Optimal. Leaf size=48 \[ \frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {b e x}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6107, 12, 5916, 321, 206} \[ \frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {b e x}{2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 321
Rule 5916
Rule 6107
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e x}{2}+\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {b e x}{2}-\frac {b e \tanh ^{-1}(c+d x)}{2 d}+\frac {e (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 1.60 \[ \frac {e \left (2 a c^2+4 a c d x+2 a d^2 x^2+b \log (-c-d x+1)-b \log (c+d x+1)+2 b (c+d x)^2 \tanh ^{-1}(c+d x)+2 b c+2 b d x\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 73, normalized size = 1.52 \[ \frac {2 \, a d^{2} e x^{2} + 2 \, {\left (2 \, a c + b\right )} d e x + {\left (b d^{2} e x^{2} + 2 \, b c d e x + {\left (b c^{2} - b\right )} e\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 137, normalized size = 2.85 \[ \frac {{\left (\frac {{\left (d x + c + 1\right )} b e \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{d x + c - 1} + \frac {2 \, {\left (d x + c + 1\right )} a e}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b e}{d x + c - 1} - b e\right )} {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )}}{2 \, {\left (\frac {{\left (d x + c + 1\right )}^{2} d^{2}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{2}}{d x + c - 1} + d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 107, normalized size = 2.23 \[ \frac {a d e \,x^{2}}{2}+x a c e +\frac {a \,c^{2} e}{2 d}+\frac {d \arctanh \left (d x +c \right ) x^{2} b e}{2}+\arctanh \left (d x +c \right ) x b c e +\frac {\arctanh \left (d x +c \right ) b \,c^{2} e}{2 d}+\frac {b e x}{2}+\frac {b e c}{2 d}+\frac {b e \ln \left (d x +c -1\right )}{4 d}-\frac {b e \ln \left (d x +c +1\right )}{4 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 113, normalized size = 2.35 \[ \frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b c e}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 73, normalized size = 1.52 \[ \frac {b\,e\,x}{2}+a\,c\,e\,x-\frac {b\,e\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d}+\frac {a\,d\,e\,x^2}{2}+\frac {b\,c^2\,e\,\mathrm {atanh}\left (c+d\,x\right )}{2\,d}+b\,c\,e\,x\,\mathrm {atanh}\left (c+d\,x\right )+\frac {b\,d\,e\,x^2\,\mathrm {atanh}\left (c+d\,x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.66, size = 95, normalized size = 1.98 \[ \begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {atanh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {atanh}{\left (c + d x \right )} + \frac {b d e x^{2} \operatorname {atanh}{\left (c + d x \right )}}{2} + \frac {b e x}{2} - \frac {b e \operatorname {atanh}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {atanh}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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