Optimal. Leaf size=87 \[ -\frac {b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {652, 632, 212}
\begin {gather*} \frac {2 (2 c d-b 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 {-2 a e+x (2 c d-b e)+b d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 652
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 c d-b e) \int \frac {1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 (2 c d-b e)) \text {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 {b d-2 a e+(2 c d-b e) x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 (2 c d-b 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 = 0.05, size = 88, normalized size = 1.01 \begin {gather*} \frac {\frac {-b d+2 a e-2 c d x+b e x}{a+x (b+c x)}+\frac {2 (-2 c d+b e) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{b^2-4 a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 89, normalized size = 1.02
method | result | size |
default | \(\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {2 \left (-b e +2 c d \right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(89\) |
risch | \(\frac {-\frac {\left (b e -2 c d \right ) x}{4 a c -b^{2}}-\frac {2 a e -b d}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (-8 c^{2} a +2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 \ln \left (\left (8 c^{2} a -2 b^{2} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) c d}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(269\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 217 vs.
\(2 (86) = 172\).
time = 2.45, size = 454, normalized size = 5.22 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - {\left (2 \, c^{2} d x^{2} + 2 \, b c d x + 2 \, a c d - {\left (b c x^{2} + b^{2} x + a b\right )} 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^{3} - 4 \, a b c\right )} d - {\left (2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x\right )} e}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, -\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d x - 2 \, {\left (2 \, c^{2} d x^{2} + 2 \, b c d x + 2 \, a c d - {\left (b c x^{2} + b^{2} x + a b\right )} 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^{3} - 4 \, a b c\right )} d - {\left (2 \, a b^{2} - 8 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x\right )} e}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 359 vs.
\(2 (78) = 156\).
time = 0.53, size = 359, normalized size = 4.13 \begin {gather*} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} - \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {16 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e - 2 b c d}{2 b c e - 4 c^{2} d} \right )} + \frac {- 2 a e + b d + x \left (- b e + 2 c d\right )}{4 a^{2} c - a b^{2} + x^{2} \cdot \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 99, normalized size = 1.14 \begin {gather*} -\frac {2 \, {\left (2 \, c d - b e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c d x - b x e + b d - 2 \, a e}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 159, normalized size = 1.83 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\left (4\,a\,c-b^2\right )\,\left (\frac {\left (b^3-4\,a\,b\,c\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {2\,c\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{b\,e-2\,c\,d}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\frac {2\,a\,e-b\,d}{4\,a\,c-b^2}+\frac {x\,\left (b\,e-2\,c\,d\right )}{4\,a\,c-b^2}}{c\,x^2+b\,x+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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