Optimal. Leaf size=143 \[ \frac {b d^2 x}{2 c^2}+\frac {4 b d^2 x^2}{15 c}+\frac {1}{6} b d^2 x^3+\frac {1}{20} b c d^2 x^4+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {31 b d^2 \log (1-c x)}{60 c^3}+\frac {b d^2 \log (1+c x)}{60 c^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {45, 6083, 12,
1816, 647, 31} \begin {gather*} \frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {31 b d^2 \log (1-c x)}{60 c^3}+\frac {b d^2 \log (c x+1)}{60 c^3}+\frac {b d^2 x}{2 c^2}+\frac {1}{20} b c d^2 x^4+\frac {4 b d^2 x^2}{15 c}+\frac {1}{6} b d^2 x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 45
Rule 647
Rule 1816
Rule 6083
Rubi steps
\begin {align*} \int x^2 (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^3 \left (10+15 c x+6 c^2 x^2\right )}{30 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \int \frac {x^3 \left (10+15 c x+6 c^2 x^2\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{30} \left (b c d^2\right ) \int \left (-\frac {15}{c^3}-\frac {16 x}{c^2}-\frac {15 x^2}{c}-6 x^3+\frac {15+16 c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {b d^2 x}{2 c^2}+\frac {4 b d^2 x^2}{15 c}+\frac {1}{6} b d^2 x^3+\frac {1}{20} b c d^2 x^4+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \int \frac {15+16 c x}{1-c^2 x^2} \, dx}{30 c^2}\\ &=\frac {b d^2 x}{2 c^2}+\frac {4 b d^2 x^2}{15 c}+\frac {1}{6} b d^2 x^3+\frac {1}{20} b c d^2 x^4+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \int \frac {1}{-c-c^2 x} \, dx}{60 c}-\frac {\left (31 b d^2\right ) \int \frac {1}{c-c^2 x} \, dx}{60 c}\\ &=\frac {b d^2 x}{2 c^2}+\frac {4 b d^2 x^2}{15 c}+\frac {1}{6} b d^2 x^3+\frac {1}{20} b c d^2 x^4+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {31 b d^2 \log (1-c x)}{60 c^3}+\frac {b d^2 \log (1+c x)}{60 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 115, normalized size = 0.80 \begin {gather*} \frac {d^2 \left (30 b c x+16 b c^2 x^2+20 a c^3 x^3+10 b c^3 x^3+30 a c^4 x^4+3 b c^4 x^4+12 a c^5 x^5+2 b c^3 x^3 \left (10+15 c x+6 c^2 x^2\right ) \tanh ^{-1}(c x)+31 b \log (1-c x)+b \log (1+c x)\right )}{60 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 152, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{2} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+\frac {d^{2} b \arctanh \left (c x \right ) c^{5} x^{5}}{5}+\frac {d^{2} b \arctanh \left (c x \right ) c^{4} x^{4}}{2}+\frac {d^{2} b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+\frac {d^{2} b \,c^{4} x^{4}}{20}+\frac {d^{2} b \,c^{3} x^{3}}{6}+\frac {4 d^{2} b \,c^{2} x^{2}}{15}+\frac {b c \,d^{2} x}{2}+\frac {31 d^{2} b \ln \left (c x -1\right )}{60}+\frac {d^{2} b \ln \left (c x +1\right )}{60}}{c^{3}}\) | \(152\) |
default | \(\frac {d^{2} a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{2} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+\frac {d^{2} b \arctanh \left (c x \right ) c^{5} x^{5}}{5}+\frac {d^{2} b \arctanh \left (c x \right ) c^{4} x^{4}}{2}+\frac {d^{2} b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+\frac {d^{2} b \,c^{4} x^{4}}{20}+\frac {d^{2} b \,c^{3} x^{3}}{6}+\frac {4 d^{2} b \,c^{2} x^{2}}{15}+\frac {b c \,d^{2} x}{2}+\frac {31 d^{2} b \ln \left (c x -1\right )}{60}+\frac {d^{2} b \ln \left (c x +1\right )}{60}}{c^{3}}\) | \(152\) |
