Optimal. Leaf size=173 \[ \frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac {5 c (b+c x) (\text {b1} c-b \text {c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 (b+c x) (\text {b1} c-b \text {c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]
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Rubi [A] time = 0.11, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {638, 614, 618, 206} \[ \frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}-\frac {5 c (b+c x) (\text {b1} c-b \text {c1})}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 (b+c x) (\text {b1} c-b \text {c1})}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 614
Rule 618
Rule 638
Rubi steps
\begin {align*} \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}-\frac {(5 (\text {b1} c-b \text {c1})) \int \frac {1}{\left (a+2 b x+c x^2\right )^3} \, dx}{6 \left (b^2-a c\right )}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}+\frac {(5 c (\text {b1} c-b \text {c1})) \int \frac {1}{\left (a+2 b x+c x^2\right )^2} \, dx}{8 \left (b^2-a c\right )^2}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}-\frac {\left (5 c^2 (\text {b1} c-b \text {c1})\right ) \int \frac {1}{a+2 b x+c x^2} \, dx}{16 \left (b^2-a c\right )^3}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {\left (5 c^2 (\text {b1} c-b \text {c1})\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (b^2-a c\right )-x^2} \, dx,x,2 b+2 c x\right )}{8 \left (b^2-a c\right )^3}\\ &=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 c^2 (\text {b1} c-b \text {c1}) \tanh ^{-1}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 168, normalized size = 0.97 \[ \frac {\frac {15 c^2 (b \text {c1}-\text {b1} c) \tan ^{-1}\left (\frac {b+c x}{\sqrt {a c-b^2}}\right )}{\sqrt {a c-b^2}}-\frac {10 \left (b^2-a c\right ) (b+c x) (b \text {c1}-\text {b1} c)}{(a+x (2 b+c x))^2}+\frac {8 \left (b^2-a c\right )^2 (a \text {c1}-b \text {b1}+b \text {c1} x-\text {b1} c x)}{(a+x (2 b+c x))^3}+\frac {15 c (b+c x) (b \text {c1}-\text {b1} c)}{a+x (2 b+c x)}}{48 \left (b^2-a c\right )^3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 340, normalized size = 1.97 \[ \frac {8 a^3 c^2 \text {c1}+9 a^2 b^2 c \text {c1}-33 a^2 b \text {b1} c^2+33 a^2 b c^2 \text {c1} x-33 a^2 \text {b1} c^3 x-2 a b^4 \text {c1}+26 a b^3 \text {b1} c+54 a b^3 c \text {c1} x-54 a b^2 \text {b1} c^2 x+120 a b^2 c^2 \text {c1} x^2-120 a b \text {b1} c^3 x^2+40 a b c^3 \text {c1} x^3-40 a \text {b1} c^4 x^3-8 b^5 \text {b1}-12 b^5 \text {c1} x+12 b^4 \text {b1} c x+30 b^4 c \text {c1} x^2-30 b^3 \text {b1} c^2 x^2+110 b^3 c^2 \text {c1} x^3-110 b^2 \text {b1} c^3 x^3+75 b^2 c^3 \text {c1} x^4-75 b \text {b1} c^4 x^4+15 b c^4 \text {c1} x^5-15 \text {b1} c^5 x^5}{48 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )^3}+\frac {5 \left (b c^2 \text {c1}-\text {b1} c^3\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {a c-b^2}}+\frac {b}{\sqrt {a c-b^2}}\right )}{16 \left (b^2-a c\right )^3 \sqrt {a c-b^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.34, size = 1950, normalized size = 11.