Optimal. Leaf size=382 \[ \frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}-\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}-\frac {2 b^2 f (c+d x) (d e-c f) \tan ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]
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Rubi [A] time = 0.57, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5047, 4864, 4846, 260, 4852, 321, 203, 4984, 4884, 4920, 4854, 2402, 2315} \[ \frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}-\frac {2 a b f x (d e-c f)}{d^2}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}-\frac {2 b^2 f (c+d x) (d e-c f) \tan ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4864
Rule 4884
Rule 4920
Rule 4984
Rule 5047
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \tan ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \operatorname {Subst}\left (\int \left (\frac {3 f^2 (d e-c f) \left (a+b \tan ^{-1}(x)\right )}{d^3}+\frac {f^3 x \left (a+b \tan ^{-1}(x)\right )}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \tan ^{-1}(x)\right )}{d^3 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3 f}-\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int x \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}-\frac {(2 b f (d e-c f)) \operatorname {Subst}\left (\int \left (a+b \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}\\ &=-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \operatorname {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(x)\right )}{1+x^2}+\frac {f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x \left (a+b \tan ^{-1}(x)\right )}{1+x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}+\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac {\left (2 b^2 f (d e-c f)\right ) \operatorname {Subst}\left (\int \tan ^{-1}(x) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {2 b^2 f (d e-c f) (c+d x) \tan ^{-1}(c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}-\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac {\left (2 b^2 f (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \tan ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac {\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{3 d^3 f}\\ &=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac {2 b^2 f (d e-c f) (c+d x) \tan ^{-1}(c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {\left (2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac {2 b^2 f (d e-c f) (c+d x) \tan ^{-1}(c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}-\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac {2 b^2 f (d e-c f) (c+d x) \tan ^{-1}(c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {\left (2 i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac {2 b^2 f (d e-c f) (c+d x) \tan ^{-1}(c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \tan ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{3 d^3}\\ \end {align*}
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Mathematica [B] time = 4.23, size = 801, normalized size = 2.10 \[ \frac {1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac {a b \left (-d f x (6 d e-4 c f+d f x)+2 \left (f^2 c^3-3 d e f c^2+3 \left (d^2 e^2-f^2\right ) c+3 d e f+d^3 x \left (3 e^2+3 f x e+f^2 x^2\right )\right ) \tan ^{-1}(c+d x)+\left (-3 d^2 e^2+6 c d f e+\left (1-3 c^2\right ) f^2\right ) \log \left ((c+d x)^2+1\right )\right )}{3 d^3}+\frac {b^2 e^2 \left (\tan ^{-1}(c+d x) \left ((c+d x-i) \tan ^{-1}(c+d x)+2 \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )\right )-i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c+d x)}\right )\right )}{d}+\frac {b^2 e f \left (\left (-c^2+2 i c+d^2 x^2+1\right ) \tan ^{-1}(c+d x)^2-2 \left (2 \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right ) c+c+d x\right ) \tan ^{-1}(c+d x)+\log \left ((c+d x)^2+1\right )+2 i c \text {Li}_2\left (-e^{2 i \tan ^{-1}(c+d x)}\right )\right )}{d^2}+\frac {b^2 f^2 \left ((c+d x)^2+1\right )^{3/2} \left (\frac {3 (c+d x) \tan ^{-1}(c+d x)^2 c^2}{\sqrt {(c+d x)^2+1}}-3 i \tan ^{-1}(c+d x)^2 \cos \left (3 \tan ^{-1}(c+d x)\right ) c^2+6 \tan ^{-1}(c+d x) \cos \left (3 \tan ^{-1}(c+d x)\right ) \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right ) c^2+3 \tan ^{-1}(c+d x)^2 \sin \left (3 \tan ^{-1}(c+d x)\right ) c^2+\frac {6 (c+d x) \tan ^{-1}(c+d x) c}{\sqrt {(c+d x)^2+1}}+6 \cos \left (3 \tan ^{-1}(c+d x)\right ) \log \left (\frac {1}{\sqrt {(c+d x)^2+1}}\right ) c+6 \tan ^{-1}(c+d x) \sin \left (3 \tan ^{-1}(c+d x)\right ) c+\frac {3 (c+d x) \tan ^{-1}(c+d x)^2}{\sqrt {(c+d x)^2+1}}+i \tan ^{-1}(c+d x)^2 \cos \left (3 \tan ^{-1}(c+d x)\right )-2 \tan ^{-1}(c+d x) \cos \left (3 \tan ^{-1}(c+d x)\right ) \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )+\frac {\left (-9 i c^2-12 c+3 i\right ) \tan ^{-1}(c+d x)^2+2 \left (\left (9 c^2-3\right ) \log \left (1+e^{2 i \tan ^{-1}(c+d x)}\right )-2\right ) \tan ^{-1}(c+d x)+18 c \log \left (\frac {1}{\sqrt {(c+d x)^2+1}}\right )}{\sqrt {(c+d x)^2+1}}-\frac {4 i \left (3 c^2-1\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c+d x)}\right )}{\left ((c+d x)^2+1\right )^{3/2}}-\tan ^{-1}(c+d x)^2 \sin \left (3 \tan ^{-1}(c+d x)\right )+\sin \left (3 \tan ^{-1}(c+d x)\right )+\frac {c+d x}{\sqrt {(c+d x)^2+1}}\right )}{12 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} + {\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \arctan \left (d x + c\right )^{2} + 2 \, {\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\right )} \arctan \left (d x + c\right ), x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 1622, normalized size = 4.25 \[ \text {result too large to display} \]
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 \[ \text {result too large to display} \]
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 \[ \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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