Optimal. Leaf size=237 \[ -\frac {a d x}{e^2}-\frac {b x}{2 c e}+\frac {b \text {ArcTan}(c x)}{2 c^2 e}-\frac {b d x \text {ArcTan}(c x)}{e^2}+\frac {x^2 (a+b \text {ArcTan}(c x))}{2 e}-\frac {d^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {b d \log \left (1+c^2 x^2\right )}{2 c e^2}+\frac {i b d^2 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 237, 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 = {4996, 4930,
266, 4946, 327, 209, 4966, 2449, 2352, 2497} \begin {gather*} -\frac {d^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))}{e^3}+\frac {d^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^3}+\frac {x^2 (a+b \text {ArcTan}(c x))}{2 e}-\frac {a d x}{e^2}+\frac {b \text {ArcTan}(c x)}{2 c^2 e}-\frac {b d x \text {ArcTan}(c x)}{e^2}+\frac {b d \log \left (c^2 x^2+1\right )}{2 c e^2}+\frac {i b d^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}-\frac {b x}{2 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 4930
Rule 4946
Rule 4966
Rule 4996
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{e^2}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{e}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{e^2}+\frac {\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac {a d x}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {\left (b c d^2\right ) \int \frac {\log \left (\frac {2}{1-i 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+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{e^3}-\frac {(b d) \int \tan ^{-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 \tan ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}+\frac {\left (i b d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i 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 \tan ^{-1}(c x)}{2 c^2 e}-\frac {b d x \tan ^{-1}(c x)}{e^2}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e^3}+\frac {d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^3}+\frac {b d \log \left (1+c^2 x^2\right )}{2 c e^2}+\frac {i b d^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e^3}-\frac {i b d^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^3}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 404, normalized size = 1.70 \begin {gather*} \frac {-2 a d e x-\frac {b e^2 x}{c}+a e^2 x^2+\frac {b e^2 \text {ArcTan}(c x)}{c^2}+i b d^2 \pi \text {ArcTan}(c x)-2 b d e x \text {ArcTan}(c x)+b e^2 x^2 \text {ArcTan}(c x)-2 i b d^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \text {ArcTan}(c x)+i b d^2 \text {ArcTan}(c x)^2+\frac {b d e \text {ArcTan}(c x)^2}{c}-\frac {b d \sqrt {1+\frac {c^2 d^2}{e^2}} e e^{i \text {ArcTan}\left (\frac {c d}{e}\right )} \text {ArcTan}(c x)^2}{c}+b d^2 \pi \log \left (1+e^{-2 i \text {ArcTan}(c x)}\right )-2 b d^2 \text {ArcTan}(c x) \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )+2 b d^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )+2 b d^2 \text {ArcTan}(c x) \log \left (1-e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )+2 a d^2 \log (d+e x)+\frac {b d e \log \left (1+c^2 x^2\right )}{c}+\frac {1}{2} b d^2 \pi \log \left (1+c^2 x^2\right )-2 b d^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )\right )+i b d^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )-i b d^2 \text {PolyLog}\left (2,e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 337, normalized size = 1.42
method | result | size |
derivativedivides | \(\frac {-\frac {a \,c^{3} d x}{e^{2}}+\frac {a \,c^{3} x^{2}}{2 e}+\frac {a \,c^{3} d^{2} \ln \left (c e x +c d \right )}{e^{3}}-\frac {b \,c^{3} \arctan \left (c x \right ) d x}{e^{2}}+\frac {b \,c^{3} \arctan \left (c x \right ) x^{2}}{2 e}+\frac {b \,c^{3} \arctan \left (c x \right ) d^{2} \ln \left (c e x +c d \right )}{e^{3}}+\frac {i b \,c^{3} d^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {-c e x +i e}{c d +i e}\right )}{2 e^{3}}-\frac {i b \,c^{3} d^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {c e x +i e}{-c d +i e}\right )}{2 e^{3}}+\frac {i b \,c^{3} d^{2} \dilog \left (\frac {-c e x +i e}{c d +i e}\right )}{2 e^{3}}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c e x +i e}{-c d +i e}\right )}{2 e^{3}}+\frac {b \,c^{2} d \ln \left (c^{2} d^{2}-2 c d \left (c e x +c d \right )+e^{2}+\left (c e x +c d \right )^{2}\right )}{2 e^{2}}+\frac {b c \arctan \left (c x \right )}{2 e}-\frac {b \,c^{2} d}{2 e^{2}}-\frac {b \,c^{2} x}{2 e}}{c^{3}}\) | \(337\) |
default | \(\frac {-\frac {a \,c^{3} d x}{e^{2}}+\frac {a \,c^{3} x^{2}}{2 e}+\frac {a \,c^{3} d^{2} \ln \left (c e x +c d \right )}{e^{3}}-\frac {b \,c^{3} \arctan \left (c x \right ) d x}{e^{2}}+\frac {b \,c^{3} \arctan \left (c x \right ) x^{2}}{2 e}+\frac {b \,c^{3} \arctan \left (c x \right ) d^{2} \ln \left (c e x +c d \right )}{e^{3}}+\frac {i b \,c^{3} d^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {-c e x +i e}{c d +i e}\right )}{2 e^{3}}-\frac {i b \,c^{3} d^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {c e x +i e}{-c d +i e}\right )}{2 e^{3}}+\frac {i b \,c^{3} d^{2} \dilog \left (\frac {-c e x +i e}{c d +i e}\right )}{2 e^{3}}-\frac {i b \,c^{3} d^{2} \dilog \left (\frac {c e x +i e}{-c d +i e}\right )}{2 e^{3}}+\frac {b \,c^{2} d \ln \left (c^{2} d^{2}-2 c d \left (c e x +c d \right )+e^{2}+\left (c e x +c d \right )^{2}\right )}{2 e^{2}}+\frac {b c \arctan \left (c x \right )}{2 e}-\frac {b \,c^{2} d}{2 e^{2}}-\frac {b \,c^{2} x}{2 e}}{c^{3}}\) | \(337\) |
risch | \(\frac {a \,x^{2}}{2 e}+\frac {i b \ln \left (i c x +1\right ) x d}{2 e^{2}}-\frac {i b \ln \left (-i c x +1\right ) d x}{2 e^{2}}+\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c \,e^{2}}+\frac {i b \,d^{2} \dilog \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{3}}-\frac {i b \,d^{2} \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{3}}+\frac {i b \ln \left (-i c x +1\right ) x^{2}}{4 e}+\frac {i b \,d^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{3}}-\frac {b d}{c \,e^{2}}-\frac {i b \,d^{2} \dilog \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{3}}-\frac {i b \ln \left (i c x +1\right ) x^{2}}{4 e}-\frac {i a d}{c \,e^{2}}+\frac {b \arctan \left (c x \right )}{2 c^{2} e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{e^{3}}+\frac {a}{2 c^{2} e}-\frac {b x}{2 c e}\) | \(368\) |
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^{2} \left (a + b \operatorname {atan}{\left (c 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^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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