Optimal. Leaf size=62 \[ \frac {c \tanh ^{-1}\left (\frac {c+2 d x}{\sqrt {c^2-4 b d}}\right )}{b \sqrt {c^2-4 b d}}-\frac {\log \left (b+c x+d x^2\right )}{2 b}+\frac {\log (x)}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1594, 705, 29, 634, 618, 206, 628} \[ \frac {c \tanh ^{-1}\left (\frac {c+2 d x}{\sqrt {c^2-4 b d}}\right )}{b \sqrt {c^2-4 b d}}-\frac {\log \left (b+c x+d x^2\right )}{2 b}+\frac {\log (x)}{b} \]
Antiderivative was successfully verified.
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Rule 29
Rule 206
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1594
Rubi steps
\begin {align*} \int \frac {1}{b x+c x^2+d x^3} \, dx &=\int \frac {1}{x \left (b+c x+d x^2\right )} \, dx\\ &=\frac {\int \frac {1}{x} \, dx}{b}+\frac {\int \frac {-c-d x}{b+c x+d x^2} \, dx}{b}\\ &=\frac {\log (x)}{b}-\frac {\int \frac {c+2 d x}{b+c x+d x^2} \, dx}{2 b}-\frac {c \int \frac {1}{b+c x+d x^2} \, dx}{2 b}\\ &=\frac {\log (x)}{b}-\frac {\log \left (b+c x+d x^2\right )}{2 b}+\frac {c \operatorname {Subst}\left (\int \frac {1}{c^2-4 b d-x^2} \, dx,x,c+2 d x\right )}{b}\\ &=\frac {c \tanh ^{-1}\left (\frac {c+2 d x}{\sqrt {c^2-4 b d}}\right )}{b \sqrt {c^2-4 b d}}+\frac {\log (x)}{b}-\frac {\log \left (b+c x+d x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 61, normalized size = 0.98 \[ -\frac {\frac {2 c \tan ^{-1}\left (\frac {c+2 d x}{\sqrt {4 b d-c^2}}\right )}{\sqrt {4 b d-c^2}}+\log (b+x (c+d x))-2 \log (x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 211, normalized size = 3.40 \[ \left [\frac {\sqrt {c^{2} - 4 \, b d} c \log \left (\frac {2 \, d^{2} x^{2} + 2 \, c d x + c^{2} - 2 \, b d + \sqrt {c^{2} - 4 \, b d} {\left (2 \, d x + c\right )}}{d x^{2} + c x + b}\right ) - {\left (c^{2} - 4 \, b d\right )} \log \left (d x^{2} + c x + b\right ) + 2 \, {\left (c^{2} - 4 \, b d\right )} \log \relax (x)}{2 \, {\left (b c^{2} - 4 \, b^{2} d\right )}}, \frac {2 \, \sqrt {-c^{2} + 4 \, b d} c \arctan \left (-\frac {\sqrt {-c^{2} + 4 \, b d} {\left (2 \, d x + c\right )}}{c^{2} - 4 \, b d}\right ) - {\left (c^{2} - 4 \, b d\right )} \log \left (d x^{2} + c x + b\right ) + 2 \, {\left (c^{2} - 4 \, b d\right )} \log \relax (x)}{2 \, {\left (b c^{2} - 4 \, b^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 62, normalized size = 1.00 \[ -\frac {c \arctan \left (\frac {2 \, d x + c}{\sqrt {-c^{2} + 4 \, b d}}\right )}{\sqrt {-c^{2} + 4 \, b d} b} - \frac {\log \left (d x^{2} + c x + b\right )}{2 \, b} + \frac {\log \left ({\left | x \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 1.00 \[ -\frac {c \arctan \left (\frac {2 d x +c}{\sqrt {4 b d -c^{2}}}\right )}{\sqrt {4 b d -c^{2}}\, b}+\frac {\ln \relax (x )}{b}-\frac {\ln \left (d \,x^{2}+c x +b \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 213, normalized size = 3.44 \[ \frac {\ln \relax (x)}{b}-\ln \left (\left (x\,\left (6\,b\,d^2-2\,c^2\,d\right )-b\,c\,d\right )\,\left (\frac {1}{2\,b}-\frac {c\,\sqrt {c^2-4\,b\,d}}{2\,\left (b\,c^2-4\,b^2\,d\right )}\right )-c\,d-3\,d^2\,x\right )\,\left (\frac {1}{2\,b}-\frac {c\,\sqrt {c^2-4\,b\,d}}{2\,\left (b\,c^2-4\,b^2\,d\right )}\right )-\ln \left (\left (x\,\left (6\,b\,d^2-2\,c^2\,d\right )-b\,c\,d\right )\,\left (\frac {1}{2\,b}+\frac {c\,\sqrt {c^2-4\,b\,d}}{2\,\left (b\,c^2-4\,b^2\,d\right )}\right )-c\,d-3\,d^2\,x\right )\,\left (\frac {1}{2\,b}+\frac {c\,\sqrt {c^2-4\,b\,d}}{2\,\left (b\,c^2-4\,b^2\,d\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.19, size = 564, normalized size = 9.10 \[ \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) \log {\left (x + \frac {24 b^{4} d^{2} \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) + 2 b^{2} c^{4} \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (- \frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) \log {\left (x + \frac {24 b^{4} d^{2} \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} - 14 b^{3} c^{2} d \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} - 12 b^{3} d^{2} \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) + 2 b^{2} c^{4} \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right )^{2} + 3 b^{2} c^{2} d \left (\frac {c \sqrt {- 4 b d + c^{2}}}{2 b \left (4 b d - c^{2}\right )} - \frac {1}{2 b}\right ) - 12 b^{2} d^{2} + 11 b c^{2} d - 2 c^{4}}{9 b c d^{2} - 2 c^{3} d} \right )} + \frac {\log {\relax (x )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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