Optimal. Leaf size=214 \[ -\frac {d^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {a d x}{e^2}-\frac {b d \log \left (1-c^2 x^2\right )}{2 c e^2}-\frac {b \tanh ^{-1}(c x)}{2 c^2 e}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^3}-\frac {b d x \tanh ^{-1}(c x)}{e^2}+\frac {b x}{2 c e} \]
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Rubi [A] time = 0.20, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {5940, 5910, 260, 5916, 321, 206, 5920, 2402, 2315, 2447} \[ \frac {b d^2 \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e^3}-\frac {b d^2 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^3}-\frac {d^2 \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^3}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {a d x}{e^2}-\frac {b d \log \left (1-c^2 x^2\right )}{2 c e^2}-\frac {b \tanh ^{-1}(c x)}{2 c^2 e}-\frac {b d x \tanh ^{-1}(c x)}{e^2}+\frac {b x}{2 c e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 2447
Rule 5910
Rule 5916
Rule 5920
Rule 5940
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{e^2}+\frac {x \left (a+b \tanh ^{-1}(c x)\right )}{e}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac {a d x}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}+\frac {\left (b c d^2\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{e^3}-\frac {\left (b c d^2\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{e^3}-\frac {(b d) \int \tanh ^{-1}(c x) \, dx}{e^2}-\frac {(b c) \int \frac {x^2}{1-c^2 x^2} \, dx}{2 e}\\ &=-\frac {a d x}{e^2}+\frac {b x}{2 c e}-\frac {b d x \tanh ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}+\frac {\left (b d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{e^3}+\frac {(b c d) \int \frac {x}{1-c^2 x^2} \, dx}{e^2}-\frac {b \int \frac {1}{1-c^2 x^2} \, dx}{2 c e}\\ &=-\frac {a d x}{e^2}+\frac {b x}{2 c e}-\frac {b \tanh ^{-1}(c x)}{2 c^2 e}-\frac {b d x \tanh ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^3}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^3}-\frac {b d \log \left (1-c^2 x^2\right )}{2 c e^2}+\frac {b d^2 \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 e^3}-\frac {b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^3}\\ \end {align*}
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Mathematica [C] time = 3.05, size = 394, normalized size = 1.84 \[ \frac {2 a d^2 \log (d+e x)-2 a d e x+a e^2 x^2-\frac {b d e \sqrt {1-\frac {c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )}}{c}-\frac {1}{2} i \pi b d^2 \log \left (1-c^2 x^2\right )-\frac {b d e \log \left (1-c^2 x^2\right )}{c}-\frac {b e^2 \tanh ^{-1}(c x)}{c^2}-b d^2 \text {Li}_2\left (e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 b d^2 \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac {c d}{e}\right )+2 b d^2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 b d^2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 b d^2 \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )+b d^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )-b d^2 \tanh ^{-1}(c x)^2+i \pi b d^2 \tanh ^{-1}(c x)-2 b d^2 \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-i \pi b d^2 \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )+\frac {b d e \tanh ^{-1}(c x)^2}{c}-2 b d e x \tanh ^{-1}(c x)+b e^2 x^2 \tanh ^{-1}(c x)+\frac {b e^2 x}{c}}{2 e^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \operatorname {artanh}\left (c x\right ) + a x^{2}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{2}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 298, normalized size = 1.39 \[ \frac {a \,x^{2}}{2 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (c x e +c d \right )}{e^{3}}+\frac {b \arctanh \left (c x \right ) x^{2}}{2 e}-\frac {b d x \arctanh \left (c x \right )}{e^{2}}+\frac {b \arctanh \left (c x \right ) d^{2} \ln \left (c x e +c d \right )}{e^{3}}+\frac {b x}{2 c e}+\frac {b d}{2 c \,e^{2}}-\frac {b \ln \left (c x e +e \right ) d}{2 c \,e^{2}}-\frac {b \ln \left (c x e +e \right )}{4 c^{2} e}-\frac {b \ln \left (c x e -e \right ) d}{2 c \,e^{2}}+\frac {b \ln \left (c x e -e \right )}{4 c^{2} e}-\frac {b \,d^{2} \ln \left (c x e +c d \right ) \ln \left (\frac {c x e +e}{-c d +e}\right )}{2 e^{3}}-\frac {b \,d^{2} \dilog \left (\frac {c x e +e}{-c d +e}\right )}{2 e^{3}}+\frac {b \,d^{2} \ln \left (c x e +c d \right ) \ln \left (\frac {c x e -e}{-c d -e}\right )}{2 e^{3}}+\frac {b \,d^{2} \dilog \left (\frac {c x e -e}{-c d -e}\right )}{2 e^{3}} \]
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 \[ \frac {1}{2} \, a {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + \frac {1}{2} \, b \int \frac {x^{2} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{e x + d}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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