Optimal. Leaf size=170 \[ \frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{b^2}{12 d e^5 (c+d x)^2}-\frac{2 b^2 \log (c+d x)}{3 d e^5}+\frac{b^2 \log \left ((c+d x)^2+1\right )}{3 d e^5} \]
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Rubi [A] time = 0.226746, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5043, 12, 4852, 4918, 266, 44, 36, 29, 31, 4884} \[ \frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{b^2}{12 d e^5 (c+d x)^2}-\frac{2 b^2 \log (c+d x)}{3 d e^5}+\frac{b^2 \log \left ((c+d x)^2+1\right )}{3 d e^5} \]
Antiderivative was successfully verified.
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Rule 5043
Rule 12
Rule 4852
Rule 4918
Rule 266
Rule 44
Rule 36
Rule 29
Rule 31
Rule 4884
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{(c e+d e x)^5} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tan ^{-1}(x)\right )^2}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tan ^{-1}(x)\right )^2}{x^5} \, dx,x,c+d x\right )}{d e^5}\\ &=-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{b \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{b \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{2 d e^5}-\frac{b \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(x)}{x^2 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac{b \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac{b \operatorname{Subst}\left (\int \frac{a+b \tan ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1+x^2\right )} \, dx,x,c+d x\right )}{6 d e^5}\\ &=-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,(c+d x)^2\right )}{12 d e^5}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,(c+d x)^2\right )}{12 d e^5}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac{b^2}{12 d e^5 (c+d x)^2}-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac{b^2 \log (c+d x)}{6 d e^5}+\frac{b^2 \log \left (1+(c+d x)^2\right )}{12 d e^5}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,(c+d x)^2\right )}{4 d e^5}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac{b^2}{12 d e^5 (c+d x)^2}-\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac{b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac{\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac{2 b^2 \log (c+d x)}{3 d e^5}+\frac{b^2 \log \left (1+(c+d x)^2\right )}{3 d e^5}\\ \end{align*}
Mathematica [A] time = 0.270523, size = 245, normalized size = 1.44 \[ -\frac{3 a^2-2 b \tan ^{-1}(c+d x) \left (3 a \left (6 c^2 d^2 x^2+4 c^3 d x+c^4+4 c d^3 x^3+d^4 x^4-1\right )+b \left (9 c^2 d x+3 c^3+9 c d^2 x^2-c+3 d^3 x^3-d x\right )\right )-6 a b (c+d x)^3+2 a b (c+d x)-4 b^2 (c+d x)^4 \log \left (c^2+2 c d x+d^2 x^2+1\right )-3 b^2 \left (6 c^2 d^2 x^2+4 c^3 d x+c^4+4 c d^3 x^3+d^4 x^4-1\right ) \tan ^{-1}(c+d x)^2+b^2 (c+d x)^2+8 b^2 (c+d x)^4 \log (c+d x)}{12 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 242, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}}{4\,d{e}^{5} \left ( dx+c \right ) ^{4}}}-{\frac{{b}^{2} \left ( \arctan \left ( dx+c \right ) \right ) ^{2}}{4\,d{e}^{5} \left ( dx+c \right ) ^{4}}}+{\frac{{b}^{2} \left ( \arctan \left ( dx+c \right ) \right ) ^{2}}{4\,d{e}^{5}}}-{\frac{{b}^{2}\arctan \left ( dx+c \right ) }{6\,d{e}^{5} \left ( dx+c \right ) ^{3}}}+{\frac{{b}^{2}\arctan \left ( dx+c \right ) }{2\,d{e}^{5} \left ( dx+c \right ) }}+{\frac{{b}^{2}\ln \left ( 1+ \left ( dx+c \right ) ^{2} \right ) }{3\,d{e}^{5}}}-{\frac{{b}^{2}}{12\,d{e}^{5} \left ( dx+c \right ) ^{2}}}-{\frac{2\,{b}^{2}\ln \left ( dx+c \right ) }{3\,d{e}^{5}}}-{\frac{ab\arctan \left ( dx+c \right ) }{2\,d{e}^{5} \left ( dx+c \right ) ^{4}}}+{\frac{ab\arctan \left ( dx+c \right ) }{2\,d{e}^{5}}}-{\frac{ab}{6\,d{e}^{5} \left ( dx+c \right ) ^{3}}}+{\frac{ab}{2\,d{e}^{5} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.8787, size = 721, normalized size = 4.24 \begin{align*} \frac{1}{6} \,{\left (d{\left (\frac{3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} + \frac{3 \, \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{2} e^{5}}\right )} - \frac{3 \, \arctan \left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} a b + \frac{1}{12} \,{\left (2 \, d{\left (\frac{3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} + \frac{3 \, \arctan \left (\frac{d^{2} x + c d}{d}\right )}{d^{2} e^{5}}\right )} \arctan \left (d x + c\right ) - \frac{{\left (3 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \arctan \left (d x + c\right )^{2} - 4 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 8 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c\right ) + 1\right )} d^{2}}{d^{5} e^{5} x^{2} + 2 \, c d^{4} e^{5} x + c^{2} d^{3} e^{5}}\right )} b^{2} - \frac{b^{2} \arctan \left (d x + c\right )^{2}}{4 \,{\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} - \frac{a^{2}}{4 \,{\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09413, size = 937, normalized size = 5.51 \begin{align*} \frac{6 \, a b d^{3} x^{3} + 6 \, a b c^{3} +{\left (18 \, a b c - b^{2}\right )} d^{2} x^{2} - b^{2} c^{2} - 2 \, a b c + 2 \,{\left (9 \, a b c^{2} - b^{2} c - a b\right )} d x + 3 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \arctan \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \,{\left (3 \, a b d^{4} x^{4} + 3 \,{\left (4 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, a b c^{4} + 3 \, b^{2} c^{3} + 9 \,{\left (2 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} - b^{2} c +{\left (12 \, a b c^{3} + 9 \, b^{2} c^{2} - b^{2}\right )} d x - 3 \, a b\right )} \arctan \left (d x + c\right ) + 4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 8 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right )}{12 \,{\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.49382, size = 1902, normalized size = 11.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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