Optimal. Leaf size=55 \[ -\frac {2 e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {d+e x}{a+b x+c x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 55, 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 = {768, 618, 206} \begin {gather*} -\frac {2 e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {d+e x}{a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 768
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d+e x}{a+b x+c x^2}+e \int \frac {1}{a+b x+c x^2} \, dx\\ &=-\frac {d+e x}{a+b x+c x^2}-(2 e) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=-\frac {d+e x}{a+b x+c x^2}-\frac {2 e \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 = 0.03, size = 58, normalized size = 1.05 \begin {gather*} \frac {2 e \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-\frac {d+e x}{a+x (b+c x)} \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+2 c x) (d+e x)}{\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 = 0.44, size = 269, normalized size = 4.89 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} e x - {\left (c e x^{2} + b e x + a e\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 ) + {\left (b^{2} - 4 \, a c\right )} d}{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 {{\left (b^{2} - 4 \, a c\right )} e x + 2 \, {\left (c e x^{2} + b e x + a e\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 ) + {\left (b^{2} - 4 \, a c\right )} d}{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 = 57, normalized size = 1.04 \begin {gather*} \frac {2 \, \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e}{\sqrt {-b^{2} + 4 \, a c}} - \frac {x e + d}{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 = 0.05, size = 58, normalized size = 1.05 \begin {gather*} \frac {2 e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {-e x -d}{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 = 0.06, size = 73, normalized size = 1.33 \begin {gather*} \frac {2\,e\,\mathrm {atan}\left (\frac {\frac {b\,e}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,e\,x}{\sqrt {4\,a\,c-b^2}}}{e}\right )}{\sqrt {4\,a\,c-b^2}}-\frac {d+e\,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 = 1.20, size = 158, normalized size = 2.87 \begin {gather*} - e \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {- 4 a c e \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} e \sqrt {- \frac {1}{4 a c - b^{2}}} + b e}{2 c e} \right )} + e \sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x + \frac {4 a c e \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} e \sqrt {- \frac {1}{4 a c - b^{2}}} + b e}{2 c e} \right )} + \frac {- d - e x}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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