Optimal. Leaf size=289 \[ \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{3 e^6 (d+e x)^3}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac {c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac {B c^2 x}{e^5} \]
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Rubi [A] time = 0.36, antiderivative size = 287, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \begin {gather*} \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{3 e^6 (d+e x)^3}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac {c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac {B c^2 x}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx &=\int \left (\frac {B c^2}{e^5}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^5}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^4}+\frac {-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^3}+\frac {-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)^2}+\frac {c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)}\right ) \, dx\\ &=\frac {B c^2 x}{e^5}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{4 e^6 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{3 e^6 (d+e x)^3}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{2 e^6 (d+e x)^2}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^6 (d+e x)}-\frac {c (5 B c d-2 b B e-A c e) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 391, normalized size = 1.35 \begin {gather*} -\frac {A e \left (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 )+c^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+B \left (e^2 \left (a^2 e^2 (d+4 e x)+2 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+2 c e \left (3 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+c^2 \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 )\right )+12 c (d+e x)^4 \log (d+e x) (-A c e-2 b B e+5 B c d)}{12 e^6 (d+e x)^4} \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 {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.38, size = 572, normalized size = 1.98 \begin {gather*} \frac {12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - 12 \, {\left (4 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} - 6 \, {\left (42 \, B c^{2} d^{3} e^{2} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 4 \, {\left (62 \, B c^{2} d^{4} e - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 12 \, {\left (5 \, B c^{2} d^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (5 \, B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \, {\left (5 \, B c^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (5 \, B c^{2} d^{4} e - {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 719, normalized size = 2.49 \begin {gather*} {\left (x e + d\right )} B c^{2} e^{\left (-6\right )} + {\left (5 \, B c^{2} d - 2 \, B b c e - A 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 {120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac {60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {96 \, B b c d e^{23}}{x e + d} - \frac {48 \, A c^{2} d e^{23}}{x e + d} + \frac {72 \, B b c d^{2} e^{23}}{{\left (x e + d\right )}^{2}} + \frac {36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {32 \, B b c d^{3} e^{23}}{{\left (x e + d\right )}^{3}} - \frac {16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {6 \, B b c d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {12 \, B b^{2} e^{24}}{x e + d} + \frac {24 \, B a c e^{24}}{x e + d} + \frac {24 \, A b c e^{24}}{x e + d} - \frac {18 \, B b^{2} d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {36 \, B a c d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {36 \, A b c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {12 \, B b^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {24 \, B a c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {24 \, A b c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B b^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {6 \, B a c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {6 \, A b c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {12 \, B a b e^{25}}{{\left (x e + d\right )}^{2}} + \frac {6 \, A b^{2} e^{25}}{{\left (x e + d\right )}^{2}} + \frac {12 \, A a c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {16 \, B a b d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {8 \, A b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {16 \, A a c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {6 \, B a b d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {6 \, A a c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac {8 \, A a b e^{26}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac {6 \, A a b d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} 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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maple [B] time = 0.05, size = 710, normalized size = 2.46 \begin {gather*} -\frac {A \,a^{2}}{4 \left (e x +d \right )^{4} e}+\frac {A a b d}{2 \left (e x +d \right )^{4} e^{2}}-\frac {A a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}-\frac {A \,b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {A b c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {A \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {B \,a^{2} d}{4 \left (e x +d \right )^{4} e^{2}}-\frac {B a b \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}+\frac {B a c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}+\frac {B \,b^{2} d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {B b c \,d^{4}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {B \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 A a b}{3 \left (e x +d \right )^{3} e^{2}}+\frac {4 A a c d}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 A \,b^{2} d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 A b c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 A \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {B \,a^{2}}{3 \left (e x +d \right )^{3} e^{2}}+\frac {4 B a b d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 B a c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {B \,b^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {8 B b c \,d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {5 B \,c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {A a c}{\left (e x +d \right )^{2} e^{3}}-\frac {A \,b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 A b c d}{\left (e x +d \right )^{2} e^{4}}-\frac {3 A \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {B a b}{\left (e x +d \right )^{2} e^{3}}+\frac {3 B a c d}{\left (e x +d \right )^{2} e^{4}}+\frac {3 B \,b^{2} d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {6 B b c \,d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {5 B \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {2 A b c}{\left (e x +d \right ) e^{4}}+\frac {4 A \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {A \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {2 B a c}{\left (e x +d \right ) e^{4}}-\frac {B \,b^{2}}{\left (e x +d \right ) e^{4}}+\frac {8 B b c d}{\left (e x +d \right ) e^{5}}+\frac {2 B b c \ln \left (e x +d \right )}{e^{5}}-\frac {10 B \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {5 B \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {B \,c^{2} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 417, normalized size = 1.44 \begin {gather*} -\frac {77 \, B c^{2} d^{5} + 3 \, A a^{2} e^{5} - 25 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 12 \, {\left (10 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 6 \, {\left (50 \, B c^{2} d^{3} e^{2} - 18 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 4 \, {\left (65 \, B c^{2} d^{4} e - 22 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {B c^{2} x}{e^{5}} - \frac {{\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 475, normalized size = 1.64 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,c^2\,e-5\,B\,c^2\,d+2\,B\,b\,c\,e\right )}{e^6}-\frac {x^3\,\left (B\,b^2\,e^4-8\,B\,b\,c\,d\,e^3+2\,A\,b\,c\,e^4+10\,B\,c^2\,d^2\,e^2-4\,A\,c^2\,d\,e^3+2\,B\,a\,c\,e^4\right )+x^2\,\left (\frac {3\,B\,b^2\,d\,e^3}{2}+\frac {A\,b^2\,e^4}{2}-18\,B\,b\,c\,d^2\,e^2+3\,A\,b\,c\,d\,e^3+B\,a\,b\,e^4+25\,B\,c^2\,d^3\,e-9\,A\,c^2\,d^2\,e^2+3\,B\,a\,c\,d\,e^3+A\,a\,c\,e^4\right )+x\,\left (\frac {B\,a^2\,e^4}{3}+\frac {2\,B\,a\,b\,d\,e^3}{3}+\frac {2\,A\,a\,b\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {2\,A\,a\,c\,d\,e^3}{3}+B\,b^2\,d^2\,e^2+\frac {A\,b^2\,d\,e^3}{3}-\frac {44\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {65\,B\,c^2\,d^4}{3}-\frac {22\,A\,c^2\,d^3\,e}{3}\right )+\frac {B\,a^2\,d\,e^4+3\,A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4+6\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+3\,B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3-50\,B\,b\,c\,d^4\,e+6\,A\,b\,c\,d^3\,e^2+77\,B\,c^2\,d^5-25\,A\,c^2\,d^4\,e}{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 {B\,c^2\,x}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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