(2 e) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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 ) \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 )}{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 ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 d}-\frac {\left (b c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (2 e^3\right ) \operatorname {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 ) \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^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} \operatorname {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} \operatorname {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 ) \operatorname {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 ) \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^3}+\frac {\left (b c e^2\right ) \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^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 ) \operatorname {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 = 2.74, size = 394, normalized size = 0.78 \[ -\frac {3 a d^2+6 a e^2 x^2 \log (d+e x)-6 a d e x-6 a e^2 x^2 \log (x)+b \left (3 e^2 x^2 \left (\text {Li}_2\left (-\frac {\left (d c^2+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2-2 \sqrt {-d} \sqrt {e} c-e}\right )+\text {Li}_2\left (-\frac {\left (d c^2+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{d c^2+2 \sqrt {-d} \sqrt {e} c-e}\right )+2 \tanh ^{-1}\left (c \sqrt {x}\right ) \left (\log \left (\frac {\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{c^2 d-2 c \sqrt {-d} \sqrt {e}-e}+1\right )+\log \left (\frac {\left (c^2 d+e\right ) e^{2 \tanh ^{-1}\left (c \sqrt {x}\right )}}{c^2 d+2 c \sqrt {-d} \sqrt {e}-e}+1\right )-\tanh ^{-1}\left (c \sqrt {x}\right )\right )\right )-3 \tanh ^{-1}\left (c \sqrt {x}\right ) \left (d \left (c^2 x-1\right ) \left (c^2 d x+d-2 e x\right )+2 e^2 x^2 \tanh ^{-1}\left (c \sqrt {x}\right )+4 e^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+c d \sqrt {x} \left (3 c^2 d x+d-6 e x\right )+6 e^2 x^2 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )}{6 d^3 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{e x^{4} + d x^{3}}, x\right ) \]
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 \[ \int \frac {b \operatorname {artanh}\left (c \sqrt {x}\right ) + a}{{\left (e x + d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 741, normalized size = 1.46 \[ -\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 )}{2 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 )}{2 d^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{2 d \,x^{2}}+\frac {c^{4} b \ln \left (1+c \sqrt {x}\right )}{4 d}-\frac {c^{4} b \ln \left (c \sqrt {x}-1\right )}{4 d}-\frac {b \,e^{2} \dilog \left (c \sqrt {x}\right )}{d^{3}}-\frac {b \,e^{2} \dilog \left (1+c \sqrt {x}\right )}{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 )}{2 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 )}{2 d^{3}}+\frac {2 a \,e^{2} \ln \left (c \sqrt {x}\right )}{d^{3}}-\frac {a \,e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{d^{3}}+\frac {a e}{d^{2} x}-\frac {c^{2} b \ln \left (1+c \sqrt {x}\right ) e}{2 d^{2}}-\frac {a}{2 d \,x^{2}}+\frac {b c e}{d^{2} \sqrt {x}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 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 )}{2 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 )}{2 d^{3}}+\frac {b \,e^{2} \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 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 )}{2 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 )}{2 d^{3}}+\frac {c^{2} b \ln \left (c \sqrt {x}-1\right ) e}{2 d^{2}}-\frac {b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c^{2} e x +c^{2} d \right )}{d^{3}}+\frac {b \arctanh \left (c \sqrt {x}\right ) e}{d^{2} x}+\frac {2 b \arctanh \left (c \sqrt {x}\right ) e^{2} \ln \left (c \sqrt {x}\right )}{d^{3}}-\frac {b \,e^{2} \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{d^{3}}-\frac {b c}{6 d \,x^{\frac {3}{2}}}-\frac {b \,c^{3}}{2 d \sqrt {x}} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \relax (x)}{d^{3}} - \frac {2 \, e x - d}{d^{2} x^{2}}\right )} + b \int \frac {\log \left (c \sqrt {x} + 1\right )}{2 \, {\left (e x^{\frac {7}{2}} + d x^{\frac {5}{2}}\right )} \sqrt {x}}\,{d x} - b \int \frac {\log \left (-c \sqrt {x} + 1\right )}{2 \, {\left (e x^{\frac {7}{2}} + d x^{\frac {5}{2}}\right )} \sqrt {x}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
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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