Optimal. Leaf size=120 \[ \frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}+\frac {b (-c f+d e+f)^3 \log (-c-d x+1)}{6 d^3 f}-\frac {b (d e-(c+1) f)^3 \log (c+d x+1)}{6 d^3 f}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {b f x (d e-c f)}{d^2} \]
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Rubi [A] time = 0.20, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6111, 5926, 702, 633, 31} \[ \frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}+\frac {b f x (d e-c f)}{d^2}+\frac {b (-c f+d e+f)^3 \log (-c-d x+1)}{6 d^3 f}-\frac {b (d e-(c+1) f)^3 \log (c+d x+1)}{6 d^3 f}+\frac {b f^2 (c+d x)^2}{6 d^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 5926
Rule 6111
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}-\frac {b \operatorname {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^3}{1-x^2} \, dx,x,c+d x\right )}{3 f}\\ &=\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f)}{d^3}-\frac {f^3 x}{d^3}+\frac {(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}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}-\frac {b \operatorname {Subst}\left (\int \frac {(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}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}\\ &=\frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}-\frac {\left (b (d e+f-c f)^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{6 d^3 f}+\frac {\left (b (d e-(1+c) f)^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{6 d^3 f}\\ &=\frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 f}+\frac {b (d e+f-c f)^3 \log (1-c-d x)}{6 d^3 f}-\frac {b (d e-(1+c) f)^3 \log (1+c+d x)}{6 d^3 f}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 174, normalized size = 1.45 \[ \frac {2 d x \left (3 a d^2 e^2+b f (3 d e-2 c f)\right )+d^2 f x^2 (6 a d e+b f)+2 a d^3 f^2 x^3+2 b d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \tanh ^{-1}(c+d x)-b (c-1) \left (-3 (c-1) d e f+(c-1)^2 f^2+3 d^2 e^2\right ) \log (-c-d x+1)+b (c+1) \left (-3 (c+1) d e f+(c+1)^2 f^2+3 d^2 e^2\right ) \log (c+d x+1)}{6 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 242, normalized size = 2.02 \[ \frac {2 \, a d^{3} f^{2} x^{3} + {\left (6 \, a d^{3} e f + b d^{2} f^{2}\right )} x^{2} + 2 \, {\left (3 \, a d^{3} e^{2} + 3 \, b d^{2} e f - 2 \, b c d f^{2}\right )} x + {\left (3 \, {\left (b c + b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} + 2 \, b c + b\right )} d e f + {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} f^{2}\right )} \log \left (d x + c + 1\right ) - {\left (3 \, {\left (b c - b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} - 2 \, b c + b\right )} d e f + {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} f^{2}\right )} \log \left (d x + c - 1\right ) + {\left (b d^{3} f^{2} x^{3} + 3 \, b d^{3} e f x^{2} + 3 \, b d^{3} e^{2} x\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 1717, normalized size = 14.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 477, normalized size = 3.98 \[ \frac {b \arctanh \left (d x +c \right ) e^{3}}{3 f}+\frac {b \,f^{2} x^{2}}{6 d}+a f \,x^{2} e +\frac {b \,f^{2} \arctanh \left (d x +c \right ) x^{3}}{3}+\arctanh \left (d x +c \right ) x b \,e^{2}+\frac {b \ln \left (d x +c -1\right ) e^{3}}{6 f}+\frac {e^{2} b \ln \left (d x +c +1\right )}{2 d}+\frac {e^{2} b \ln \left (d x +c -1\right )}{2 d}+\frac {b \,f^{2} \ln \left (d x +c -1\right )}{6 d^{3}}+\frac {b \,f^{2} \ln \left (d x +c +1\right )}{6 d^{3}}-\frac {b \ln \left (d x +c +1\right ) e^{3}}{6 f}+\frac {a \,e^{3}}{3 f}-\frac {5 b \,f^{2} c^{2}}{6 d^{3}}+\frac {a \,f^{2} x^{3}}{3}+a x \,e^{2}-\frac {2 b \,f^{2} c x}{3 d^{2}}-\frac {b f \ln \left (d x +c +1\right ) c^{2} e}{2 d^{2}}+\frac {b f \ln \left (d x +c -1\right ) c^{2} e}{2 d^{2}}-\frac {b f \ln \left (d x +c -1\right ) c e}{d^{2}}-\frac {b f \ln \left (d x +c +1\right ) c e}{d^{2}}+\frac {b \,f^{2} \ln \left (d x +c +1\right ) c}{2 d^{3}}+\frac {b \ln \left (d x +c +1\right ) c \,e^{2}}{2 d}-\frac {b \ln \left (d x +c -1\right ) c \,e^{2}}{2 d}+\frac {b \,f^{2} \ln \left (d x +c +1\right ) c^{2}}{2 d^{3}}+\frac {b \,f^{2} \ln \left (d x +c +1\right ) c^{3}}{6 d^{3}}-\frac {b \,f^{2} \ln \left (d x +c -1\right ) c}{2 d^{3}}-\frac {b f \ln \left (d x +c +1\right ) e}{2 d^{2}}+b f \arctanh \left (d x +c \right ) e \,x^{2}+\frac {b f c e}{d^{2}}+\frac {b f e x}{d}+\frac {b \,f^{2} \ln \left (d x +c -1\right ) c^{2}}{2 d^{3}}-\frac {b \,f^{2} \ln \left (d x +c -1\right ) c^{3}}{6 d^{3}}+\frac {b f \ln \left (d x +c -1\right ) e}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 207, normalized size = 1.72 \[ \frac {1}{3} \, a f^{2} x^{3} + a e f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b e f + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b f^{2} + a e^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{2}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 381, normalized size = 3.18 \[ x^2\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{6\,d}-\frac {a\,c\,f^2}{d}\right )-\ln \left (1-d\,x-c\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )-x\,\left (\frac {2\,c\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{3\,d}-\frac {2\,a\,c\,f^2}{d}\right )}{d}-\frac {3\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+3\,a\,d^2\,e^2+3\,b\,d\,e\,f-3\,a\,f^2}{3\,d^2}+\frac {a\,f^2\,\left (3\,c^2-3\right )}{3\,d^2}\right )+\ln \left (c+d\,x+1\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )+\frac {a\,f^2\,x^3}{3}+\frac {\ln \left (c+d\,x-1\right )\,\left (\frac {b\,f^2}{6}+d\,\left (\frac {b\,e\,f\,c^2}{2}-b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}-\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}-\frac {b\,c^3\,f^2}{6}-\frac {b\,c\,f^2}{2}\right )}{d^3}+\frac {\ln \left (c+d\,x+1\right )\,\left (\frac {b\,f^2}{6}-d\,\left (\frac {b\,e\,f\,c^2}{2}+b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}+\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}+\frac {b\,c^3\,f^2}{6}+\frac {b\,c\,f^2}{2}\right )}{d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.57, size = 369, normalized size = 3.08 \[ \begin {cases} a e^{2} x + a e f x^{2} + \frac {a f^{2} x^{3}}{3} + \frac {b c^{3} f^{2} \operatorname {atanh}{\left (c + d x \right )}}{3 d^{3}} - \frac {b c^{2} e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} + \frac {b c^{2} f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {b c^{2} f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + \frac {b c e^{2} \operatorname {atanh}{\left (c + d x \right )}}{d} - \frac {2 b c e f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {2 b c e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} - \frac {2 b c f^{2} x}{3 d^{2}} + \frac {b c f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + b e^{2} x \operatorname {atanh}{\left (c + d x \right )} + b e f x^{2} \operatorname {atanh}{\left (c + d x \right )} + \frac {b f^{2} x^{3} \operatorname {atanh}{\left (c + d x \right )}}{3} + \frac {b e^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e^{2} \operatorname {atanh}{\left (c + d x \right )}}{d} + \frac {b e f x}{d} + \frac {b f^{2} x^{2}}{6 d} - \frac {b e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} + \frac {b f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{3 d^{3}} - \frac {b f^{2} \operatorname {atanh}{\left (c + d x \right )}}{3 d^{3}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {atanh}{\relax (c )}\right ) \left (e^{2} x + e f x^{2} + \frac {f^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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