Optimal. Leaf size=227 \[ \frac {2 c^3 x}{e^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^6 (d+e x)^2}-\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)}-\frac {5 c^2 (2 c d-b e) \log (d+e x)}{e^6} \]
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Rubi [A]
time = 0.14, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {785}
\begin {gather*} -\frac {4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac {5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac {2 c^3 x}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {2 c^3}{e^5}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5 (d+e x)^4}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^5 (d+e x)^3}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^5 (d+e x)^2}-\frac {5 c^2 (2 c d-b e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {2 c^3 x}{e^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^6 (d+e x)^2}-\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)}-\frac {5 c^2 (2 c d-b e) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 292, normalized size = 1.29 \begin {gather*} -\frac {2 c^3 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )+b e^3 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+2 c e^2 \left (a^2 e^2 (d+4 e x)+3 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+6 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+c^2 e \left (12 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+60 c^2 (2 c d-b e) (d+e x)^4 \log (d+e x)}{12 e^6 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.94, size = 335, normalized size = 1.48
method | result | size |
norman | \(\frac {-\frac {3 a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-125 b \,c^{2} d^{4} e +250 d^{5} c^{3}}{12 e^{6}}+\frac {2 c^{3} x^{5}}{e}-\frac {4 \left (e^{2} c^{2} a +b^{2} e^{2} c -5 d e b \,c^{2}+10 c^{3} d^{2}\right ) x^{3}}{e^{3}}-\frac {\left (6 c \,e^{3} a b +12 d \,e^{2} c^{2} a +b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +180 c^{3} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+220 d^{4} c^{3}\right ) x}{3 e^{5}}}{\left (e x +d \right )^{4}}+\frac {5 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{6}}\) | \(329\) |
default | \(\frac {2 c^{3} x}{e^{5}}-\frac {4 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{6} \left (e x +d \right )}+\frac {5 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{6}}-\frac {6 c \,e^{3} a b -12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{2 e^{6} \left (e x +d \right )^{2}}-\frac {a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}}{4 e^{6} \left (e x +d \right )^{4}}-\frac {2 e^{4} a^{2} c +2 a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a -2 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-20 d^{3} e b \,c^{2}+10 d^{4} c^{3}}{3 e^{6} \left (e x +d \right )^{3}}\) | \(335\) |
risch | \(\frac {2 c^{3} x}{e^{5}}+\frac {\left (-4 a \,c^{2} e^{4}-4 b^{2} c \,e^{4}+20 d \,e^{3} b \,c^{2}-20 d^{2} e^{2} c^{3}\right ) x^{3}-\frac {e \left (6 c \,e^{3} a b +12 d \,e^{2} c^{2} a +b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +100 c^{3} d^{3}\right ) x^{2}}{2}+\left (-\frac {2}{3} e^{4} a^{2} c -\frac {2}{3} a \,b^{2} e^{4}-2 a b c d \,e^{3}-4 d^{2} e^{2} c^{2} a -\frac {1}{3} b^{3} d \,e^{3}-4 b^{2} c \,d^{2} e^{2}+\frac {110}{3} d^{3} e b \,c^{2}-\frac {130}{3} d^{4} c^{3}\right ) x -\frac {3 a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-125 b \,c^{2} d^{4} e +154 d^{5} c^{3}}{12 e}}{e^{5} \left (e x +d \right )^{4}}+\frac {5 c^{2} \ln \left (e x +d \right ) b}{e^{5}}-\frac {10 c^{3} \ln \left (e x +d \right ) d}{e^{6}}\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 336, normalized size = 1.48 \begin {gather*} 2 \, c^{3} x e^{\left (-5\right )} - 5 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-6\right )} \log \left (x e + d\right ) - \frac {154 \, c^{3} d^{5} - 125 \, b c^{2} d^{4} e + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{3} + 48 \, {\left (5 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + b^{2} c e^{5} + a c^{2} e^{5}\right )} x^{3} + 3 \, a^{2} b e^{5} + {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{2} + 6 \, {\left (100 \, c^{3} d^{3} e^{2} - 90 \, b c^{2} d^{2} e^{3} + b^{3} e^{5} + 6 \, a b c e^{5} + 12 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d\right )} x^{2} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d + 4 \, {\left (130 