Optimal. Leaf size=173 \[ \frac {20 c^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {5 c (b+2 c x) (2 c d-b e)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {5 (b+2 c x) (2 c d-b e)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {-2 a e+x (2 c d-b e)+b d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]
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Rubi [A] time = 0.07, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {638, 614, 618, 206} \begin {gather*} \frac {20 c^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 )^{7/2}}-\frac {5 c (b+2 c x) (2 c d-b e)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {5 (b+2 c x) (2 c d-b e)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {-2 a e+x (2 c d-b e)+b d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 614
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {(5 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 (2 c d-b e) (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {(5 c (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 (2 c d-b e) (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {5 c (2 c d-b e) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (10 c^2 (2 c d-b e)\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 (2 c d-b e) (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {5 c (2 c d-b e) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (20 c^2 (2 c d-b e)\right ) \operatorname {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 )^3}\\ &=-\frac {b d-2 a e+(2 c d-b e) x}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {5 (2 c d-b e) (b+2 c x)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {5 c (2 c d-b e) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {20 c^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 )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 168, normalized size = 0.97 \begin {gather*} \frac {\frac {120 c^2 (b e-2 c d) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}+\frac {2 \left (b^2-4 a c\right )^2 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^3}+\frac {30 c (b+2 c x) (b e-2 c d)}{a+x (b+c x)}}{6 \left (b^2-4 a c\right )^3} \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 {d+e x}{\left (a+b x+c x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 1960, normalized size = 11.33
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 378, normalized size = 2.18 \begin {gather*} -\frac {20 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {120 \, c^{5} d x^{5} - 60 \, b c^{4} x^{5} e + 300 \, b c^{4} d x^{4} - 150 \, b^{2} c^{3} x^{4} e + 220 \, b^{2} c^{3} d x^{3} + 320 \, a c^{4} d x^{3} - 110 \, b^{3} c^{2} x^{3} e - 160 \, a b c^{3} x^{3} e + 30 \, b^{3} c^{2} d x^{2} + 480 \, a b c^{3} d x^{2} - 15 \, b^{4} c x^{2} e - 240 \, a b^{2} c^{2} x^{2} e - 6 \, b^{4} c d x + 108 \, a b^{2} c^{2} d x + 264 \, a^{2} c^{3} d x + 3 \, b^{5} x e - 54 \, a b^{3} c x e - 132 \, a^{2} b c^{2} x e + 2 \, b^{5} d - 26 \, a b^{3} c d + 132 \, a^{2} b c^{2} d + a b^{4} e - 18 \, a^{2} b^{2} c e - 64 \, a^{3} c^{2} e}{6 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 369, normalized size = 2.13 \begin {gather*} -\frac {10 b \,c^{2} e x}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )}-\frac {20 b \,c^{2} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{3} d x}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )}+\frac {40 c^{3} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {7}{2}}}-\frac {5 b^{2} c e}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )}+\frac {10 b \,c^{2} d}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )}-\frac {5 b c e x}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {10 c^{2} d x}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}-\frac {5 b^{2} e}{6 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {5 b c d}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{2}}+\frac {-2 a e +b d +\left (-b e +2 c d \right ) x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{3}} \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 = 1.14, size = 633, normalized size = 3.66 \begin {gather*} \frac {\frac {10\,c^4\,x^5\,\left (b\,e-2\,c\,d\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}-\frac {-64\,e\,a^3\,c^2-18\,e\,a^2\,b^2\,c+132\,d\,a^2\,b\,c^2+e\,a\,b^4-26\,d\,a\,b^3\,c+2\,d\,b^5}{6\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (44\,a^2\,c^2+18\,a\,b^2\,c-b^4\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^3\,\left (11\,b^2\,c+16\,a\,c^2\right )\,\left (b\,e-2\,c\,d\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^2\,\left (b^3+16\,a\,c\,b\right )\,\left (b\,e-2\,c\,d\right )}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {25\,b\,c^3\,x^4\,\left (b\,e-2\,c\,d\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {20\,c^2\,\mathrm {atan}\left (\frac {\left (\frac {20\,c^3\,x\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {10\,c^2\,\left (b\,e-2\,c\,d\right )\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{20\,c^3\,d-10\,b\,c^2\,e}\right )\,\left (b\,e-2\,c\,d\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.41, size = 1062, normalized size = 6.14 \begin {gather*} 10 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {- 2560 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 2560 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 960 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 160 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 10 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{2} c^{2} e - 20 b c^{3} d}{20 b c^{3} e - 40 c^{4} d} \right )} - 10 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) \log {\left (x + \frac {2560 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 2560 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 960 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) - 160 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (b e - 2 c d\right ) + 10 b^{2} c^{2} e - 20 b c^{3} d}{20 b c^{3} e - 40 c^{4} d} \right )} + \frac {- 64 a^{3} c^{2} e - 18 a^{2} b^{2} c e + 132 a^{2} b c^{2} d + a b^{4} e - 26 a b^{3} c d + 2 b^{5} d + x^{5} \left (- 60 b c^{4} e + 120 c^{5} d\right ) + x^{4} \left (- 150 b^{2} c^{3} e + 300 b c^{4} d\right ) + x^{3} \left (- 160 a b c^{3} e + 320 a c^{4} d - 110 b^{3} c^{2} e + 220 b^{2} c^{3} d\right ) + x^{2} \left (- 240 a b^{2} c^{2} e + 480 a b c^{3} d - 15 b^{4} c e + 30 b^{3} c^{2} d\right ) + x \left (- 132 a^{2} b c^{2} e + 264 a^{2} c^{3} d - 54 a b^{3} c e + 108 a b^{2} c^{2} d + 3 b^{5} e - 6 b^{4} c d\right )}{384 a^{6} c^{3} - 288 a^{5} b^{2} c^{2} + 72 a^{4} b^{4} c - 6 a^{3} b^{6} + x^{6} \left (384 a^{3} c^{6} - 288 a^{2} b^{2} c^{5} + 72 a b^{4} c^{4} - 6 b^{6} c^{3}\right ) + x^{5} \left (1152 a^{3} b c^{5} - 864 a^{2} b^{3} c^{4} + 216 a b^{5} c^{3} - 18 b^{7} c^{2}\right ) + x^{4} \left (1152 a^{4} c^{5} + 288 a^{3} b^{2} c^{4} - 648 a^{2} b^{4} c^{3} + 198 a b^{6} c^{2} - 18 b^{8} c\right ) + x^{3} \left (2304 a^{4} b c^{4} - 1344 a^{3} b^{3} c^{3} + 144 a^{2} b^{5} c^{2} + 36 a b^{7} c - 6 b^{9}\right ) + x^{2} \left (1152 a^{5} c^{4} + 288 a^{4} b^{2} c^{3} - 648 a^{3} b^{4} c^{2} + 198 a^{2} b^{6} c - 18 a b^{8}\right ) + x \left (1152 a^{5} b c^{3} - 864 a^{4} b^{3} c^{2} + 216 a^{3} b^{5} c - 18 a^{2} b^{7}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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