Optimal. Leaf size=281 \[ -\frac {x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac {\log (d+e x) \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6}+\frac {\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac {c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac {B c^2 x^3}{3 e^3} \]
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Rubi [A] time = 0.44, antiderivative size = 280, normalized size of antiderivative = 1.00, 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 {x \left (A c e (3 c d-2 b e)-B \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^5}-\frac {\log (d+e x) \left (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 )\right )}{e^6}-\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 )}{e^6 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{2 e^6 (d+e x)^2}-\frac {c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}+\frac {B c^2 x^3}{3 e^3} \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)^3} \, dx &=\int \left (\frac {-A c e (3 c d-2 b e)+B \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5}+\frac {c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac {B c^2 x^2}{e^3}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx\\ &=-\frac {\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) x}{e^5}-\frac {c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac {B c^2 x^3}{3 e^3}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{2 e^6 (d+e x)^2}-\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^6 (d+e x)}-\frac {\left (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 )\right ) \log (d+e x)}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 262, normalized size = 0.93 \begin {gather*} \frac {6 e x \left (B \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+A c e (2 b e-3 c d)\right )+6 \log (d+e x) \left (A e \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+B \left (6 c d e (2 b d-a e)+b e^2 (2 a e-3 b d)-10 c^2 d^3\right )\right )-\frac {6 \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )}{d+e x}+\frac {3 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)+2 B c^2 e^3 x^3}{6 e^6} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 620, normalized size = 2.21 \begin {gather*} \frac {2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 21 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 9 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - {\left (5 \, B c^{2} d e^{4} - 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (10 \, B c^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \, {\left (21 \, B c^{2} d^{3} e^{2} - 11 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (B c^{2} d^{4} e + {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 2 \, {\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 - 6 \, {\left (10 \, B c^{2} d^{5} - 6 \, {\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 (10 \, B c^{2} d^{3} e^{2} - 6 \, {\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} + 2 \, {\left (10 \, B c^{2} d^{4} e - 6 \, {\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}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 430, normalized size = 1.53 \begin {gather*} -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, B a c d e^{2} + 6 \, A b c d e^{2} - 2 \, B a b e^{3} - A b^{2} e^{3} - 2 \, A a c e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, B a c x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac {{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, B a c d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 2 \, A a b d e^{4} + A a^{2} e^{5} + 2 \, {\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, B a c d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} + B a^{2} e^{5} + 2 \, A a b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 654, normalized size = 2.33 \begin {gather*} \frac {B \,c^{2} x^{3}}{3 e^{3}}-\frac {A \,a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {A a b d}{\left (e x +d \right )^{2} e^{2}}-\frac {A a c \,d^{2}}{\left (e x +d \right )^{2} e^{3}}-\frac {A \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {A b c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {A \,c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {A \,c^{2} x^{2}}{2 e^{3}}+\frac {B \,a^{2} d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {B a b \,d^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {B a c \,d^{3}}{\left (e x +d \right )^{2} e^{4}}+\frac {B \,b^{2} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {B b c \,d^{4}}{\left (e x +d \right )^{2} e^{5}}+\frac {B b c \,x^{2}}{e^{3}}+\frac {B \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {3 B \,c^{2} d \,x^{2}}{2 e^{4}}-\frac {2 A a b}{\left (e x +d \right ) e^{2}}+\frac {4 A a c d}{\left (e x +d \right ) e^{3}}+\frac {2 A a c \ln \left (e x +d \right )}{e^{3}}+\frac {2 A \,b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {A \,b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 A b c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {6 A b c d \ln \left (e x +d \right )}{e^{4}}+\frac {2 A b c x}{e^{3}}+\frac {4 A \,c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 A \,c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 A \,c^{2} d x}{e^{4}}-\frac {B \,a^{2}}{\left (e x +d \right ) e^{2}}+\frac {4 B a b d}{\left (e x +d \right ) e^{3}}+\frac {2 B a b \ln \left (e x +d \right )}{e^{3}}-\frac {6 B a c \,d^{2}}{\left (e x +d \right ) e^{4}}-\frac {6 B a c d \ln \left (e x +d \right )}{e^{4}}+\frac {2 B a c x}{e^{3}}-\frac {3 B \,b^{2} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 B \,b^{2} d \ln \left (e x +d \right )}{e^{4}}+\frac {B \,b^{2} x}{e^{3}}+\frac {8 B b c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {12 B b c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {6 B b c d x}{e^{4}}-\frac {5 B \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {10 B \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {6 B \,c^{2} d^{2} x}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 397, normalized size = 1.41 \begin {gather*} -\frac {9 \, B c^{2} d^{5} + A a^{2} e^{5} - 7 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 3 \, {\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} + 2 \, {\left (5 \, B c^{2} d^{4} e - 4 \, {\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} - 2 \, {\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}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, B c^{2} e^{2} x^{3} - 3 \, {\left (3 \, B c^{2} d e - {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (6 \, B c^{2} d^{2} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac {{\left (10 \, B c^{2} d^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\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 = 2.39, size = 468, normalized size = 1.67 \begin {gather*} x^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{2\,e^3}-\frac {3\,B\,c^2\,d}{2\,e^4}\right )-\frac {x\,\left (B\,a^2\,e^4-4\,B\,a\,b\,d\,e^3+2\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2-4\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )+\frac {B\,a^2\,d\,e^4+A\,a^2\,e^5-6\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4+10\,B\,a\,c\,d^3\,e^2-6\,A\,a\,c\,d^2\,e^3+5\,B\,b^2\,d^3\,e^2-3\,A\,b^2\,d^2\,e^3-14\,B\,b\,c\,d^4\,e+10\,A\,b\,c\,d^3\,e^2+9\,B\,c^2\,d^5-7\,A\,c^2\,d^4\,e}{2\,e}}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^3}-\frac {3\,B\,c^2\,d}{e^4}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{e^3}+\frac {3\,B\,c^2\,d^2}{e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-3\,B\,b^2\,d\,e^2+A\,b^2\,e^3+12\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2+2\,B\,a\,b\,e^3-10\,B\,c^2\,d^3+6\,A\,c^2\,d^2\,e-6\,B\,a\,c\,d\,e^2+2\,A\,a\,c\,e^3\right )}{e^6}+\frac {B\,c^2\,x^3}{3\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 28.80, size = 534, normalized size = 1.90 \begin {gather*} \frac {B c^{2} x^{3}}{3 e^{3}} + x^{2} \left (\frac {A c^{2}}{2 e^{3}} + \frac {B b c}{e^{3}} - \frac {3 B c^{2} d}{2 e^{4}}\right ) + x \left (\frac {2 A b c}{e^{3}} - \frac {3 A c^{2} d}{e^{4}} + \frac {2 B a c}{e^{3}} + \frac {B b^{2}}{e^{3}} - \frac {6 B b c d}{e^{4}} + \frac {6 B c^{2} d^{2}}{e^{5}}\right ) + \frac {- A a^{2} e^{5} - 2 A a b d e^{4} + 6 A a c d^{2} e^{3} + 3 A b^{2} d^{2} e^{3} - 10 A b c d^{3} e^{2} + 7 A c^{2} d^{4} e - B a^{2} d e^{4} + 6 B a b d^{2} e^{3} - 10 B a c d^{3} e^{2} - 5 B b^{2} d^{3} e^{2} + 14 B b c d^{4} e - 9 B c^{2} d^{5} + x \left (- 4 A a b e^{5} + 8 A a c d e^{4} + 4 A b^{2} d e^{4} - 12 A b c d^{2} e^{3} + 8 A c^{2} d^{3} e^{2} - 2 B a^{2} e^{5} + 8 B a b d e^{4} - 12 B a c d^{2} e^{3} - 6 B b^{2} d^{2} e^{3} + 16 B b c d^{3} e^{2} - 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac {\left (2 A a c e^{3} + A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e + 2 B a b e^{3} - 6 B a c d e^{2} - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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