Optimal. Leaf size=150 \[ -\frac {\left (c d^2-b d e+a e^2\right )^2}{4 e^5 (d+e x)^4}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^3}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{2 e^5 (d+e x)^2}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {c^2 \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.08, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {c^2 \log (d+e x)}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^5}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^4}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^3}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^2}+\frac {c^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {\left (c d^2-b d e+a e^2\right )^2}{4 e^5 (d+e x)^4}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^3}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{2 e^5 (d+e x)^2}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {c^2 \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 170, normalized size = 1.13 \begin {gather*} \frac {c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-e^2 \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 \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 193, normalized size = 1.29
method | result | size |
risch | \(\frac {-\frac {2 c \left (b e -2 c d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+6 b c d e -18 c^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (2 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+6 d^{2} e b c -22 c^{2} d^{3}\right ) x}{3 e^{4}}-\frac {3 a^{2} e^{4}+2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+6 d^{3} e b c -25 c^{2} d^{4}}{12 e^{5}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}\) | \(182\) |
norman | \(\frac {-\frac {3 a^{2} e^{4}+2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+6 d^{3} e b c -25 c^{2} d^{4}}{12 e^{5}}-\frac {2 \left (b c e -2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+6 b c d e -18 c^{2} d^{2}\right ) x^{2}}{2 e^{3}}-\frac {\left (2 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+6 d^{2} e b c -22 c^{2} d^{3}\right ) x}{3 e^{4}}}{\left (e x +d \right )^{4}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}\) | \(184\) |
default | \(-\frac {2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{4 e^{5} \left (e x +d \right )^{4}}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 208, normalized size = 1.39 \begin {gather*} c^{2} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {25 \, c^{2} d^{4} - 6 \, b c d^{3} e + 24 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} - 2 \, a b d e^{3} - {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 6 \, {\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - b^{2} e^{4} - 2 \, a c e^{4}\right )} x^{2} - 3 \, a^{2} e^{4} + 4 \, {\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} - {\left (b^{2} e^{3} + 2 \, a c e^{3}\right )} d\right )} x}{12 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 244, normalized size = 1.63 \begin {gather*} \frac {25 \, c^{2} d^{4} - {\left (24 \, b c x^{3} + 8 \, a b x + 6 \, {\left (b^{2} + 2 \, a c\right )} x^{2} + 3 \, a^{2}\right )} e^{4} + 2 \, {\left (24 \, c^{2} d x^{3} - 18 \, b c d x^{2} - a b d - 2 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + {\left (108 \, c^{2} d^{2} x^{2} - 24 \, b c d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 2 \, {\left (44 \, c^{2} d^{3} x - 3 \, b c d^{3}\right )} e + 12 \, {\left (c^{2} x^{4} e^{4} + 4 \, c^{2} d x^{3} e^{3} + 6 \, c^{2} d^{2} x^{2} e^{2} + 4 \, c^{2} d^{3} x e + c^{2} d^{4}\right )} \log \left (x e + d\right )}{12 \, {\left (x^{4} e^{9} + 4 \, d x^{3} e^{8} + 6 \, d^{2} x^{2} e^{7} + 4 \, d^{3} x e^{6} + d^{4} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 18.70, size = 238, normalized size = 1.59 \begin {gather*} \frac {c^{2} \log {\left (d + e x \right )}}{e^{5}} + \frac {- 3 a^{2} e^{4} - 2 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 6 b c d^{3} e + 25 c^{2} d^{4} + x^{3} \left (- 24 b c e^{4} + 48 c^{2} d e^{3}\right ) + x^{2} \left (- 12 a c e^{4} - 6 b^{2} e^{4} - 36 b c d e^{3} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 8 a b e^{4} - 8 a c d e^{3} - 4 b^{2} d e^{3} - 24 b c d^{2} e^{2} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \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. 304 vs.
\(2 (147) = 294\).
time = 1.44, size = 304, normalized size = 2.03 \begin {gather*} -c^{2} e^{\left (-5\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 {48 \, c^{2} d e^{15}}{x e + d} - \frac {36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac {16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac {3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac {24 \, b c e^{16}}{x e + d} + \frac {36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac {24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac {6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac {6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} - \frac {12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac {8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} + \frac {16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {8 \, a b e^{18}}{{\left (x e + d\right )}^{3}} + \frac {6 \, a b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac {3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.71, size = 186, normalized size = 1.24 \begin {gather*} \frac {c^2\,\ln \left (d+e\,x\right )}{e^5}-\frac {x^2\,\left (\frac {b^2\,e^4}{2}+3\,b\,c\,d\,e^3-9\,c^2\,d^2\,e^2+a\,c\,e^4\right )+x\,\left (\frac {b^2\,d\,e^3}{3}+2\,b\,c\,d^2\,e^2+\frac {2\,a\,b\,e^4}{3}-\frac {22\,c^2\,d^3\,e}{3}+\frac {2\,a\,c\,d\,e^3}{3}\right )-x^3\,\left (4\,c^2\,d\,e^3-2\,b\,c\,e^4\right )+\frac {a^2\,e^4}{4}-\frac {25\,c^2\,d^4}{12}+\frac {b^2\,d^2\,e^2}{12}+\frac {a\,b\,d\,e^3}{6}+\frac {b\,c\,d^3\,e}{2}+\frac {a\,c\,d^2\,e^2}{6}}{e^5\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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