Optimal. Leaf size=94 \[ -\frac {e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 c e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {-d-e x}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 91, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {768, 614, 618, 206} \begin {gather*} -\frac {e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 c e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {d+e x}{2 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 614
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 )^3} \, dx &=-\frac {d+e x}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{2} e \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d+e x}{2 \left (a+b x+c x^2\right )^2}-\frac {e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(c e) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {d+e x}{2 \left (a+b x+c x^2\right )^2}-\frac {e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c e) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac {d+e x}{2 \left (a+b x+c x^2\right )^2}-\frac {e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 c e \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 93, normalized size = 0.99 \begin {gather*} \frac {1}{2} \left (-\frac {e (b+2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {4 c e \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac {d+e x}{(a+x (b+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+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 681, normalized size = 7.24 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{3} + 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e x^{2} + 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e x + 2 \, {\left (c^{3} e x^{4} + 2 \, b c^{2} e x^{3} + 2 \, a b c e x + a^{2} c e + {\left (b^{2} c + 2 \, a c^{2}\right )} e x^{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 ) + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d + {\left (a b^{3} - 4 \, a^{2} b c\right )} e}{2 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}}, -\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e x^{3} + 3 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e x^{2} + 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e x - 4 \, {\left (c^{3} e x^{4} + 2 \, b c^{2} e x^{3} + 2 \, a b c e x + a^{2} c e + {\left (b^{2} c + 2 \, a c^{2}\right )} e x^{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 ) + {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d + {\left (a b^{3} - 4 \, a^{2} b c\right )} e}{2 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 122, normalized size = 1.30 \begin {gather*} -\frac {2 \, c \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c^{2} x^{3} e + 3 \, b c x^{2} e + 2 \, b^{2} x e - 2 \, a c x e + b^{2} d - 4 \, a c d + a b e}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 146, normalized size = 1.55 \begin {gather*} \frac {2 c e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {\frac {c^{2} e \,x^{3}}{4 a c -b^{2}}+\frac {3 b c e \,x^{2}}{2 \left (4 a c -b^{2}\right )}-\frac {\left (a c -b^{2}\right ) e x}{4 a c -b^{2}}+\frac {a b e -4 a c d +b^{2} d}{8 a c -2 b^{2}}}{\left (c \,x^{2}+b x +a \right )^{2}} \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 = 1.91, size = 213, normalized size = 2.27 \begin {gather*} \frac {\frac {d\,b^2+a\,e\,b-4\,a\,c\,d}{2\,\left (4\,a\,c-b^2\right )}+\frac {c^2\,e\,x^3}{4\,a\,c-b^2}-\frac {e\,x\,\left (a\,c-b^2\right )}{4\,a\,c-b^2}+\frac {3\,b\,c\,e\,x^2}{2\,\left (4\,a\,c-b^2\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-\frac {2\,c\,e\,\mathrm {atan}\left (-\frac {\left (\frac {2\,c^2\,e\,x}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {c\,e\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (4\,a\,c-b^2\right )}{c\,e}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.40, size = 377, normalized size = 4.01 \begin {gather*} - c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {- 16 a^{2} c^{3} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c e}{2 c^{2} e} \right )} + c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x + \frac {16 a^{2} c^{3} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c^{2} e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} c e \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c e}{2 c^{2} e} \right )} + \frac {a b e - 4 a c d + b^{2} d + 3 b c e x^{2} + 2 c^{2} e x^{3} + x \left (- 2 a c e + 2 b^{2} e\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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