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 363, normalized size = 2.10 \[ -\frac {5 \, {\left (b_{1} c^{3} - b c^{2} c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{16 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} \sqrt {-b^{2} + a c}} - \frac {15 \, b_{1} c^{5} x^{5} - 15 \, b c^{4} c_{1} x^{5} + 75 \, b b_{1} c^{4} x^{4} - 75 \, b^{2} c^{3} c_{1} x^{4} + 110 \, b^{2} b_{1} c^{3} x^{3} + 40 \, a b_{1} c^{4} x^{3} - 110 \, b^{3} c^{2} c_{1} x^{3} - 40 \, a b c^{3} c_{1} x^{3} + 30 \, b^{3} b_{1} c^{2} x^{2} + 120 \, a b b_{1} c^{3} x^{2} - 30 \, b^{4} c c_{1} x^{2} - 120 \, a b^{2} c^{2} c_{1} x^{2} - 12 \, b^{4} b_{1} c x + 54 \, a b^{2} b_{1} c^{2} x + 33 \, a^{2} b_{1} c^{3} x + 12 \, b^{5} c_{1} x - 54 \, a b^{3} c c_{1} x - 33 \, a^{2} b c^{2} c_{1} x + 8 \, b^{5} b_{1} - 26 \, a b^{3} b_{1} c + 33 \, a^{2} b b_{1} c^{2} + 2 \, a b^{4} c_{1} - 9 \, a^{2} b^{2} c c_{1} - 8 \, a^{3} c^{2} c_{1}}{48 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 206, normalized size = 1.19
method | result | size |
default | \(\frac {\left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) x +2 b \mathit {b1} -2 a \mathit {c1}}{3 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 \left (-2 b \mathit {c1} +2 \mathit {b1} c \right ) \left (\frac {2 c x +2 b}{2 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +2 b}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {2 c \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\right )}{4 a c -4 b^{2}}\right )}{3 \left (4 a c -4 b^{2}\right )}\) | \(206\) |
risch | \(\frac {-\frac {5 c^{4} \left (b \mathit {c1} -\mathit {b1} c \right ) x^{5}}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {25 c^{3} \left (b \mathit {c1} -\mathit {b1} c \right ) b \,x^{4}}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {5 \left (4 a c +11 b^{2}\right ) c^{2} \left (b \mathit {c1} -\mathit {b1} c \right ) x^{3}}{24 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {5 b \left (4 a c +b^{2}\right ) c \left (b \mathit {c1} -\mathit {b1} c \right ) x^{2}}{8 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {\left (11 a^{2} b \,c^{2} \mathit {c1} -11 a^{2} \mathit {b1} \,c^{3}+18 a \,b^{3} c \mathit {c1} -18 a \,b^{2} \mathit {b1} \,c^{2}-4 b^{5} \mathit {c1} +4 b^{4} \mathit {b1} c \right ) x}{16 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}-\frac {8 a^{3} c^{2} \mathit {c1} +9 a^{2} b^{2} c \mathit {c1} -33 a^{2} b \mathit {b1} \,c^{2}-2 a \,b^{4} \mathit {c1} +26 a \,b^{3} \mathit {b1} c -8 b^{5} \mathit {b1}}{48 \left (c^{3} a^{3}-3 b^{2} c^{2} a^{2}+3 b^{4} c a -b^{6}\right )}}{\left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 c^{2} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) b \mathit {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{3} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) \mathit {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{2} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) b \mathit {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}+\frac {5 c^{3} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) \mathit {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}\) | \(792\) |
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 640, normalized size = 3.70 \[ \frac {\frac {5\,c^4\,x^5\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}-\frac {-8\,c_{1}\,a^3\,c^2-9\,c_{1}\,a^2\,b^2\,c+33\,b_{1}\,a^2\,b\,c^2+2\,c_{1}\,a\,b^4-26\,b_{1}\,a\,b^3\,c+8\,b_{1}\,b^5}{48\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (11\,a^2\,c^2+18\,a\,b^2\,c-4\,b^4\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^3\,\left (11\,b^2\,c+4\,a\,c^2\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{24\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^2\,\left (b^3+4\,a\,c\,b\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {25\,b\,c^3\,x^4\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}}{x^3\,\left (8\,b^3+12\,a\,c\,b\right )+x^2\,\left (3\,c\,a^2+12\,a\,b^2\right )+x^4\,\left (12\,b^2\,c+3\,a\,c^2\right )+a^3+c^3\,x^6+6\,b\,c^2\,x^5+6\,a^2\,b\,x}-\frac {5\,c^2\,\mathrm {atan}\left (\frac {16\,\left (\frac {5\,c^3\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}}+\frac {5\,c^2\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (-32\,a^3\,b\,c^3+96\,a^2\,b^3\,c^2-96\,a\,b^5\,c+32\,b^7\right )}{512\,{\left (a\,c-b^2\right )}^{7/2}\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}\right )\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}{5\,b_{1}\,c^3-5\,b\,c^2\,c_{1}}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.06, size = 1027, normalized size = 5.94 \[ \frac {5 c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {- 5 a^{4} c^{6} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 20 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 30 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 20 a b^{6} c^{3} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 5 b^{8} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{2} c^{2} c_{1} - 5 b b_{1} c^{3}}{5 b c^{3} c_{1} - 5 b_{1} c^{4}} \right )}}{32} - \frac {5 c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {5 a^{4} c^{6} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 20 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 30 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) - 20 a b^{6} c^{3} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{8} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{7}}} \left (b c_{1} - b_{1} c\right ) + 5 b^{2} c^{2} c_{1} - 5 b b_{1} c^{3}}{5 b c^{3} c_{1} - 5 b_{1} c^{4}} \right )}}{32} + \frac {- 8 a^{3} c^{2} c_{1} - 9 a^{2} b^{2} c c_{1} + 33 a^{2} b b_{1} c^{2} + 2 a b^{4} c_{1} - 26 a b^{3} b_{1} c + 8 b^{5} b_{1} + x^{5} \left (- 15 b c^{4} c_{1} + 15 b_{1} c^{5}\right ) + x^{4} \left (- 75 b^{2} c^{3} c_{1} + 75 b b_{1} c^{4}\right ) + x^{3} \left (- 40 a b c^{3} c_{1} + 40 a b_{1} c^{4} - 110 b^{3} c^{2} c_{1} + 110 b^{2} b_{1} c^{3}\right ) + x^{2} \left (- 120 a b^{2} c^{2} c_{1} + 120 a b b_{1} c^{3} - 30 b^{4} c c_{1} + 30 b^{3} b_{1} c^{2}\right ) + x \left (- 33 a^{2} b c^{2} c_{1} + 33 a^{2} b_{1} c^{3} - 54 a b^{3} c c_{1} + 54 a b^{2} b_{1} c^{2} + 12 b^{5} c_{1} - 12 b^{4} b_{1} c\right )}{48 a^{6} c^{3} - 144 a^{5} b^{2} c^{2} + 144 a^{4} b^{4} c - 48 a^{3} b^{6} + x^{6} \left (48 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 144 a b^{4} c^{4} - 48 b^{6} c^{3}\right ) + x^{5} \left (288 a^{3} b c^{5} - 864 a^{2} b^{3} c^{4} + 864 a b^{5} c^{3} - 288 b^{7} c^{2}\right ) + x^{4} \left (144 a^{4} c^{5} + 144 a^{3} b^{2} c^{4} - 1296 a^{2} b^{4} c^{3} + 1584 a b^{6} c^{2} - 576 b^{8} c\right ) + x^{3} \left (576 a^{4} b c^{4} - 1344 a^{3} b^{3} c^{3} + 576 a^{2} b^{5} c^{2} + 576 a b^{7} c - 384 b^{9}\right ) + x^{2} \left (144 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 1296 a^{3} b^{4} c^{2} + 1584 a^{2} b^{6} c - 576 a b^{8}\right ) + x \left (288 a^{5} b c^{3} - 864 a^{4} b^{3} c^{2} + 864 a^{3} b^{5} c - 288 a^{2} b^{7}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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