Optimal. Leaf size=96 \[ \frac {2 (b+2 c x) \log (b+2 c x)}{\sqrt {\frac {b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac {2 (b+2 c x) \log (d+e x)}{\sqrt {\frac {b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {646, 36, 31} \[ \frac {2 (b+2 c x) \log (b+2 c x)}{\sqrt {\frac {b^2}{c}+4 b x+4 c x^2} (2 c d-b e)}-\frac {2 (b+2 c x) \log (d+e x)}{\sqrt {\frac {b^2}{c}+4 b x+4 c x^2} (2 c d-b e)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt {\frac {b^2}{4 c}+b x+c x^2}} \, dx &=\frac {\left (\frac {b}{2}+c x\right ) \int \frac {1}{\left (\frac {b}{2}+c x\right ) (d+e x)} \, dx}{\sqrt {\frac {b^2}{4 c}+b x+c x^2}}\\ &=\frac {\left (2 c \left (\frac {b}{2}+c x\right )\right ) \int \frac {1}{\frac {b}{2}+c x} \, dx}{(2 c d-b e) \sqrt {\frac {b^2}{4 c}+b x+c x^2}}-\frac {\left (2 e \left (\frac {b}{2}+c x\right )\right ) \int \frac {1}{d+e x} \, dx}{(2 c d-b e) \sqrt {\frac {b^2}{4 c}+b x+c x^2}}\\ &=\frac {2 (b+2 c x) \log (b+2 c x)}{(2 c d-b e) \sqrt {\frac {b^2}{c}+4 b x+4 c x^2}}-\frac {2 (b+2 c x) \log (d+e x)}{(2 c d-b e) \sqrt {\frac {b^2}{c}+4 b x+4 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.53 \[ \frac {2 (b+2 c x) (\log (b+2 c x)-\log (d+e x))}{\sqrt {\frac {(b+2 c x)^2}{c}} (2 c d-b e)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 294, normalized size = 3.06 \[ \left [-\frac {2 \, \sqrt {c} \log \left (\frac {16 \, c^{3} e^{2} x^{3} + 4 \, b c^{2} d^{2} + b^{3} e^{2} + 16 \, {\left (c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \, {\left (4 \, c^{3} d^{2} + 4 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x + {\left (4 \, c^{2} d^{2} - b^{2} e^{2} + 4 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{3} + b^{2} d + 4 \, {\left (c^{2} d + b c e\right )} x^{2} + {\left (4 \, b c d + b^{2} e\right )} x}\right )}{2 \, c d - b e}, -\frac {4 \, \sqrt {-c} \arctan \left (-\frac {{\left (4 \, c e x + 2 \, c d + b e\right )} \sqrt {-c} \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{2 \, b c d - b^{2} e + 2 \, {\left (2 \, c^{2} d - b c e\right )} x}\right )}{2 \, c d - b e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 58, normalized size = 0.60 \[ \frac {2 \left (2 c x +b \right ) \left (-\ln \left (2 c x +b \right )+\ln \left (e x +d \right )\right )}{\sqrt {\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{c}}\, \left (b e -2 c d \right )} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {2}{\left (d+e\,x\right )\,\sqrt {4\,b\,x+4\,c\,x^2+\frac {b^2}{c}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \int \frac {1}{d \sqrt {\frac {b^{2}}{c} + 4 b x + 4 c x^{2}} + e x \sqrt {\frac {b^{2}}{c} + 4 b x + 4 c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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