Optimal. Leaf size=49 \[ 2 d^2 (b+2 c x)-2 d^2 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {692, 618, 206} \begin {gather*} 2 d^2 (b+2 c x)-2 d^2 \sqrt {b^2-4 a c} \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx &=2 d^2 (b+2 c x)+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=2 d^2 (b+2 c x)-\left (2 \left (b^2-4 a c\right ) d^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 d^2 (b+2 c x)-2 \sqrt {b^2-4 a c} d^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.96 \begin {gather*} d^2 \left (4 c x-2 \sqrt {4 a c-b^2} \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 132, normalized size = 2.69 \begin {gather*} \left [4 \, c d^{2} x + \sqrt {b^{2} - 4 \, a c} d^{2} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ), 4 \, c d^{2} x - 2 \, \sqrt {-b^{2} + 4 \, a c} d^{2} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 57, normalized size = 1.16 \begin {gather*} 4 \, c d^{2} x + \frac {2 \, {\left (b^{2} d^{2} - 4 \, a c d^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 1.80 \begin {gather*} -\frac {8 a c \,d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {2 b^{2} d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+4 c \,d^{2} x \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 81, normalized size = 1.65 \begin {gather*} 2\,d^2\,\mathrm {atan}\left (\frac {b\,d^2\,\sqrt {4\,a\,c-b^2}+2\,c\,d^2\,x\,\sqrt {4\,a\,c-b^2}}{b^2\,d^2-4\,a\,c\,d^2}\right )\,\sqrt {4\,a\,c-b^2}+4\,c\,d^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 99, normalized size = 2.02 \begin {gather*} 4 c d^{2} x + d^{2} \sqrt {- 4 a c + b^{2}} \log {\left (x + \frac {b d^{2} - d^{2} \sqrt {- 4 a c + b^{2}}}{2 c d^{2}} \right )} - d^{2} \sqrt {- 4 a c + b^{2}} \log {\left (x + \frac {b d^{2} + d^{2} \sqrt {- 4 a c + b^{2}}}{2 c d^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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