Optimal. Leaf size=160 \[ -\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3}+\frac {3 b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^6}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {b \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^6} \]
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Rubi [A]
time = 0.30, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {45, 6127,
6037, 308, 212, 327, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^6}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {3 b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^6}-\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx &=2 \text {Subst}\left (\int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \text {Subst}\left (\int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^2}+\frac {2 \text {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {2 \text {Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^4}+\frac {2 \text {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^4}+\frac {b \text {Subst}\left (\int \frac {x^4}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 c}\\ &=-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt {x}\right )}{c^5}+\frac {b \text {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {b \text {Subst}\left (\int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 c}\\ &=-\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{2 c^5}+\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^5}-\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^5}\\ &=-\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3}+\frac {3 b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^6}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {(2 b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c \sqrt {x}}\right )}{c^6}\\ &=-\frac {3 b \sqrt {x}}{2 c^5}-\frac {b x^{3/2}}{6 c^3}+\frac {3 b \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^6}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^6}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^6}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^6}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 130, normalized size = 0.81 \begin {gather*} -\frac {9 b c \sqrt {x}+6 a c^2 x+b c^3 x^{3/2}+3 a c^4 x^2-6 b \tanh ^{-1}\left (c \sqrt {x}\right )^2+3 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (-3+2 c^2 x+c^4 x^2-4 \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+6 a \log \left (1-c^2 x\right )+6 b \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{6 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 244, normalized size = 1.52
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {a \,c^{4} x^{2}}{4}+\frac {a \,c^{2} x}{2}+\frac {a \ln \left (c \sqrt {x}-1\right )}{2}+\frac {a \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) c^{4} x^{2}}{4}+\frac {b \arctanh \left (c \sqrt {x}\right ) c^{2} x}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {b \dilog \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {b \,c^{3} x^{\frac {3}{2}}}{12}+\frac {3 b c \sqrt {x}}{4}+\frac {3 b \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 b \ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{6}}\) | \(244\) |
default | \(-\frac {2 \left (\frac {a \,c^{4} x^{2}}{4}+\frac {a \,c^{2} x}{2}+\frac {a \ln \left (c \sqrt {x}-1\right )}{2}+\frac {a \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) c^{4} x^{2}}{4}+\frac {b \arctanh \left (c \sqrt {x}\right ) c^{2} x}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {b \dilog \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{8}+\frac {b \,c^{3} x^{\frac {3}{2}}}{12}+\frac {3 b c \sqrt {x}}{4}+\frac {3 b \ln \left (c \sqrt {x}-1\right )}{8}-\frac {3 b \ln \left (1+c \sqrt {x}\right )}{8}\right )}{c^{6}}\) | \(244\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 208, normalized size = 1.30 \begin {gather*} -\frac {1}{2} \, a {\left (\frac {c^{2} x^{2} + 2 \, x}{c^{4}} + \frac {2 \, \log \left (c^{2} x - 1\right )}{c^{6}}\right )} - \frac {{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b}{c^{6}} + \frac {3 \, b \log \left (c \sqrt {x} + 1\right )}{4 \, c^{6}} - \frac {3 \, b \log \left (c \sqrt {x} - 1\right )}{4 \, c^{6}} - \frac {2 \, b c^{3} x^{\frac {3}{2}} + 3 \, b \log \left (c \sqrt {x} + 1\right )^{2} - 3 \, b \log \left (-c \sqrt {x} + 1\right )^{2} + 18 \, b c \sqrt {x} + 3 \, {\left (b c^{4} x^{2} + 2 \, b c^{2} x\right )} \log \left (c \sqrt {x} + 1\right ) - 3 \, {\left (b c^{4} x^{2} + 2 \, b c^{2} x + 2 \, b \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{12 \, c^{6}} \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 {a x^{2}}{c^{2} x - 1}\, dx - \int \frac {b x^{2} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x - 1}\, 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 = -0.01 \begin {gather*} \int -\frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{c^2\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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