risch | \(\frac {d^{2} b \,x^{3} \left (6 c^{2} x^{2}+15 c x +10\right ) \ln \left (c x +1\right )}{60}-\frac {d^{2} c^{2} x^{5} b \ln \left (-c x +1\right )}{10}+\frac {d^{2} c^{2} x^{5} a}{5}-\frac {d^{2} c \,x^{4} b \ln \left (-c x +1\right )}{4}+\frac {d^{2} c \,x^{4} a}{2}+\frac {b c \,d^{2} x^{4}}{20}-\frac {d^{2} x^{3} b \ln \left (-c x +1\right )}{6}+\frac {d^{2} x^{3} a}{3}+\frac {b \,d^{2} x^{3}}{6}+\frac {4 b \,d^{2} x^{2}}{15 c}+\frac {b \,d^{2} x}{2 c^{2}}+\frac {b \,d^{2} \ln \left (c x +1\right )}{60 c^{3}}+\frac {31 b \,d^{2} \ln \left (-c x +1\right )}{60 c^{3}}\) | \(186\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 184, normalized size = 1.29 \begin {gather*} \frac {1}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{2} \, a c d^{2} x^{4} + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c d^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 146, normalized size = 1.02 \begin {gather*} \frac {12 \, a c^{5} d^{2} x^{5} + 3 \, {\left (10 \, a + b\right )} c^{4} d^{2} x^{4} + 10 \, {\left (2 \, a + b\right )} c^{3} d^{2} x^{3} + 16 \, b c^{2} d^{2} x^{2} + 30 \, b c d^{2} x + b d^{2} \log \left (c x + 1\right ) + 31 \, b d^{2} \log \left (c x - 1\right ) + {\left (6 \, b c^{5} d^{2} x^{5} + 15 \, b c^{4} d^{2} x^{4} + 10 \, b c^{3} d^{2} x^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{60 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 177, normalized size = 1.24 \begin {gather*} \begin {cases} \frac {a c^{2} d^{2} x^{5}}{5} + \frac {a c d^{2} x^{4}}{2} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{2} d^{2} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c d^{2} x^{4} \operatorname {atanh}{\left (c x \right )}}{2} + \frac {b c d^{2} x^{4}}{20} + \frac {b d^{2} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {b d^{2} x^{3}}{6} + \frac {4 b d^{2} x^{2}}{15 c} + \frac {b d^{2} x}{2 c^{2}} + \frac {8 b d^{2} \log {\left (x - \frac {1}{c} \right )}}{15 c^{3}} + \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{30 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs.
\(2 (125) = 250\).
time = 0.42, size = 525, normalized size = 3.67 \begin {gather*} \frac {4}{15} \, c {\left (\frac {{\left (\frac {15 \, {\left (c x + 1\right )}^{4} b d^{2}}{{\left (c x - 1\right )}^{4}} - \frac {15 \, {\left (c x + 1\right )}^{3} b d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {20 \, {\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {10 \, {\left (c x + 1\right )} b d^{2}}{c x - 1} + 2 \, b d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} - \frac {5 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {10 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} + \frac {\frac {30 \, {\left (c x + 1\right )}^{4} a d^{2}}{{\left (c x - 1\right )}^{4}} - \frac {30 \, {\left (c x + 1\right )}^{3} a d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {40 \, {\left (c x + 1\right )}^{2} a d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {20 \, {\left (c x + 1\right )} a d^{2}}{c x - 1} + 4 \, a d^{2} + \frac {13 \, {\left (c x + 1\right )}^{4} b d^{2}}{{\left (c x - 1\right )}^{4}} - \frac {36 \, {\left (c x + 1\right )}^{3} b d^{2}}{{\left (c x - 1\right )}^{3}} + \frac {41 \, {\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {23 \, {\left (c x + 1\right )} b d^{2}}{c x - 1} + 5 \, b d^{2}}{\frac {{\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} - \frac {5 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} - \frac {10 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )} c^{4}}{c x - 1} - c^{4}} - \frac {2 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{4}} + \frac {2 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{4}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 134, normalized size = 0.94 \begin {gather*} \frac {\frac {4\,b\,c^2\,d^2\,x^2}{15}-\frac {d^2\,\left (30\,b\,\mathrm {atanh}\left (c\,x\right )-16\,b\,\ln \left (c^2\,x^2-1\right )\right )}{60}+\frac {b\,c\,d^2\,x}{2}}{c^3}+\frac {d^2\,\left (20\,a\,x^3+10\,b\,x^3+20\,b\,x^3\,\mathrm {atanh}\left (c\,x\right )\right )}{60}+\frac {c^2\,d^2\,\left (12\,a\,x^5+12\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{60}+\frac {c\,d^2\,\left (30\,a\,x^4+3\,b\,x^4+30\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{60} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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