Optimal. Leaf size=374 \[ \frac {b \sqrt {x}}{c e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}+\frac {2 d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{e^2}-\frac {b d \text {PolyLog}\left (2,1-\frac {2}{1+c \sqrt {x}}\right )}{e^2}+\frac {b d \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}-\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^2}+\frac {b d \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} \sqrt {x}\right )}{\left (c \sqrt {-d}+\sqrt {e}\right ) \left (1+c \sqrt {x}\right )}\right )}{2 e^2} \]
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Rubi [A]
time = 0.37, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {45, 6127,
6037, 327, 212, 6191, 6057, 2449, 2352, 2497} \begin {gather*} \frac {2 d \log \left (\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e^2}-\frac {d \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 )}{e^2}-\frac {d \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 )}{e^2}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e}-\frac {b d \text {Li}_2\left (1-\frac {2}{\sqrt {x} c+1}\right )}{e^2}+\frac {b d \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 e^2}+\frac {b d \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 e^2}+\frac {b \sqrt {x}}{c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 6037
Rule 6057
Rule 6127
Rule 6191
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{d+e x} \, dx &=2 \text {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \text {Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x^2} \, dx,x,\sqrt {x}\right )}{e}\\ &=\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}-\frac {(b c) \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \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 )}{e}\\ &=\frac {b \sqrt {x}}{c e}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}+\frac {d \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{e^{3/2}}-\frac {d \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx,x,\sqrt {x}\right )}{e^{3/2}}-\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c e}\\ &=\frac {b \sqrt {x}}{c e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}+\frac {2 d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^2}-\frac {d \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 )}{e^2}-\frac {d \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 )}{e^2}-2 \frac {(b c d) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{e^2}+\frac {(b c d) \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 )}{e^2}+\frac {(b c d) \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 )}{e^2}\\ &=\frac {b \sqrt {x}}{c e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}+\frac {2 d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^2}-\frac {d \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 )}{e^2}-\frac {d \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 )}{e^2}+\frac {b d \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 e^2}+\frac {b d \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 e^2}-2 \frac {(b d) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c \sqrt {x}}\right )}{e^2}\\ &=\frac {b \sqrt {x}}{c e}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2 e}+\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{e}+\frac {2 d \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1+c \sqrt {x}}\right )}{e^2}-\frac {d \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 )}{e^2}-\frac {d \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 )}{e^2}-\frac {b d \text {Li}_2\left (1-\frac {2}{1+c \sqrt {x}}\right )}{e^2}+\frac {b d \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 e^2}+\frac {b d \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 e^2}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 337, normalized size = 0.90 \begin {gather*} \frac {2 a e x-2 a d \log (d+e x)+\frac {2 b \left (c e \sqrt {x}+c^2 d \tanh ^{-1}\left (c \sqrt {x}\right )^2+\tanh ^{-1}\left (c \sqrt {x}\right ) \left (-e+c^2 e x+2 c^2 d \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )-c^2 d \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )}{c^2}-b d \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 )}{2 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 589, normalized size = 1.57
method | result | size |
derivativedivides | \(\frac {\frac {a \,c^{4} x}{e}-\frac {a \,c^{4} d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+\frac {b \,c^{4} \arctanh \left (c \sqrt {x}\right ) x}{e}-\frac {b \,c^{4} \arctanh \left (c \sqrt {x}\right ) d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}+\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}+\frac {b \,c^{3} \sqrt {x}}{e}+\frac {b \,c^{2} \ln \left (c \sqrt {x}-1\right )}{2 e}-\frac {b \,c^{2} \ln \left (1+c \sqrt {x}\right )}{2 e}}{c^{4}}\) | \(589\) |
default | \(\frac {\frac {a \,c^{4} x}{e}-\frac {a \,c^{4} d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+\frac {b \,c^{4} \arctanh \left (c \sqrt {x}\right ) x}{e}-\frac {b \,c^{4} \arctanh \left (c \sqrt {x}\right ) d \ln \left (c^{2} e x +c^{2} d \right )}{e^{2}}+\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (1+c \sqrt {x}\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}-e \left (1+c \sqrt {x}\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}+e \left (1+c \sqrt {x}\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}-\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (c^{2} e x +c^{2} d \right )}{2 e^{2}}+\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}-e \left (c \sqrt {x}-1\right )-e}{c \sqrt {-d e}-e}\right )}{2 e^{2}}+\frac {b \,c^{4} d \dilog \left (\frac {c \sqrt {-d e}+e \left (c \sqrt {x}-1\right )+e}{c \sqrt {-d e}+e}\right )}{2 e^{2}}+\frac {b \,c^{3} \sqrt {x}}{e}+\frac {b \,c^{2} \ln \left (c \sqrt {x}-1\right )}{2 e}-\frac {b \,c^{2} \ln \left (1+c \sqrt {x}\right )}{2 e}}{c^{4}}\) | \(589\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )}{d + e x}\, dx \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 {x\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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