Optimal. Leaf size=131 \[ -\frac {b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {6 c (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {652, 628, 632,
212} \begin {gather*} \frac {3 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {-2 a e+x (2 c d-b e)+b d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {6 c (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 )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 628
Rule 632
Rule 652
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(3 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {(3 c (2 c d-b e)) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(6 c (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 )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {6 c (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 )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 128, normalized size = 0.98 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right ) (-b d+2 a e-2 c d x+b e x)}{(a+x (b+c x))^2}+\frac {3 (2 c d-b e) (b+2 c x)}{a+x (b+c x)}-\frac {12 c (-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}}}{2 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 137, normalized size = 1.05
method | result | size |
default | \(\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {3 \left (-b e +2 c d \right ) \left (\frac {2 c x +b}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )}+\frac {4 c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (4 a c -b^{2}\right )}\) | \(137\) |
risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {9 b c \left (b e -2 c d \right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) \left (b e -2 c d \right ) x}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {8 a^{2} c e +a \,b^{2} e -10 a b c d +b^{3} d}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {3 c \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {6 c^{2} \ln \left (\left (32 a^{2} c^{3}-16 b^{2} c^{2} a +2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) d}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) b e}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {6 c^{2} \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} c^{2} a -2 c \,b^{4}\right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) d}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(479\) |
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. 541 vs.
\(2 (127) = 254\).
time = 4.32, size = 1102, normalized size = 8.41 \begin {gather*} \left [\frac {12 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{3} + 18 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d x^{2} + 4 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} d x - 6 \, {\left (2 \, c^{4} d x^{4} + 4 \, b c^{3} d x^{3} + 4 \, a b c^{2} d x + 2 \, a^{2} c^{2} d + 2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{2} - {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} + 2 \, a b^{2} c x + a^{2} b c + {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{2}\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^{5} - 14 \, a b^{3} c + 40 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2} + 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3} + 9 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x^{2} + 2 \, {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} x\right )} e}{2 \, {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )}}, \frac {12 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d x^{3} + 18 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d x^{2} + 4 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} d x - 12 \, {\left (2 \, c^{4} d x^{4} + 4 \, b c^{3} d x^{3} + 4 \, a b c^{2} d x + 2 \, a^{2} c^{2} d + 2 \, {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{2} - {\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} + 2 \, a b^{2} c x + a^{2} b c + {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{2}\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^{5} - 14 \, a b^{3} c + 40 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} + 4 \, a^{2} b^{2} c - 32 \, a^{3} c^{2} + 6 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3} + 9 \, {\left (b^{4} c - 4 \, a b^{2} c^{2}\right )} x^{2} + 2 \, {\left (b^{5} + a b^{3} c - 20 \, a^{2} b c^{2}\right )} x\right )} e}{2 \, {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )}}\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. 651 vs.
\(2 (122) = 244\).
time = 1.12, size = 651, normalized size = 4.97 \begin {gather*} 3 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 192 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 144 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 36 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{2} c e - 6 b c^{2} d}{6 b c^{2} e - 12 c^{3} d} \right )} - 3 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {192 a^{3} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 144 a^{2} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 36 a b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 3 b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{2} c e - 6 b c^{2} d}{6 b c^{2} e - 12 c^{3} d} \right )} + \frac {- 8 a^{2} c e - a b^{2} e + 10 a b c d - b^{3} d + x^{3} \left (- 6 b c^{2} e + 12 c^{3} d\right ) + x^{2} \left (- 9 b^{2} c e + 18 b c^{2} d\right ) + x \left (- 10 a b c e + 20 a c^{2} d - 2 b^{3} e + 4 b^{2} c d\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \cdot \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \cdot \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \cdot \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 206, normalized size = 1.57 \begin {gather*} \frac {6 \, {\left (2 \, c^{2} d - b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} d x^{3} - 6 \, b c^{2} x^{3} e + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c x^{2} e + 4 \, b^{2} c d x + 20 \, a c^{2} d x - 2 \, b^{3} x e - 10 \, a b c x e - b^{3} d + 10 \, a b c d - a b^{2} e - 8 \, a^{2} c e}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\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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Mupad [B]
time = 0.30, size = 353, normalized size = 2.69 \begin {gather*} \frac {6\,c\,\mathrm {atan}\left (\frac {\left (\frac {6\,c^2\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {3\,c\,\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{6\,c^2\,d-3\,b\,c\,e}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\frac {8\,c\,e\,a^2+e\,a\,b^2-10\,c\,d\,a\,b+d\,b^3}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (b^2+5\,a\,c\right )\,\left (b\,e-2\,c\,d\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {3\,c^2\,x^3\,\left (b\,e-2\,c\,d\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {9\,b\,c\,x^2\,\left (b\,e-2\,c\,d\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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