Optimal. Leaf size=62 \[ -\frac {4 c d^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {d^2 (b+2 c x)}{a+b x+c x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 62, 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 = {686, 618, 206} \begin {gather*} -\frac {4 c d^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {d^2 (b+2 c x)}{a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 686
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^2 (b+2 c x)}{a+b x+c x^2}+\left (2 c d^2\right ) \int \frac {1}{a+b x+c x^2} \, dx\\ &=-\frac {d^2 (b+2 c x)}{a+b x+c x^2}-\left (4 c d^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=-\frac {d^2 (b+2 c x)}{a+b x+c x^2}-\frac {4 c d^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 1.05 \begin {gather*} d^2 \left (\frac {4 c \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {-b-2 c x}{a+b x+c x^2}\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}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 309, normalized size = 4.98 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x + {\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2}\right )} \sqrt {b^{2} - 4 \, a c} \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 )}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x}, -\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x + {\left (b^{3} - 4 \, a b c\right )} d^{2} + 4 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 1.06 \begin {gather*} \frac {4 \, c d^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c d^{2} x + b d^{2}}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 77, normalized size = 1.24 \begin {gather*} -\frac {2 c \,d^{2} x}{c \,x^{2}+b x +a}+\frac {4 c \,d^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {b \,d^{2}}{c \,x^{2}+b x +a} \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 = 0.44, size = 88, normalized size = 1.42 \begin {gather*} \frac {4\,c\,d^2\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {b\,d^2}{c\,x^2+b\,x+a}-\frac {2\,c\,d^2\,x}{c\,x^2+b\,x+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 211, normalized size = 3.40 \begin {gather*} - 2 c d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {- 8 a c^{2} d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + 2 b^{2} c d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + 2 b c d^{2}}{4 c^{2} d^{2}} \right )} + 2 c d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {8 a c^{2} d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} - 2 b^{2} c d^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + 2 b c d^{2}}{4 c^{2} d^{2}} \right )} + \frac {- b d^{2} - 2 c d^{2} x}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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