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 2 \, a b^{2} e^{5} + 2 \, a^{2} c e^{5} + 12 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{2} + {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d\right )} x}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.37, size = 432, normalized size = 1.90 \begin {gather*} -\frac {154 \, c^{3} d^{5} - {\left (24 \, c^{3} x^{5} - 48 \, {\left (b^{2} c + a c^{2}\right )} x^{3} - 3 \, a^{2} b - 6 \, {\left (b^{3} + 6 \, a b c\right )} x^{2} - 8 \, {\left (a b^{2} + a^{2} c\right )} x\right )} e^{5} - 2 \, {\left (48 \, c^{3} d x^{4} + 120 \, b c^{2} d x^{3} - 36 \, {\left (b^{2} c + a c^{2}\right )} d x^{2} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d x - {\left (a b^{2} + a^{2} c\right )} d\right )} e^{4} + {\left (96 \, c^{3} d^{2} x^{3} - 540 \, b c^{2} d^{2} x^{2} + 48 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x + {\left (b^{3} + 6 \, a b c\right )} d^{2}\right )} e^{3} + 4 \, {\left (126 \, c^{3} d^{3} x^{2} - 110 \, b c^{2} d^{3} x + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3}\right )} e^{2} + {\left (496 \, c^{3} d^{4} x - 125 \, b c^{2} d^{4}\right )} e + 60 \, {\left (2 \, c^{3} d^{5} - b c^{2} x^{4} e^{5} + 2 \, {\left (c^{3} d x^{4} - 2 \, b c^{2} d x^{3}\right )} e^{4} + 2 \, {\left (4 \, c^{3} d^{2} x^{3} - 3 \, b c^{2} d^{2} x^{2}\right )} e^{3} + 4 \, {\left (3 \, c^{3} d^{3} x^{2} - b c^{2} d^{3} x\right )} e^{2} + {\left (8 \, c^{3} d^{4} x - b c^{2} d^{4}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{10} + 4 \, d x^{3} e^{9} + 6 \, d^{2} x^{2} e^{8} + 4 \, d^{3} x e^{7} + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs.
\(2 (226) = 452\).
time = 1.79, size = 525, normalized size = 2.31 \begin {gather*} 2 \, {\left (x e + d\right )} c^{3} e^{\left (-6\right )} + 5 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {240 \, c^{3} d^{2} e^{22}}{x e + d} - \frac {120 \, c^{3} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {40 \, c^{3} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {6 \, c^{3} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {240 \, b c^{2} d e^{23}}{x e + d} + \frac {180 \, b c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {80 \, b c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {15 \, b c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {48 \, b^{2} c e^{24}}{x e + d} + \frac {48 \, a c^{2} e^{24}}{x e + d} - \frac {72 \, b^{2} c d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {72 \, a c^{2} d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {48 \, b^{2} c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {48 \, a c^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {12 \, b^{2} c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {12 \, a c^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {6 \, b^{3} e^{25}}{{\left (x e + d\right )}^{2}} + \frac {36 \, a b c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {8 \, b^{3} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {48 \, a b c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {3 \, b^{3} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {18 \, a b c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {8 \, a b^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac {8 \, a^{2} c e^{26}}{{\left (x e + d\right )}^{3}} - \frac {6 \, a b^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a^{2} c d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, a^{2} b e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 369, normalized size = 1.63 \begin {gather*} \frac {2\,c^3\,x}{e^5}-\frac {x\,\left (\frac {2\,a^2\,c\,e^4}{3}+\frac {2\,a\,b^2\,e^4}{3}+2\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+\frac {b^3\,d\,e^3}{3}+4\,b^2\,c\,d^2\,e^2-\frac {110\,b\,c^2\,d^3\,e}{3}+\frac {130\,c^3\,d^4}{3}\right )+x^2\,\left (\frac {b^3\,e^4}{2}+6\,b^2\,c\,d\,e^3-45\,b\,c^2\,d^2\,e^2+3\,a\,b\,c\,e^4+50\,c^3\,d^3\,e+6\,a\,c^2\,d\,e^3\right )+x^3\,\left (4\,b^2\,c\,e^4-20\,b\,c^2\,d\,e^3+20\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )+\frac {3\,a^2\,b\,e^5+2\,a^2\,c\,d\,e^4+2\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-125\,b\,c^2\,d^4\,e+154\,c^3\,d^5}{12\,e}}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}-\frac {\ln \left (d+e\,x\right )\,\left (10\,c^3\,d-5\,b\,c^2\,e\right )}{e^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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